Calculo MATEM

Escuela de Matemática Proyecto MATEM MA-1001 Cálculo Serie: CABÉCAR Cálculo I 1a. edición Recopilado por: Lizeth Sancho Mora 2014 2012 515 C144c Cálculo I / recopilado por Lizeth Sancho Mora. – 1.ed., 2.ª reimpr. – San José, C.R. : Sección de Impresión del SIEDIN, 2014. 361 p. : il. – (Cabécar) A la cabeza de la port.: Universidad de Costa Rica. Escuela de Matemática. Proyecto MATEM. MA-1001 Cálculo. ISBN 978-9977-15-209-7 1. CÁLCULO – PROBLEMAS, EJERCICIOS, ETC. I. Sancho Mora, Lizeth, recopiladora. II. Serie. CIP/2703 CC/SIBDI.UCR Universidad de Costa Rica © Escuela de Matemática Ciudad Universitaria Rodrigo Facio, Costa Rica. Primera edición: 2011 Segunda reimpresión: 2014 Prohibida la reproducción total o parcial. Todos los derechos reservados. Hecho el depósito de ley. Este libro se terminó de imprimir en la Sección de Impresión del SIEDIN, en octubre de 2014. Universidad de Costa Rica. Ciudad Universitaria Rodrigo Facio. 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ȱ ¤ \ =ȱ [ ȱǻŗǰŗǼǯȱȱ ȱ ȱ àǱȱ ȱȱǰȱȱȱȱȱȱȱȱȱȱȱ ȱ ȱ ȱ ȱ ȱ \ =ȱ [ ȱ ȱ ȱ ǻŗǰŗǼǯȱ ȱ ǰȱ ȱ ȱ ȱȱȱȱȱȱȱȱȱȱȱǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡àȱ ȱ ȱ ȱ ȃȄȱ ȱ ȱ 4ȱ [[ ȱ ȱ ȱ ȱ ȱ ǻŗǰŗǼǲȱ ¢ȱ ȱ ·ȱ ȱ ȱ Pȱ34 ȱȱȱȱȱǻȱȱȱȱȱȱȱ¢ȱǼǰȱ [ − ǯȱ [ȱ ≠ ǰȱȱȱ 4 ≠ȱ 3ǰȱ P34ȱ = [− ȱȱȱȱ ȱ ȱ Pȱ34 ȱ ȱ ȱȱ ¡ȱ ȱȱŗǯȱ ¡ Pȱ34 ¡ Pȱ34 Řȱ ŗǰśȱ ŗǰŗȱ ŗǰŖŗȱ ŗǰŖŖŗȱ řȱ Řǰśȱ Řǰŗȱ ŘǰŖŗȱ ŘǰŖŖŗȱ Ŗȱ Ŗǰśȱ Ŗǰşȱ Ŗǰşşȱ Ŗǰşşşȱ ŗȱ ŗǰśȱ ŗǰşȱ ŗǰşşȱ ŗǰşşşȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱǰȱȱȱȱȱȱȱȱȱȱ¡ȱȱŗȱȱ Pȱ34 ¤ȱ ¤ȱ à¡ȱ ȱ Řǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Pȱ = ǯȱ ȱȱȱȱȱȱȱȱȱȱȱÇȱȱȱȱȱ ȱ ȱ ȱ ¢ǰȱ àǰȱ ¡ȱ ȱ ȱ OLPȱ P34 = Pǰȱ 4 →3 P34 = [ − I [ − I = [− [− I( [) − I() [ − P= OLP = OLP = [ȱ→ [→ [− [− ȱȱȱȱȱȱǰȱȱȱȱàȱȱȱ ȱ \ = ȱ[+ E = ()+ E − = E − = E Ǳȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ \ =ȱ [ \ȱ = [− ȱ ȱ ȱ ǻŗǰŗǼȱ ȱȱǰȱ·ȱȱȱàȱȱȱ ȱȱȱ ȱ àǰȱ ȱ Ƕàȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ǵǰȱ ȱ àȱȱȱȱȱȱȱàȱÇȱ¡ǯȱȱȱȱ ȱȱȱȱȱȱȱȱȱǯȱ ǰȱȱȱȱ ȱȱȱȱȱȱȱȱȱǯȱ Ǳȱ ȱ£ȱǻȱǼȱȱ ȱÇȱȱȱȱȱ ȱ ȱȱ ¡ȱ V =ȱ W ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ¤ȱ Wȱ = ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱȱ ȱ ȱ ȱ ȱ ȱ ¡àȱ ȱ ȱ ȱ ¤ȱ ȱ ⎛ W ⎞ ȱȱ 4ȱ⎜⎜ W ⎟⎟ ȱȱȱȱȱȱ£ȱȱ ⎝ ⎠ ⎛ ⎞ ⎜ȱ ⎟ ǯȱ ⎝ ⎠ ȱȱȱȱ YȱP ǰȱȱ Wȱ ≠ W − V(W) − V() W − = YPȱ = = W− W− (W− ) ȱ ȱ ȱ ȱ ȱ ȱ YȱP ȱ ȱ ȱ ȱ ȱ ȱȱŗǯȱ Ŗǰśȱ ŖǰŞȱ Ŗǰşȱ Ŗǰşşȱ Ŗǰşşşȱ Ŗǰşşşşȱ YȱP YȱP ŖǰŘşŗŜȱ ŖǰŚŖŜŜŜȱ ŖǰŚśŗŜŜȱ ŖǰŚşśŖŗŜȱ ŖǰŚşşśȱ ŖǰŚşşşśȱ ŗǰśȱ ŗǰřȱ ŗǰŗȱ ŗǰŖŗȱ ŗǰŖŖŗȱ ŗǰŖŖŖŗȱ ŖǰŝşŗŜȱ ŖǰŜŜśȱ ŖǰśśŗŜŜȱ ŖǰśŖśŖŗŜȱ ŖǰśŖŖśŖŖŗŜȱ ŖǰśŖŖŖśŖŖŖŗȱ ȱȱȱȱȱȱȱȱȱȱȱȱȱȱŗȱȱ ȱ ȱ ȱ ¤ȱ ȱ à¡ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ Yȱ = ǯȱ ȱȱȱȱàȱȱ£ǰȱȱȱȱȱȱ¤ȱȱ ȱ Çȱ ȱ ȱ ȱ ȱ ¢ǰȱ àǰȱ ȱ ¡ȱ ȱ ȱ I(W) − I(D) ǯȱȱ YLQV (D) = OLPȱ W→D W− D Ǳȱ ȱ ȱ ȱ ȱ Wȱ = ȱ ȱ Çȱ ȱ ȱ ȱ ȱ ȱ W ȱ £ I (W)=ȱ ȱȱ Ȧǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Çȱȱȱàȱ Ǳȱ ȱȱȱàȱȱÇȱȱȱàǰȱȱȱȱ ǰȱȱ·ȱȱȱ¤ȱȱȱȱ¢ȱȱȱȱǯȱȱ ȱȱȱÇȱȱȱàȱȱȱȱȱȱȱȱ ȱȱȱ¤ȱȱȱȱȱ¤ȱŗǱȱȱǰȱȱȱ ¢ ȱȱȱȱàǯȱ ȱȱȱȱàȱȱȱ ȱȱȱ¡àǱ OLPȱ I( [)= O [→D àǱȱ OLPȱ I( [)= O ¢ȱȱȃȱÇȱ Iȱ( [)ǰȱȱ¡ȱȱȱǰȱȱ [→D ȱ O Ȅǰȱȱȱȱȱȱȱ Iȱ( [) Oȱ ǻȱ ȱǼȱȱ¡ȱȱȱȱȱǰȱȱȱȱȱȃȄǯ ȱŗǱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱȱà Iȱ( [)= [ − [+ ȱȱȱȱŘǯȱ ¡ ŗǰŖȱ ŗǰśȱ ŗǰŞȱ ŗǰşȱ ŗǰşśȱ ŗǰşşȱ ŗǰşşśȱ ŗǰşşşȱ Iȱ( [) ¡ řȱ Řǰśȱ ŘǰŘȱ Řǰŗȱ ŘǰŖśȱ ŘǰŖŗȱ ŘǰŖŖśȱ ŘǰŖŖŗȱȱ Iȱ [ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ OLP I( [)ǯȱ [→ȱ OLP I( [)= ǲȱȱȱȱȱȱȱȱȱȱȱȱ [ȱ→ ȱ Iȱ( [)D ȱȱȱǰȱȱøȱȱȱȱȱȱȱ¡ȱȱ ȱȱȱŘȱȱ ŘǱȱ ȱǰȱ OLP [ȱ→ [− [ȱ→ [ − OLP = [− ¢ȱ¡ȱȱȱ ǯȱ [ − ¢ȱȱȱȱȱȱȱ Iȱ [ = [− [ − ȱ ȱȱȱǰȱȱøȱȱȱȱȱȱȱ¡ȱȱȱ ȱȱŗǯȱ ȱ ȱ ȱ ǰȱ ȱ OLP [ȱ→ [− ȱ ȱ ȱ [ȱ = ǰȱ [ − [ȱ− ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ [ − ȱȱȱȱȱ [ ≠ȱ D ȱȱàȱȱÇǯȱ ȱ ȱ ŗȱ ȱ ȱ ¡à ȱ [ ≠ȱD ȱȱȱȱȱÇȱ Iȱ( [) ȱ¡ȱȱȱǰȱ ȱ [ =ȱDǯȱȱȱǰȱȱȱȱȱŘȱȱȱȱ Iȱ( [) ·ȱȱ [ =ȱ Dǯȱȱøȱȱȱȱàȱ¤ȱȱȱ ȱȱǯȱ ǯȱ ȱȱȱȱ ȱȱȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ¡ȱȱȱÇȱ Ǳȱ ȱ ȱ ȱ ¡ȱ ȱ Çȱ ȱ ȱ àǰȱ ȱ ȱ ȱ ǰȱ ȱ ·ȱȱàȱȱȱȱȱȱȱȱ¤ǯȱ ȱÇȱȱȱ¤ȱȱȱàǯ Ǳȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ȱ I ȱ[ = [ [ ǰ J [ =ȱ [ ¢ K ȱ[ =VHQ⎛⎜ ⎞⎟ ȱȱǯȱ ⎝ [⎠ ȱȱȱȱȱȱȱȱȱȱȱȱȱ [ ȱ¡ȱȱȱÇȱȱ¡ȱȱȱǯȱ [→ȱ [ Ǽ OLP Ǽ OLP ȱ¡ȱȱȱȱȱȱàȱȱ ǯȱ [ ǻǱȱÇȱȱȱȱȱàȱȱǼȱ ⎛ ⎞ Ǽ OLPVHQ⎜ ⎟ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ [→ȱ ⎝ [⎠ ȱȱǱ Ȭŗȱ¢ȱŗǯȱȱ [→ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¤ȱ ȱȱ¤ȱȱȱȱȱÇȱǯȱȱȱȱȱȱ ¡ȱȱȱȱǯȱ Ǽ OLP I [ ȱ Ǽ OLP I [ ȱ Ǽ OLP I [ ȱ Ǽ OLPI [ ȱ [→− [→− [→− [→ Ǽ OLP I [ ȱ [→ àǱȱ Ǽ OLP I [ ȱ ȱ¡ȱȱȱȱÇȱȱȱǱ [→− OLP I [ = ¢ OLP I [ = − [ȱ→− − [ȱ→− + Ǽ OLP I [ ȱ ¡ȱ ȱ ȱ ȱ Ȭŗȱ ȱ àȱ ȱ ȱ ȱ [→ȱ− ǯȱ Ǽ OLP I [ ƽŖȱ [→ Ǽ OLP I [ ƽȱȬŗȱ [→− Ǽ OLPI [ ƽȱȬŗ [→ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱ¤ȱȱÇȱ Ǳȱ ȱȱȱ¤ȱȱÇǯ Ǳȱ ȱȱȱȱǰȱ OLP [→ȱ [ + − [ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ¡ȱ ȱ ȱ àȱ ȱ ȱ ȱ ǰȱ [ȱ = ȱ OLP [ȱ→ [ + − = [ ǯȱ ȱȱȱȱȱȱȱ ȱ ¤ȱà¡ȱȱǰȱȱ ȱȱȱ ¡ [ +ȱ− [ ŖǰŖŖŖśȱ ŖǰŖŖŖŗȱ ŖǰŖŖŖŖŗȱ ŖǰŖŖŖŖŖŗȱ ŖǰŖŖŖŖŖŖŗȱ ŖǰŖŖŖŖŖŖŖŗȱ ŖǰŖŖŖŖŖŖŖŖŗȱ ŖǰŖŖŖŖŖŖŖŖŖŗȱ ŖǰŖŖŖŖŖŖŖŖŖŖŗȱ ŖǰŖŖŖŖŖŖŖŖŖŖŖŗȱ [ + − ǰȱȱȱȱȱȱȱ OLP [ [ȱ→ = ǯȱ Ƕ·ȱȱȱȱȱȱȱȱǵȱ ȱȱȱȱȱȱ ȱ ¢ȱ Ûȱ ȱ ¡ǰ [ȱ + ȱ ¢ȱ à¡ȱ ȱ řȱ ¢ȱ ȱ ȱ ȱ Ç [ȱ + = ȱȱøȱȱÇȱȱȱȱ£ȱȱ¡ǰȱ ȱ ȱ ȱ ȱ ȱ [ȱ + = ǰȱ ȱ ȱ ȱȱȱǰȱȱȱ¤ȱ·ǰȱ OLP [ȱ→ OLP [ [ȱ→ [ + − [ [ + − = ǯǯȱȱ = ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱǰȱȱ·ȱȱȱȱȱ£ȱȱȱǰȱȱȱ ȱȱȱȱ¤ȱǰȱȱȱȱȱȱȱȱȱǯȱ ¤ȱ ȱǰȱ ȱȱȱȱÇǯȱ ȱȱȱÇȱ ȱȱȱȱȱȱ¢ȱȱ OLPȱ I( [) ¢ OLPȱJ( [) ¡ǯȱȱ ŗǯ OLPȱ( I( [) + J( [)) = OLP I( [) + OLPJ( [) [→ D [→ D [→D [→D [→ D Řǯ OLPȱ( I( [) −J( [)) = OLP I( [) − OLPJ( [) [→ D [→ D [→ D řǯ OLPȱ(FI( [)) =FOLP I( [) [→ D [→ D Śǯ OLPȱ( I([) ⋅J([)) = OLP I([) ⋅ OLPJ([) [→ D [→ D [→ D I( [) I( [) OLP = [→D [→ D J( [) OLP J( [) śǯ OLPȱ OLP J([)≠ [ȱ→D [→ D ( ) Ŝǯ OLPȱ( I( [)) = OLP I( [) P [→ D [→ D P P∈ Ζ + ǻȘǼȱ ŝǯ OLPȱF= F ǻȱȱȱȱȱȱ¤Ǽǯȱ Şǯ OLPȱ[= D ǻȱȱȱȱȱȱ¤Ǽǯȱȱ [→D [→D şǯȱ ȱȱȱŜȱ¢ȱŞȱȱȱ OLPȱ[Q = DQ VL P∈ Ζ + ǻȘǼȱ [→ D ŗŖǯ OLPȱQ I ( [) = Q OLP I( [) ȱȱȱȱȱȱǻȱȱȱǰȱȱ [→ D [→ D OLPȱ I([)≥ RǼǯ ǻȘǼȱ [→D ŗŗǯ ȱȱȱŞȱ¢ȱŗŖȱȱȱ OLPȱQ [ = Q D ǯȱȱȱȱȱ [→D ȱȱǁȱŖǯȱȱ ǻȘǼȱ ǻȘǼȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ǻȱ ȱ ȱ Çȱ ȱ ȱ ·ȱ ȱ Ŝȱ ¢ȱ şǰȱ ȱ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ŗŖȱ ¢ȱ ŗŗǼǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱȱ Ǳȱ ȱȱȱȱȱÇȱǱȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ OLPȱ S( [) = S(D)ǯȱ [→D Ǽȱȱȱȱȱàȱȱ¢ȱȱȱȱȱȱȱȱȱ OLPȱ S( [) = S(D)ǯȱȱ [→D Ǳȱ ȱȱȱȱȱÇȱ¢ȱȱȱ ȱǱȱ ( ) OLP[ + [− Ǽ [ȱ→ Ǽ OLP [ȱ→ [+ [ + [ − Ǽ [OLP → − − [ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ ¤ȱ ǯȱȱ OLP I [ = ¢ OLP J [ = −ǯȱ ȱ [ȱ→F [ȱ→F Ǽ OtP[ I [ ⋅ J [ ] Ǽ OtP[I [ ⋅ J [ ] Ǽ OtP[ I [ + J [ ] ⎛ I [ ⎞ ⎟ Ǽ OtP⎜⎜ [ȱ→F I [ − J [ ⎟ ⎝ ⎠ [→ȱF [→ȱF [→ȱF àǱȱ ( )( ) Ǽ OtP[ I [ ⋅ J [ ] ƽ OtP I [ OtPJ [ ƽ ⋅ȱ−= − [→ȱF [→ F [→ȱF Ǽ OtP[ I [ + J [ ]ƽ OtP I [ + OtPJ [ = + −= − [→ȱF [ȱ→F ( [→ F )( ) Ǽ OtP[ I [ ⋅ J [ ] = OtP I [ OtPJ [ = ⋅ ⋅ −= − [ȱ→F Ǽ OtP [ȱ→F [→ F [→F OtP I [ I [ [→ F = = = I [ − J [ OtP( I [ − J [ ) + [→F ǯȱȱȱàȱȱ ŗǯȱ OtP [ȱ→ I [ − = ǰȱ OtP I [ [→ȱ [− àǱ ( ) OtP I [ − OtP I [ − I [ − [→ [→ = ⇔ = ⇔ = OtP [→ [− OtP [− ( ) [→ ⇔ OtP I [ − = ⇔ OtP I [ = [ȱ→ Řǯȱ OtP [ȱ→− [→ I [ I [ = ǰȱ OtP I [ ¢ OtP [→ȱ− [→ȱ− [ [ àǱ OtP I [ = ǰȱ¢ OtP [ȱ→− [→− I [ = − [ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 &iOFXORGHOtPLWHVGHDOJXQDVIRUPDVLQGHWHUPLQDGDVȱ Ǳȱ ȱÇȱȱ OLPȱ I( [) ȱ Iȱ( [) ȱȱàȱȱ [→D ȱ [ →ȱDǰȱȱȱȱȱàȱ¢ȱ£àǯ ȱ ȱ ¤ȱ OLP [ȱ→ [ − ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ [− ǰȱ OLP( [− )= ǰȱȱȱȱ ȱàȱ¤ȱ [ȱ→ OLP([ − )= ȱȱ¤ȱȱȱȱȱȱ [ȱ→ ȱ ǯȱ [ − ȱ ȱ OLP ǰȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ [ȱ→ [− ȱȱȱà [ȱ − ǰȱȱȱŗǯȱ ȱȱȱȱȱ [− à Jȱ ȱȱȱȱȱŗȱȱȱǰȱȱȱ¤ȱȱ ȱÇȱȱȱ OLP I( [) =ȱOLPJ( [)ǯȱ [→ [→ ȱȱȱǱ I ≠ȱ Jǰȱȱȱȱȱŗȱȱȱȱ¢ȱ ȱȱȱȱǰȱȱȱȱȱȱȱȱ Iȱ( [) 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¢ Jȱ( [) ȱ Řȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ¡ȱ ȱ ȱ ȱ ȱ ŗǰȱ ȱ ȱ ȱȱŗǰȱȱ OLP I( [) = OLPJ( [)= ǯȱ [ȱ→ [→ ȱ Jȱ( [) ȱȱ Iȱ( [) ǰȱȱȱȱȱȱ¤ǰȱȱȱȱ ȱȱǰȱȱ¤ȱȱȱȱ¤ǯ [ȱ − ([− )([+ ) = ȱ ȱ ȱ Çȱ ȱ ȱ ȱ ȱ ȱ [− [− ¢ȱȱȱȱȱȱǰȱȱ [ȱ− ≠ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ( [ȱ−) ¢ȱ ȱ ȱ Çȱ ȱ ȱ ȱ Ǳ ([− )([+ ) = OLP([+ )ǰȱ ȱ ȱ ȱ Çȱ ȱ ȱ ȱ ȱ [ − OLP = OLP [ȱ→ [− [→ [→ [− [ − = ǯ ȱȱȱÇȱ¢ȱȱ OLP [ȱ→ [− 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¤ȱ ȱȱȱÇȱȱ [ + [+ ȱ ŗǯ OLP Ǳȱȱȱȱàȱȱ [+ [ȱ→− àǱ [ + [+ [+ ƽ OLP ƽ OLP [+ ƽȱŖȱ [+ [ȱ→− [ȱ→− [+ [ȱ→− OLP [ − [+ ȱ àȱȱ [ȱ→− [ − [+ Řǯ OLP àǱ [ − [+ OLP [ȱ→ [ − [+ [− [ + [− [ȱ→ [− [ + [ + [− ƽ OLP £àȱ [ȱ −[+ £àȱ [ −[+ [− [ + [− [ȱ→ [− [ + [ + [− OLP [ + [− ȱ Ǳȱàȱȱ [ȱ→ [ + [ + [− ƽ OLP [+ [− [ȱ→ [− [ + [+ ƽ OLP [+ [ȱ→ [ + [+ ƽ OLP ƽ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ [ − ȱ Ǳȱàȱȱ [ȱ→ [ − řǯ OLP àǱ [ − [− [ + [+ [ + [+ ȱ ƽ ƽ ƽ OLP OLP OLP [− [+ [+ [ȱ→ [ − [ȱ→ [ȱ→ Śǯ OLP [→ȱ [+ − ȱ Ǳȱàȱȱ [ àǱ OLP [→ȱ [+ − [ȱ+ + [+ − ƽ OLP • [ [ [ + + [→ [+ − ƽ OLP [ȱ→ [ [+ + ƽ OLP [ȱ→ [ [ [+ + ƽ OLP [ȱ→ [+ − [ȱ→ [ [+ + ƽ OLP [+ + ƽ ȱ − [ àȱȱ ȱ śǯ OLP [ȱ→ [− àǱ − [ − − [ OLP [ ƽ OLP [ ƽ OLP [ȱ→ [ [− [ȱ→ [− [ȱ→ [− − ƽ − [→ȱ [ ƽ OLP /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Ŝǯ OLP [ȱ→ − [ − − [ − /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ àȱȱ ȱ àǱ ⎛ − [− − [ + − [ + ⎞ ⎟ ƽ OLP⎜ • • [ȱ→ − [ − [ȱ→⎜⎝ − [ − − [+ − [+ ⎟⎠ OLP − [ − − [− − [+ − [ − [+ ƽ OLP [ȱ→ − [− − [ + [ȱ→ − [ − [ + ƽ OLP − [+ = = − [+ ƽ OLP [ȱ→ [ − ȱ Ǳȱàȱȱ [ȱ→ [ − ŝǯ OLP àǱȱ ȱȱȱȱȱ Ǳ X =ȱ [ [ȱ→ Xȱ → ⇒ȱX = [ ¤Ǳ ⇒ [ = X ¢ [ = X [ =ȱ X [ = X [ − X − X+ ƽ OLP ƽ OLP ƽ [ȱ→ [ − Xȱ→ X − Xȱ→ X + X+ OLP 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Şǯ OLP + F[ − [ [→ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Ǳȱàȱȱ ȱ àǱ OLP + F[ − [→ȱ X =ȱ + F[ [ ⇒X ȱ = + F[ X− X→ X − F F X − ƽ OLP Xȱ→ X − X + X+ ƽ OLP ⇒ [ȱ→ Xȱ → F Fȱ ƽ Xȱ→ X + X+ ƽ OLP [ − [ − ȱ şǯ OLP Ǳȱàȱȱ [ȱ→ [ + [− àǱ [ − [ − ƽȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ£Ǳ [ −[ − OLP [ȱ→ [ + [− [− [ + [+ ƽ [− [+ [ȱ→ OLP X − = [ F [ + [+ ȱ ƽ [+ [ȱ→ OLP [ − [ ȱ Ǳȱàȱȱ [ȱ→ [− − [+ ŗŖǯ OLP à [ − [ [ − [ [− + [+ ƽ OLP • OLP [ȱ→ [− − [+ [ȱ→ [− − [+ [− + [+ [ − [ [− + [+ ƽ OLP [→ [ [− − [+ ] 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ [ − [ [− + [+ [ȱ→ [ − [+ − [− ƽ OLP [ ] [ − [ [− + [+ [→ȱ [ − [ ƽ OLP ƽ OLP [− + [ȱ→ [+ ƽ −ȱ+ ƽ −ȱ+ = ⎧ [ − [+ VL ŗŗǯȱ I [ = ⎨ VL ⎩− [ [ȱ< ǰȱ OLP I [ [≥ [ →ȱ àǱ OLP I [ = OLP [ − [+ = [→ − [→− OLP I [ = OLP − [ = [→+ ⇒ȱOLP I [ ȱ¡ȱ [→ [→+ ŗŘǼ OLP [− − [+ [ [→ȱ àǱ OLP [− − [+ [ [→ȱ ȉȱ [→ ⇒ [− < ⇒ [− = − [− OLP − [− − [+ [ [→ȱ ȉȱ [→ ⇒ [+ > ⇒ [+ = [+ OLP − [+ − [− [ [→ OLP [→ − [ [ = − /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD [− [ − ȱ [→ [ + [ − /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ W − W→W + W − Ȧ ȱ [+ − [ȱ→ [− Ȧ ȱ − [ [→ȱ − [ ȦȱȱŖȱ [ + [− [ȱ→ [+ − ȦȱȱŘŖȱ Ȧ −ȱ ŘŗǼ OLPȱ [ − [ − [ − ȦȱȱȱȱŘȱ ŘŘǼ OLP [+ + [ ŗśǼ OLPȱ [→− [ + ȱ Ȧ ŘřǼ OLP [ + [− ŗŜǼ OLPȱ [→ [ − ȱ Ȧ ŘŚǼ OLP ȦȱȱŘŚȱ − ŘśǼ OLP ⎢ ⎥ ⎦ Kȱ→K⎣ + K ŗřǼ OLP ŗŚǼ OLPȱ [→ [ − [− [→ [ + − ŗŝǼ OLPȱ [ − [→ [ −[ [ − ŗşǼ OLP [ȱ→ [ − [− ŗŞǼ OLPȱ [ − [− [+ [ȱ→− ŘŖǼ OLP ȱ ȱ Ȧ Ȧ ⎡ ŘŜǼ OLPȱ [→ ŘŝǼ OLP ⎤ [+ − [ [ + [→− [ ȱ + [+ [→−[ + ¢ ȱ ȦȱȱŖȱ [+ = [+ [ȱ→− + OLP − [− ŘşǼ OLPȱ [→ [ − Ȧ −ȱ ⎡ ⎤ − řŖǼ OLPȱ⎢ [→ ⎣ [− [ + [ − ⎥ ⎦ Ȧ ȱ Ȧ − [→ Ȧȱȱ¡ǰȱȱÇȱȱ¡ȱȱ ȱȱǱ [+ =− OLP [+ [ȱ→− − [+ − [ [ − Ȧ −ȱ Ȧȱ Ȭśȱ ŘŞǼ OLPȱ řŗǼ OLPȱ Ȧ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD [ − [− [→−[ + [ + [ + Ȧȱ Ȭŗȱ [ − [ − [− [ȱ→ [ − [ + [− Ȧ ȱ + [ − [→ [ − Ȧ ȱ řŘǼ OLPȱ řřǼ OLP řŚǼ OLPȱ [ + [− řśǼ OLPȱ [→[ − [ + řŜǼ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ⎛ ⎞⎟ − [ȱ→ ⎝ − [ − [⎟⎠ OLP ⎜⎜ [− [+ − [+ [− řŝǼ OLP [− [+ [ȱ→ − + [ řŞǼ OLPȱ [→ − −[ řşǼ OLP + [ − [ȱ→ [+ − Ȧȱ Ȭřȱ Ȧȱ Ȭŗȱ Ȧ −ȱ Ȧ −ȱ Ȧ ȱ ŚŖǼ OLP [+ + [+ Ȧ ȱ [ − [ − ŚŗǼ OLP [ȱ→ [ − [− Ȧ ȱ ŚŘǼ OLP Ȧ ȱ [ − [ ŚřǼ OLPȱ [→ − [ Ȧȱȱř [ȱ→ [− [− [ȱ→ [ − 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ [ − [+ ŚŚǼ OLP [ȱ→ [ − [+ ŚśǼ OLP [ȱ→ + [ − + [ − Ȧ ȱ Ȧ ȱ ȱȱȱȱȱ ŗǯȱ I ȱ[ = Ǽ OLP I [ [→ȱ+ [ [ + [ Ǽ OLP I [ [→ȱ− àǯȱ ȱǱ [ = Ǽ OLP I [ = OLP [→+ [→+ Ǽ OLP I [ = OLP [ȱ→− Ǳȱ [→− ǼȱǶ¡ OLP I [ ǵȱǶȱ·ǵȱȱ [→ȱ [ VL [ȱ≥ −[ VL [< [ [ + [ [ [ + [ = OLP [ = OLP = [ [+ [ȱ→+ [+ = OLP − [ − ƽ OLP = − [ [+ [ȱ→− [+ [→+ [→− Ǽ OLP I [ ȱ¡ǰȱȱȱÇȱȱȱǯȱ [→ȱ Řǯȱ ⎧ [ + [ [ȱ≤ I ȱ[ = ⎨ [ > ⎩N + [ ȱȱȱ Nȱ ȱ OLP I [ ¡ǯ [→ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱ OLP I [ = OLP N+ [ = N+ ⎫⎪ [ȱ→+ [→ + ⎪ OLP I [ = OLP [→ − [→ − [ + [ = ȱȱȱǰȱǱ ⎪ ⎬ ⎪ ⎪ ⎪⎭ ȱ OLP I [ ¡ȱȱÇȱ [→ȱ Nȱ ƸŘȱƽȱŜȱ ǰȱ Nȱ ƽȱŚȱ [ + D[+ D+ ¡ǵȱȱȱȱÇǰȱȱȱ [ȱ→ [ − [− řǼȱǶ ¢ȱȱøȱȃȄȱȱ OLP ȱȱȃȄȱ¢ȱȱÇǯȱȱ àǱ [ + D[+ D+ OLP [ȱ→ [− [− [ + D[+ D+ OLP [ + D[+ D+ = N≠ OLP ȱ ¡Çȱ [ȱ→ [ȱ→ [− [− ȱ ȱ ȱ à Iȱ [ = [+ D[+ D+ Çȱ ȱ Çȱ ȱ [ − [− ȱ ȱ ¡ȱƽȱ Řȱ ȱ ǻȱ ȱ ȱ Nȱ ¢ȱ ȱ ȱ ȱ ¡ȱ ȱ Řȱ ȱ ȱ ȱȱȱȱ£Ǽǯȱ [ + D[+ D+ ¡ȱȱȱȱǱ [ȱ→ [ − [− ȱȱȱ OLP OLP [ + D[+ D+ = [ȱ→ +D ȱ+ D+ = D = − D = − 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ¤ȱȱÇǱ [− [− [ + −[− + = OLP OLP [→ [→ [ − [− [ − [− OLP [→ [+ [− [+ = OLP = [− [+ [→ [+ ŚǼȱȱȱȱȱȱ¢ȱȱȱ OLP [ȱ→ D[+ E− = [ àǱȱ Ǽȱȱ OLP [→ȱ D[+ E− ¡ȱȱȱ OLP D[+ E− = ǲȱ [ȱ→ [ ȱ OLP[= [→ OLP D[+ E− = E− = ⇔ E= [ȱ→ Ǽ OLP [→ ⇔ D[+ E− D[+ − D[+ − = ⇔ OLP = ⇔ OLP = [→ [→ [ D[+ + [ [ Dȱ + = ⇔ D= śǼȱ I ȱ[ =[ [] + [ − []ǰȱȱ OLP I [ ¡ȱȱȱȱȱ Iȱ [→ȱ àǱ I =[ ] + [ − ]= OLP I [ =ȱ OLP [ [] + [ − [] ƽ + −= − [→ − [→ − OLP I [ = OLP [ [] + [ − [] = + − = − [→+ [→ + 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ OLP I [ = − ȱ I ≠ȱ OLP I [ [ȱ→ ŜǼȱOLP [ȱ→ [→ I [ − = ǰȱ OLP I [ [→ȱ [− àǱȱ OLP [− = ¢ OLP [ȱ→ [ȱ→ I [ − ¡ȱ OLP I [ − = [ȱ→ [− OLP Iȱ [ − = OLP I [ − OLP ƽ ⎛⎜ OLP I [ ⎞⎟ − [→ [→ [→ ⎠ ⎝ [ȱ→ OLP I [ = [→ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱ àȱ ȱ £ȱ ȱ ȱ ȱ àȱ ȱ OLPȱ I( [)ǰȱ Iȱ( [) ȱ [→D àȱ ȃ¤Ȅȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱȱȱǯ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱȱȱȱȱȱȱȱȱȱǯȱ ȱ ǰȱ ȱ OLP [VHQ⎛⎜ ⎞⎟ ȱ ȱ ȱ ȱ ȱ ¢ȱ [→ȱ ⎝ [⎠ ⎛ ⎞ ǰȱ ȱ ȱ ȱ ȱ OLPVHQ⎜ ⎟ ȱ ¡ǯȱ ȱ ȱ ȱ [→ȱ ⎝ [⎠ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ £àȱ ¢ȱ £àǯȱ ȱ ȱ ȱ ȱȱÇȱȱȱȱȱȱȱǯȱ ȱȱàȱ I ( [) ≤ȱ J( [) ≤ K( [) ȱ ¡ȱ ȱ ȱ ȱ ȱ ǻ¡ȱ £¤ȱ ȱ Ǽȱ ¢ OLPȱ I( [) = OLPK( [)= O OLPȱJ( [)= Oǯȱȱ [→D [→ D [→D ȱȱȱàȱȱǰȱȱȱȱȱ¤ȱȱȱ ǯȱ ⎛ ⎞ Ǽȱ OLP[VHQ⎜ ⎟ ǯȱ [→ȱ ⎝ [⎠ ⎛ ȱ⎞ ȱȱȱ¤ȱȱȱà W( [) = VHQ⎜ ⎟ [−ȱ]ǲȱȱ ȱ ⎝ [⎠ ⎛ ⎞ ȱ − ≤ VHQȱ⎜ ⎟ ≤ ǯȱ ¤ȱ ȱ ȱ ȱ [ȱ ≠ ȱ ȱ ⎝ [⎠ [ȱ > ¢ȱȱȱǱ ⎛ ⎞ ⎛ ⎞ [ (−ȱ) ≤ [VHQ⎜ ⎟ ≤ [ ǰȱȱ −ȱ[ ≤ [VHQ⎜ ⎟ ≤ [ ⎝ [⎠ ⎝ [⎠ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ⎛⎞ ȱ [ȱVHQ⎜ ⎟ ǰȱȱ I ([)=ȱ − [ ¢ K( [)=ȱ [ ǰȱ¤ OLP(− [ )= ¢ [ȱ→ ⎝ [⎠ ⎛ ⎞ OLP([)= ȱȱȱȱȱȱ OLP[VHQ⎜ ⎟ = ǯȱȱ [ȱ→ [ȱ→ ⎝ [⎠ ¤ȱ ,ǼȱȱȱÇȱȱ ŗǼȱ≤ Iȱ [ ≤ [ + [+ ȱȱ¡ǰȱ OLP I [ [→ȱ− àǱȱ ȱȱ¡ȱȱȱ ≤ Iȱ [ ≤ [ + [+ ǰ OLP= [ȱ→− ȱȱȱȱȱà OLP [ + [+ = − + = [ȱ→− ȱȱȱ OLP I [ = [ȱ→− ŘǼȱ [ȱ≤ I [ ≤ [ + ≤[ȱ≤ ǰȱ· OLP I [ [→ȱ Ȧȱřȱ řǼȱȱȱ¡ȱ ȱ ȱǁŖȱȱ I ȱ[ < 0 ȱȱ¡ǯȱȱ OLP[I [ = ǯȱȱ [ȱ→ àǱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ I ȱ[ < 0 −0 < I ȱ[ < 0 ǯȱ ȱ¡ǁŖȱ −0[ < [I [ < 0[ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ − 0[= OLP [ ȱ→ + ȱȱȱȱà OLP[I [ = [ȱ→ + 0[= OLP [ →+ ȱ¡ǀŖȱ −0[ > [I ȱ[ > 0[ OLP− 0[= [ȱ→ − ȱ ȱȱà OLP[I [ = [ȱ→ − OLP0[= [→− ȱ ȱ Çȱ ȱ ¡ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ OLP[I [ = ǯȱ [ȱ→ ŚǼȱ OLPJ [ ǰȱȱ Jȱ [ − < [+ ȱȱ¡ǯȱ [→ȱ− Ȧȱřȱ śǼȱ I [ ≤ȱ . [− D ǰȱȱ [ȱ ≠ ǰȱȱȱǰȱ OLPI [ [→ȱD ȦȱŖȱ ŜǼȱ ⎛ ⎞ [ FRV⎜ ⎟ OLPȱ [ ⎝ [⎠ ȦȱŖȱ → ŝǼȱ ⎛ ⎞ [VHQ⎜ ⎟ OLP ⎝ [⎠ [ ȦŖȱ → ȱ ŞǼȱ E − [− D ≤ I ȱ[ ≤ E+ [− D ǰȱȱȱ¡ǰȱ OLPI [ [→ȱD Ȧȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Çȱ ȱ ȱ ȱÇȱǱȱ Ǽȱȱȱàȱȱȱȱȱàǯȱ Ǽȱȱȱȱȱȱàǯȱ ȱȱÇȱȱȱȱàȱǱȱ Ǽȱȱàȱȱȱȱàǯȱ Ǽȱȱȱȱȱàǯ ȱȱȱȱ£à OLP ¢ȱȱ¢àȱȱȱÇȱȱ¡Çǰȱ [ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ǯȱ ǰȱ ȱ ȱ ȱ [→ȱ = +∞ ǯȱ ȱ ȱ ȱ OLPȱ I( [) = ∞ ȱ [→F [ ȱȱȱȱ¡ǰȱȱ¤ȱȱ ∞ȱ ȱ ȱ øǯȱȱȱ £ȱȱȱȱȱȱ Iȱ( [) ȱ¡ ȱȱŖǯȱ ȱ ȱ ¤ OLP [ȱ→ ȱ OLP = ∞ ȱȱȱ¤ȱȱȱȱȱà I ( [)=ȱ [ [ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱȱŖǯȱ [ȱ→ ȱ ȱ ¤ǰȱ ȱ Ç OLPȱ I( [) = −∞ ȱ Iȱ( [) ȱ [→F ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ¡ȱ ȱ ȱȱȱǯȱ Ǳȱ ȱȱÇȱǯȱ Ǽ OLP [ȱ→ [− ([− ) àǱȱ OLP([− )= − ¢ OLP( [− ) = ǰȱȱȱȱȱȱȱ [ȱ→ [ȱ→ ¢ȱȱȱŘȱȱàȱ¤ȱǰȱȱ¢ȱǻȱǼȱ ȱȱȱǱȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ ŗǰŞȱ Ȭśśȱ ŘǰŘȱ ŗǰşȱ ȬŘŗŖȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱŘǰŗȱ ŗǰşşȱ ȬŘŖŗŖŖȱ ŘǰŖŗȱ ŗǰşşşȱȬŘŖŖŗŖŖŖȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱŘǰŖŖŗȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȬŚśȱ ȬŗşŖȱ ȬŗşşŖŖȱ ȬŗşşşŖŖŖȱ [− = −∞ ǯȱ [ȱ→ ( [− ) ȱȱǰȱȱȱ OLP [+ [→ȱ − [ Ǽ OLP àǱȱ OLP [+ = ¢ OLP(− [ȱ→ [ȱ→ [)= ǰȱȱȱȱȱȱȱ ¢ȱȱȱřȱȱàȱȱȱȱȱȱȱȱȱ (ȱ−[) ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ řǰȱ ȱ ȱ ¢ȱȱȱȱȱȱȱȱǯȱ OLP([+ )= ¢ OLP(− [) = − ȱ ȱ ȱ ȱ ¡ǁřȱ [ȱ→+ [ȱ→+ −[ȱ< ǯ [+ = −∞ ǯȱȱȱǰ OLP− ([+ )= ¢ [ȱ→ − [ + OLP−(− [) = ȱȱȱ [ȱ< −[ȱ > ǰȱȱȱȱȱ¢ȱ ȱȱȱȱ¢ȱ OLP [→ȱ + [ȱ→ OLP− [→ȱ [+ = +∞ ǯȱ − [ ȱǰȱȱȱǱȱ OLP I( [)= N > [ȱ→D OLP I( [)= N < [ȱ→D I( [) = +∞ OLPJ( [) =+ [→D J( [) [→ D I( [) OLPȱ = −∞ OLPJ( [) = − [→D J( [) [ȱ→ D OLPȱ I( [) = +∞ OLPJ( [) = − [→D J( [) [→ D I( [) OLPȱ = −∞ OLPJ( [) = + [→D J( [) [→ D OLPȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ àǯȱ ȱ [ =ȱD ȱȱÇȱȱ ȱȱ \ =ȱ I( [) ȱȱȱȱ ȱȱȱȱȱȱ OLP I( [) = ∞ OLP I( [) = ∞ OLP I( [) = ∞ [→D− [→D OLP I( [) = −∞ [→ D OLP I( [) = −∞ [→ D− [→D+ OLP I( [) = −∞ [→ D+ ȱ àȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £ȱ ¤ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ Çȱ ȱ ȱ £ȱ ȱ ¤ȱȱȱǯȱ ȱ ǰȱ ȱ £ȱ ȱ ¤ȱ OQȱ[ǰ OLP I [ = −∞ [ȱ→+ ȱ ȱ ȱ àǰ [ȱ= ȱ ȱ àȱ ȱ ȱ Çȱ ȱȱ \ =ȱOQ [ǯȱ ȱ ȱ ¤ȱ ȱ ȱ à I( [)=ȱ WDQ [ǰȱ ȱ ȱ ȱ ȱ øȱ π ȱ ȱ Çȱ ǰȱ ȱ ȱ ȱ ȱ à [ = + Nπ Nȱ ȱ ȱ øȱǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ Çȱȱȱ Ǳȱ ȱÇȱȱǰ OLPȱ I( [) ¢ OLPȱ I( [) ǰȱǱȱ [→∞ [→−∞ Ǽȱȱàȱȱȱȱàǯȱ Ǽȱȱȱȱȱàǯȱ ȱȱÇȱ£ȱȱȱàȱǱȱ Ǽȱȱàȱȱȱȱàǯȱ Ǽȱȱȱȱȱàǯ ȱ ȱ ȱ ȱ ȱ £àȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ǰȱ ȱ ȱ àȱ ¤ȱ ȱ ȱ ȱ ȱ ȱàȱȱȱȱ¡ȱȱȱȱ¢ȱȱǻȱ¢ȱȱȱȱ Ǽǯȱ ȱȱ ȱ¤ OLPȱ I([) OLPȱ I( [)ǯȱȱ [→−∞ [→∞ ǯȱ Ǳȱ [ + [+ ŗǯ OLP [ȱ→∞ [ − [− [ + [+ Řǯ OLP [ȱ→∞ [+ [+ řǯ OLP [ȱ→∞ [ − [− àǱ ȱ ȱ OLP([ + [+ ) = ∞ ¢ OLP([ − [− ) = ∞ [ȱ→∞ [ȱ→∞ ȱȱȱȱȱȱȱȱÇȱ¡ȱ¢ȱȱ ∞ȱ ȱ ∞ȱ ȱøȱǯȱȱȱ¤ȱ ȱȱàȱȱ ǯȱ ∞ ȱ ¡ȱ ȱ ∞ȱ ȱ ȱ ȱ [ȱ ≠ ȱ ȱ £ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ¡ǰȱ ȱ ȱ ǰȱ ¢ȱ ȱ Ǳȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ⎛ + [ + [+ [ ⎝ = OLP OLP [ȱ→∞ [ − [− [→ ∞ ⎛ [⎜ − − [ ⎝ [⎜ + ⎞ + + ⎟ [ ⎠ [ = OLP ⎞ [→∞ − − ⎟ [ [ ⎠ [ = [ [ + [+ = " [ȱ→∞ [ − [− Ƕ·ȱȱ OLP ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ Iȱ( [) ȱ ¡ȱ ȱ ȱ ȱȱ ȱȱȱȱȱ¡ǯ [ + [+ OLP [ȱ→∞ [+ ȱȱǰȱȱȱȱȱ ȱàȱȱ ∞ȱ ǰȱȱ ∞ £¤ȱ ȱȱȱ ŗȱ¢ȱȱ¤Ǳ ⎛ OLP ⎝ ⎞ ⎞ ⎛ [⎜ + + ⎟ ⎟ [ [ ⎠ [ [ ⎠ = OLP ⎝ [ → ∞ ⎞ ⎛ + [⎜ + ⎟ [ [⎠ ⎝ [ ⎜ + [→ȱ∞ + ⎛ ⎝ OLP[⎜ + [ȱ→∞ ⎞ ⎛ ⎞ + OLPȱ ⎜ ⎟ ƽŚǯȱ = ∞ ¢ ⎟ [→∞ [⎠ [ [ ⎠ ⎝ + ȱ ȱ àȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ àȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ⎛ ⎝ ⎞ ⎟ [ [ȱ ⎠ = ∞ ǯȱ + [⎜ + OLP [→∞ + [ [ + [+ =∞ ǯ OLP [ȱ→∞ [+ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD OLP [ȱ→∞ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 [+ [ − [− ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ∞ȱ ȱ ȱ ∞ ¤ȱȱȱȱȱȱȱ ȱǱ + [+ [ OLP = OLP [ȱ→∞ [ − [− [→ ∞ ⎛ ⎞ [⎜ − − ⎟ [ [ ⎠ ⎝ ⎛ ⎝ ⎜ + OLP [ȱ→∞ ⎞ ⎞ ⎛ ⎟ = ¢ OLPȱ[⎜ − − ⎟ = ∞ [→∞ [⎠ [ [⎠ ⎝ [+ = ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ [ȱ→∞ [ − [− ȱȱȱ ȱȱȱȱ¡ȱȱǯȱȱ ȱ ȱ OLP ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ 3( [)= DQ [Q + DQ−ȱ[Q− + + D[ + D 3( [) ǯȱ [→∞ 4( [) OLPȱ Ǽȱȱȱȱȱȱȱȱȱȱȱȱǯȱ Ǽȱȱȱȱȱȱȱ¢ȱȱȱȱȱǯȱ Ǽȱȱȱȱȱȱȱȱȱȱȱȱǯȱȱ Ǳȱ ǰȱȱ¡ǰȱȱȱÇ [ + OLP [→−∞ [− 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱ OLP [ + = ∞ ¢ OLP ([− ) = −∞ ǯȱ [ȱ→−∞ [ȱ→−∞ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ∞ȱ −∞ ǯȱ ȱ ȱ ȱ ȱ £ȱ¤ǰȱ¤ȱȱȱȱȱǲȱȱǰȱȱȱ¡àǰȱ ȱȱ¡ȱȱ −ȱ∞ ǰȱȱȱ [ȱ ≠ ǯ [ + = OLP [→ −∞ [− ȱ OLP [→−∞ ⎞ ⎛ ⎟ [ ⎠ ⎝ ⎞ ⎛ [⎜ − ⎟ [⎠ ⎝ [⎜ + ȱ [ ȱ = [ ⎛ ⎝ ⎛ [⎜ − ⎝ ⎞ [ + ⎟ [ ⎠ [ = OLP OLP ¢ȱ ȱ ¡ȱ ȱ −ȱ∞ ȱ ȱ ¤ȱ [→−∞ [ → −∞ ⎞ ⎞ ⎛ [⎜ − ⎟ ⎟ [⎠ [⎠ ⎝ ȱ ȱ ȱ ȱ ȱ Ŗǲȱ [ =ȱ − [ [ȱ⎜ + [ + OLP [ȱ→−∞ ⎛ [ = OLP ⎞ [⎜ − ⎟ [⎠ ⎝ [→ −∞ − [ + ⎛ [ = OLP ⎞ [⎜ − ⎟ [⎠ ⎝ − + [→ −∞ − [ = − ǯȱȱ [ Ǳȱ ǰȱȱ¡ǰȱȱȱÇǱ ( ) OLP [ȱ + [+ − [ [→∞ à Ǳ OLP [ + [+ = ∞ ¢ OLPȱ[ = ∞ [ȱ→∞ [→∞ ȱȱȱȱȱȱ ∞ȱ ȱȱøȱȱȱȱ Çȱ ¡ǰȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ∞ȱ− ∞ ǯȱ ȱ ∞ȱ ȱȱȱȱȱȱȱȱàȱȱ ǰȱȱȱ ∞ ȱȱȱǯȱȱȱȱȱȱ£ȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ( ) ⎡ OLP [ + [+ − [ = OLP⎢ [→∞ [→ ∞ ⎣⎢ = OLP [ + [+ − [ [ + [+ + [ [→∞ ⎛ = OLP [ →∞ ( [ + [+ − [)( = OLP [→ ∞ ⎞ ⎞ [ ⎜+ + ⎟ + [ [ [ ⎠ ⎝ ⎦⎥ [+ [ + [+ + [ ⎞ [⎜ + ⎟ [⎠ ⎝ = OLP [→ ∞ )⎥ [ + [+ + [ ⎤ [ + [+ + [ ⎛ [⎜ + ⎟ [⎠ ⎝ ⎛ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 [ + + [ [ + [ ⎛ ⎞ ⎞ ⎛ [⎜ + ⎟ ⎟ [⎠ [⎠ ⎝ ⎝ = OLP = OLP [→∞ [→ ∞ ⎛ ⎞ [ + + + [ [⎜ + + + ⎟ ⎜ ⎟ [ [ [ [ ⎝ ⎠ [⎜ + + OLP [ȱ→∞ + [ + [ [ = + ǯȱȱ Ǳȱ ǰȱȱ¡ǰȱȱȱÇ ( ) OLP [+ [ + [ [→ȱ−∞ àǱ OLPȱ [ = −∞ ¢ OLP [ȱ + [ = ∞ ǯȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ [→−∞ [→−∞ àȱ ȱ −ȱ∞ + ∞ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ∞ȱ ȱȱȱàȱȱ ǰȱȱȱȱȱȱȱ£ȱȱ ∞ ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ( ) OLP [+ [ + [ = OLP [→ −∞ = OLP [→−∞ = OLP [→ −∞ = OLP [→−∞ = OLP [ȱ→−∞ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ([+ [ − ([ + [) − [ ⎛ ⎞ − [ [+ [ + + + [→ −∞ [→ −∞ = OLP [→ −∞ [ − = OLP = OLP [− [ ⎜+ ⎟ [⎠ ⎝ ) [− [ + [ [→ −∞ [− [ + [ )( [ + [ [− [ + [ [ − [ − [ [− [ + [ − [ [− [ + [ − [ ⎛ ⎞ [⎜⎜+ + ⎟⎟ [⎠ ⎝ = − [ àȱ ȱ \ =ȱO ȱȱÇȱ£ȱȱȱ \ =ȱ I( [) ȱȱ ȱȱȱȱȱ OLP I( [) = O R OLP I( [)= O [→∞ [→ −∞ ȱ àȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £ȱ ¤ȱ ȱ ǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¤ȱ ȱȱȱÇȱȱ [− ŗǼ OLP [ȱ→ + àǱȱ [ȱ→ + ȱ ȱ ȱ ¤ȱ ȱ ȱ ¢ȱȱȱřȱȱ¢ȱȱřǰȱȱǻ¡ȬřǼȱȱȱȱȱȱȱ = +∞ ¢ȱȱǯȱ OLP [ȱ→ [− + [− ŘǼ OLP [ȱ→ − àǱȱ [ȱ→− ȱ ȱ ȱ ¤ȱ ȱ ȱ ¢ȱ ȱ ȱ Řȱ ȱ ȱ ȱ Řǰȱ ȱ ǻ¡ȬŘǼȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ = −∞ OLP [ȱ→ [− − řǼ OLP [ȱ→∞ [ + [− [ + [− àǱ [ + [− OLP [ȱ→∞ [ + [− [ + [− ƽOLP [ȱ→∞ [ + [− ⎞ ⎛ [⎜ + − ⎟ [ [ ⎠ ƽ OLP ⎝ ⎞ ⎛ [→∞ [⎜ + − ⎟ [ [ ⎠ ⎝ Ǳȱ àȱȱ ∞ȱ ∞ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD + /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ − Nȱ ƽOLP [ [ =ȱ ȱȱ [ȱ→∞ → ȱƾŖȱ [ [→∞ + − [ [ [ + [− ∞ȱ ŚǼ OLP Ǳ àȱȱ ∞ [ + [ȱ→∞ àǱ ⎛ ⎝ ⎞ ⎟ [ [ ⎠ ⎞ ⎛ [⎜ + ⎟ [ ⎠ ⎝ [⎜ + OLP [ ȱ→∞ − Ŗȱ Ŗȱ [ + − [ [ ƽOLP [→ȱ∞ + [ Ŗ =ȱ∞ śǼ OLP [ȱ→∞ − [ − [+ [ + [ − àǱ ⎛ ⎝ OLP ⎛ [ȱ→∞ [⎜ + ⎝ ⎞ ⎟ [ [ ⎠ ⎞ − ⎟ [ [ ⎠ ŖȱȱȱȱȱȱŖ − − + [⎜ − − OLP [ ȱ→∞ + [ [ ⎛ ⎝ [⎜ + [ − ⎞ ⎟ [ ⎠ ŖȱȱȱȱŖ = Ǳ àȱȱ −∞ȱ ∞ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ŜǼ ( OLPȱ [ ȱ ) [ + [+ − [ →∞ Ǳ àȱȱ ∞ȱ− ∞ àǱ ) ( OLPȱ [ [ + [+ − [ →∞ ⎡ ⎛ [ + [+ + [⎞ ⎤ ⎜ ⎟⎥ ⎢⎛ ⎠ ⎞ ⎝ ⎢⎜ [ + [+ − [⎟ ⋅ ⎥ ⎠ ⎝ ⎛ ⎞ ⎢ ⎜ [ + [+ + [⎟ ⎥ ⎝ ⎠⎦ ⎣ ƽ OLPȱ [→∞ ƽOLPȱ [→∞ ƽOLPȱ [ + [+ − [ [ + [+ + [ [+ ⎞ ⎟ + [ [ [ ⎠ [+ ǰȱ [ →ȱ∞ ⇒ [ = [ ƽOLPȱ [→∞ [ + + + [ [ [ [+ ƽOLPȱ [→∞ [ + + + [ [ [ [+ ƽ OLP ⎛ ⎞ [ȱ→∞ [⎜⎜ + + + ⎟⎟ [ [ ⎝ ⎠ ⎛ ⎞ [⎜+ ⎟ [⎠ ⎝ ƽ OLP ⎛ ⎞ [ȱ→∞ [⎜⎜ + + + ⎟⎟ [ [ ⎝ ⎠ + [ = ƽOLP ƽ + [ȱ→∞ + + + [→∞ ⎛ ⎝ ǲ [ =ȱ [ [⎜+ + [ [ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ŝǼ ([+ OLPȱ [ ) [ + [ →−∞ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ àȱȱ −ȱ∞ + ∞ àǱ ⎛ [+ [ + [⎞⎛ [− [ + [⎞ ⎟ ⎟⎜ ⎜ ⎠ ⎠⎝ ⎝ OLPȱ [ →−∞ [− [ + [ [ − ([ + [) ƽ OLPȱ [→−∞ [− [ + [ − [ ƽOLP [→ȱ−∞ [− [ + [ − [ ƽOLP [→ȱ−∞ [+ [ + [ − [ ƽ OLPȱ [→−∞ ⎛ ⎞ [⎜⎜ + + ⎟⎟ [ ⎠ ⎝ − − ƽOLP = [ȱ→−∞ + + [ [ȱ→−∞ [ =ȱ − [ − [ ŞǼ OLP [→ȱ − [+ [ àǱȱ − [ ƽOLP [→ȱ − [+ [ − [ ƽ OLP [→ȱ (− [) =ȱ∞ ƽOLP [→ − [ [ − [− şǼ OLP [+ [ȱ→−+ Ǳȱ àȱȱ ȱ ȱ [ =ȱ( [) Ȧ − ∞ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD [− [ȱ→ [ − [+ ŗŖǼ OLP Ȧ +ȱ∞ [ + ŗŗǼ OLP Ȧ −ȱ [ȱ − [ + [→ −∞ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ) ( ŗŘǼ OLP [+ [ + [ Ȧ −ȱ ŗřǼ OLP ⎛⎜ [ + [− ȱ+ [⎞⎟ ⎠ [→ −∞ ⎝ Ȧ −ȱ [→ȱ−∞ ŗŚǼ OLP [ȱ + [+ [ [→ −∞ Ȧ −ȱ ŗśǼ OLP ⎛⎜ [+ [ȱ − [ ⎞⎟ ⎠ [→ −∞ ⎝ Ȧ [ + ȱ ŗŜǼ OLP ȱ [→ −∞ [ + ȦȱŖȱ ŗŝǼ OLP ⎛⎜ [ȱ + [ + [⎞⎟ ⎠ [→ −∞ ⎝ Ȧ −ȱ [− ŗŞǼ OLP [→ −∞ Ȧ [ ȱ + [ ŗşǼ OLP [→ −∞ [→ −∞ ( [ȱ + −[ Ȧ ȱ [ + [+ ȱ+[ Ȧ −ȱ ) Ȧ − ŘŘǼ OLP [+ [ + [+ [ȱ→−∞ −ȱ Ȧ −ȱ ŘŖǼ OLP ⎛⎜ Wȱ − W + W ⎞⎟ ⎠ [→ −∞ ⎝ ŘŗǼ OLP 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱÇȱ£ȱ¢ȱȱȱȱ¤ȱȱȱàȱ [ + ŗǼ Iȱ [ = [− àǱȱ ǼȱÇȱǯȱ ȱàȱȱȱ [ = [ + OLP + [− OLP [ + = [ȱ→ [→ ¢ + OLP ([− ) = [ + = +∞ [− ∴ OLP + + [→ [ȱ→ ȱ¢ȱÇȱ ȱȱȱȱà [ȱ = ǼȱÇȱ£ ⎛ [ + = OLP [− [→∞ ȱ OLP [→∞ [⎜⎜ + ⎝ ⎞ ⎛ [⎜ − ⎟ [⎠ ⎝ ⎛ OLPȱ [→−∞ [ + = OLP [→ −∞ [− − + OLP [→−∞ − [ [ = ⎞⎟ ⎟ [⎠ − [⎜⎜ + [ + = OLP [→∞ ⎞⎟ ⎟ [⎠ ⎝ ⎞ ⎛ [⎜ − ⎟ [⎠ ⎝ ⎛ [ ⎞ [⎜ − ⎟ [⎠ ⎝ + ƽ OLP − [ + = OLP [→ −∞ ⎞ ⎜ − ⎟ [⎠ ⎝ [→∞ ⎛ [ ⎞ ⎛ [⎜ − ⎟ [⎠ ⎝ [ ƽ = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ¢ȱÇȱ£ȱ¢ȱȱȱȱȱà \ȱ = \ȱ =− ¢ ŘǼ Iȱ [ = [ − [− [ − àǱȱ Ǽȱ Çȱȱ ȱàȱȱȱȱ¡ ƽȱřȱ¢ȱ¡ȱƽȱȬř • • OLP [ − [− [− [ȱ→+ OLP [− [− [ȱ→− [ − ([− )([+ ) [− ƽ OLP = −∞ ( )( ) [ + [ − [ − + + [→ [→ = OLP [− = [→ − [− = OLP ȱ¢ȱÇȱȱȱȱȱȱà ¡ȱƽȱřȱ ǼȱÇȱ£ OLP [→∞ [− [− [− = OLP [ȱ→−∞ [− [− [− = ¢ȱÇȱ£ȱȱȱȱȱà ¢ȱƽȱŗȱ [ [ − [+ ȦȱÇȱ£ ȱȱȱȱàȱ¢ȱƽȱŖȱ Çȱȱȱȱȱ ȱà ¡ȱƽȱŗȱ ¢ȱ ¡ȱƽȱŘȱ řǼ Iȱ [ = ŚǼ Iȱ [ = [ + [− [ − [− Ȧȱ Çȱ£ ȱȱȱȱàȱ¢ȱƽȱŗǯȱ Çȱȱȱȱȱ ȱàȱ¡ȱƽř 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱȱȱǯȱ ȱȱȱȱȱȱ ȱȱȱȱȱȱǯȱ ȱȱȱȱȱàȱȱȱȱǯȱ ȱ¢ ȱȱȱȱǯ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ·ȱ ȱ ȱ ¤ȱ ȱ ȱ àǰȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ¤ȱ ¢ȱ ȱ ȱ ȱ ȱ¤ȱȱȱàȱȱȱȱȱǯȱ àȱ ȱàȱȱȱȱȱ ȱȱ OLPȱ I( [) = I(D) [→D ȱȱȱȱȱȱȃȄȱ ȱȱȱ ȱȱ ȱ ǯȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ àȱ ȱ ȱ ȱ ǰȱ ȱ ȱ £ȱ ȱ ¤ȱ ȱ ȱ¤ǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ àȱ ·ȱ ȱ ȱ ǰȱ ȱ Iȱ(D) ȱ ¡ǰȱ ȱ ȱ ȱ ȱ Çȱ ȱȱȱ OLPȱ I([) = I(D)ǯȱ [→D ȱȱȱ¤ȱǰȱȱȱȱȱ¤ȱȱ ǰȱȱ ȱȱȱȱ ǯȱ ȱȱȱȱ ȱȱ ¤ȱȱ ȱ OLPȱ I([) ≠ I(D)ǯȱ [→D ȱȱȱ ȱ OLPȱ I( [) = I(D) [→D ȱȱȱȱ ȱȱ ¤ȱȱ ȱ OLPȱ I([) ȱ¡ǯ [→D 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱàȱǰȱȱȱȱ·ȱȱǰȱȱȱȱÇȱ ȱȱȱ¡ǰȱȱ OLPȱ I( [) ¡ǰȱȱȱȱȱÇȱȱȱȱ [→D OLPȱ I([) = I(D)ǯȱ [→D ȱȱȱȱ¤ǰȱȱȱǰȱȱȱȱȱ OLPȱ I( [) ¡ǰȱȱȱȱȱ ǯȱ [→D ȱȱȱȱ ȱ OLPȱ I( [) ¢ I(D) H[LVWHQSHURQR [→D VRQLJXDOHVȱ ȱȱȱ ȱ OLPȱ I( [) = I(D) [→D ȱȱȱȱ ȱ OLPȱ I( [) ¡ȱ I(D) QR [→D HVWiGHILQLGD 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱǰȱȱȱÇȱȱȱàȱȱȱȱ ȱȱȱøȱȱȱǱȱ ŗǼ Iȱ(D) ¤ȱǯȱ ŘǼ OLPȱ I( [) ¡ǰȱ¢ȱ [→D řǼ OLPȱ I([) = I(D) [→D ȱ ȱ ŗǼȱ ¢ȱ ŘǼȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱȱȱȱȱŗǼǰȱŘǼȱ¢ȱřǼȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱȱǰȱȱǰȱȱȱàȱȱȱȱȱǯȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ¢ OLPȱ I( [) ¡ǰȱ ȱ ȱ ȱ ȱ [→D ȱȱȱȱǯȱȱȱǱȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ¢ OLPȱ I([) = ∞ OLPȱ I([) = ∞ ǰȱ ȱ ȱ ȱ [→D+ [→D− ¡ȱ ȱ ȱ ǰȱ ¢ȱ ȱ ȱ ȱ ¢ȱ ȱ Çȱ ǯȱ ȱǱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Dȱ ȱ Ǳȱ ȱàȱ ȱ ȱ I [ = + ȱ [ − ȱȱȱŞǯȱȱȱȱȱ [− ȱȱǯȱȱ àǱ Iȱ ȱ ȱȱȱȱȱȱȱŞǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD • /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 + [ − OLP [− [ȱ→ (X ) − = [ [ȱ→ Xȱ → X− ƽ OLP X = ȱ+ [ X → ⎛⎜X− ⎞⎟ − ⎝ ⎠ X− = OLP ⎛ ⎞ ⎛ Xȱ→ ⎛⎜ ⎞ ⎟ ⎞ ⎞⎜ ⎛ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ X − − ⎟⎜ ⎜ X − ⎟ + ⎜ X − ⎟ + ⎟ = OLP X→ ⎠ ⎠⎜ ⎝ ⎝ ⎝ ⎠ ⎝ X− ⎟ ⎠ ⎛ ⎛ ⎞ ⎞⎟ ⎜ ⎛ ⎞ X− X+ ⎜ ⎜⎜ X − ⎟⎟ + ⎜⎜ X − ⎟⎟ + ⎟ ⎜⎝ ⎠ ⎝ ⎟ ⎠ ⎝ ⎠ ( )( ) = ⎛ ⎞ ⎛ ⎜⎛ ⎞ ⎟ ⎞ X+ )⎜ ⎜⎜ X− ⎟⎟ + ⎜⎜ X− ⎟⎟ + ⎟ = OLP X→ ( ⎜⎝ ⎝ ⎠ ⎝ ⎠ ⎟ ⎠ ȱȱǰ ȱȱȱǰȱȱ OLP I( [) ¡ȱȱ [→ȱ àǯȱ ȱà ȱȱȱȱȱȱȱȱøȱȱ OLPȱ I( [) = I(D)ǯȱ [→D+ ȱà ȱȱȱȱȱ£ȱȱȱøȱȱ OLPȱ I( [) = I(D)ǯȱȱ [→D− àȱ Ǽȱȱȱȱȱȱ ȱ ȱ ȱ ȱ ȱ Ιȱ= (D E) ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱȱǯȱ Ǽȱȱȱȱȱȱ ȱȱȱȱȱ [D ȱE] ȱȱǱ •ȱ ȱȱȱ (D E) •ȱ ȱȱȱȱȱ£ȱȱ •ȱ ȱȱȱȱȱȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱ ȱ¢ ȱȱȱ ȱ¢ ȱȱȱǰȱȱȱȱ ȱ·ȱȱȱ Ǳȱ ŗǼ I +ȱJ ŘǼ I −ȱJ řǼ I •ȱJ Iȱ ŚǼ Jȱ(D)≠ J śǼ FI àȱ ŗǼȱ ȱ ȱ ȱ I +ȱ Jȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱȱȱȱȱȱ ȱ¢ ȱȱȱ ǯȱ ȱ I +ȱ J ȱ ȱ ȱ ǰȱ ȱ àǰȱ ȱ ȱ OLPȱ( I + J)( [) = ( I + J)(D)ǯȱ [→D ȱ ȱ OLPȱ( I + J)( [) = OLP( I( [) + J( [)) ȱ ȱ Çȱ ¡ǰȱ ¢ȱ ȱ ȱ ȱ [→D [→ D ȱ ȱ¢ ȱȱȱ ȱ OLPȱ I( [) = I(D) ¢ OLPȱJ( [) = J(D)ǯȱ [→D [→D ȱȱ£ȱȱȱȱ OLPȱ I( [) ¡ȱ¢ OLPȱJ( [) ¡ǰȱȱȱ [→D [→D ȱȱȱȱȱÇȱȱȱȱȱǱ OLP ( I( [) +J( [)) = OLP I( [) + OLP J( [) [ → D [→ D [→ D =ȱ I (D) + J(D) = ( I + J)(D) ȱȱ I +ȱ J ȱȱȱǯȱȱ Ǳȱ ȱŘǼǰȱřǼǰȱŚǼȱ¢ȱśǼȱ ȱȱǯȱ ǱȱȱȱȱȱȱȱȱȱȱȱȱŗǼȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ Ǳȱȱȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯ Ǽȱ øȱ ȱ ȱ ȱ àǰȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ £ȱȱȱȱȱȱȱȱ I ( [)=ȱ [Q Qȱ∈Ζ ǻȱ ¡ ≠ȱ ( ) Ŗȱ ȱ ≤ȱ ŖǼǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ OLPȱ[Q = OLP[ Q [→ D ȱ ȱ ¤ȱ ȱ ȱ ǯȱ àǼȱ ȱ [→ D ǻȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ Ǽȱ·ȱȱàȱȱȱà I ( [)=ȱ Q [ ȱȱȱȱǯȱ ȱȱȱȱȱ OLPȱQ [ = Q OLP [ ǯȱȱȱȱǰȱȱȱ (OLP [)> ¢ȱ ȱȱǰ (OLPȱ[) ȱȱȱøȱǯȱ [→ D [ȱ→D [→ D [→D Ǽȱȱȱȱ¤ȱȱȱȱ·ǰȱ·ȱ ǰȱ¡ȱ¢ȱÇǰȱȱȱȱȱ¤ȱȱ ǻȬŖŗŘśǼǰȱȱȱȱȱȱȱȱȱȱȱȱ ȱǯȱǻȱȱǰȱȱȱȱȱȱàǼȱȱ Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ Jȱ(D)ǰȱ ȱ ȱ àȱ ( IRJ )([) =ȱ I(J([)) ȱȱ ǯȱ Ǳȱ ǻȱ ȱ ǰȱ ȱ ȱ ǰȱ ȱ ȱ àȱ ȱ ȱ Ǽȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤¡ǯȱ ȱ ȱ Çȱȱǰȱ£ȱȱȱȱȱȱȱȱ à ȱǯȱ Q[ ȱ + WDQ− [ [ − Ǽ I ( [)=Hȱ+ [ Ǽ I( [)= ( ) Ǽ I ( [) =ȱ VHQ [ Ǽ I ( [) = Q(+ VHQ[) 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ Ǳȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱȱȱȱȱ ⎧ D[ + VL[< ⎪ ⎪⎪ [− VL [= Kȱ [ =⎨E ⎪D[− VL[> ⎪ ⎪⎩ àǱȱ Ǳ ȱȱȱȱȱ ȱȱȱȱȱȱøȱǰȱȱ ȱȱȱȱ ȱȱȱȱȱȱȱàȱ ȱ ȱøȱǯ D[ + ȱȱàȱǰȱȱȱȱȱ ¡ȱƽȱ [− řǯȱ ∉ȱ(− ∞)ǰȱ ȱȱȱȱ (−ȱ∞)ǰȱ ȱȱȱȱ • Kȱ [ = ǯ • Kȱ ( [)=D[− ȱ ȱàȱǰȱȱȱȱȱȱ ȱȱ ȱȱȱȱȱ (ȱ+∞) ȱȱȱȱǯȱ ȱȱȱȱȱȱȱ ¡ȱƽȱŘǯ • K (ȱ)= E • D[+ D+ = = −(D+)⎪⎪ OLP K [ = OLP ⎪ − [ →− [→ − [− ⎬ ⎪ OLP K [ = OLP (D[− ) = (D− ) ⎪ + + ⎪⎭ [→ [→ ǻȱȱàȱ¤ȱȱȱŘǼ ⎫ ȱ¢ȱȱ −(D + )= D− ⇔ − D −= D− ⇔ −D= − ⇔ D= • HQWRQFHV OLP K [ ¡ȱ [ →ȱ OLP K [ = K()⇔ ⋅ − = E⇔ E= − [ȱ→ ȱȱȱ ȱȱȱȱ ȱ D ȱ = ¢ E = − 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ [D ȱE] ¢ȱ ȱ ȱ ȱøȱȱ Iȱ(D) ¢ Iȱ(E)ǯȱȱ¡ȱȱ øȱȱ (D ȱE) ȱ I (F)=ȱ 1 ǯ Ǳȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ǯȱȱȱȱǰȱȱȱȱ¢ȱȱȱǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ǰȱ ȱ ȱ ȱȱȱȱàǰȱȱ ȱȱȱǯȱ ȱȱȱȱȱ¤Ǳȱ ȱȱȱȱȱȱȱȱȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ¡ȱ ȱ øȱ ȱ ȱ ȱ ȱ àǰȱ ȱ ȱ ¤ȱ ȱ ȱ ¡ȱ ȱ øȱ øǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ øǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱǱȱ Ƕ·ȱȱȱȱȱȱàȱȱȱǵȱȱǰȱ ȱȱȱǯȱ ȱȱȱàȱȱȱȱȱȱȱ¤ǰȱȱȱ ȱȱȱȱȱ Iȱ(D) ¢ Iȱ(E)ǯȱȱȱȱȱ¡ȱ F∈ȱ(D E) ȱ I (F)=ȱ 1 ǯȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱȱȱȱȱȱȱàǯȱȱ Ǳȱ ȱȱȱȱȱȱȱȱȱ¡ȱȱÇ£ȱ ȱȱà [ + =ȱ [ ȱȱ ] ȱ[ àǱȱ ȱ [ +ȱ= ȱ [ȱ +− [ [ ȱȱàȱȱȱ = ȱàȱȱȱ ] [ ] ȱ[ȱȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱà I( [)=ȱ [ + − ¡ȱƽȱŖȱ R ∉ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ [ ǯȱȱȱȱȱȱȱ ] ȱ[ ȱȱ [ȱ∈⎤⎥⎦ ⎡⎢⎣ ȱȱǯȱǰȱȱȱȱ ⎡ȱ ⎤ ȱȱ ⎢ ⎥ ǯȱ ⎣ ⎦ − ȱ I⎛⎜ ⎞⎟ = \ Iȱ()= ȱ ȱ ƽȱ Ŗȱ ȱ ȱ øȱ ȱ ȱ ⎝ ⎠ ⎛⎞ ⎛ ⎞ ǯȱȱȱȱȱȱȱ¡ Fȱ∈⎜ ⎟ ȱ Iȱ⎜ ⎟ ¢ Iȱ ⎝ ⎠ ⎝ ⎠ ȱ Iȱ F =ǰȱȱǰȱȱȱȱàȱ [ȱ +− [ = ȱȱ ] ȱ[ǯȱȱ ¤ȱ ȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àǱ ⎧ [+ VL[ ≤ I [ = ⎨ ⎩F[+ VLȱ[ > Ǳȱ ȱȱȱȱȱȱȱøȱǯ Iȱ ȱȱȱ ȱȱȱ Fȱ = ǯ ⎧P[− Q VL[ < ⎪ ȱ ȱȱàȱ ȱȱ I [ = ⎨ VL[ = ǯȱ ⎪P[+ Q VL[ > ȱ ⎩ ȱȱȱȱȱ¢ ȱȱȱȱ ȱȱȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǳȱ ȱȱȱȱȱȱȱøȱǯ − ¢ Q= Iȱ ȱȱȱ ȱ Pȱ = ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱȱȱ ǯ ⎧ D[ + [+ VL[≤ ⎪ Kȱ [ =⎨ D[− VL[ > ⎪ ⎩ [ Ǳȱ ȱȱȱȱȱȱȱøȱǯȱ ȱȱȱȱ ȱȱȱƽȱŘȱàȱȱƽȱř ȱ ȱȱàǱ ⎧ ⎪ ⎪ Kȱ [ =⎨ ⎪ D[ − D[+ ⎪⎩ VL[< VL≤ [≤ VL[> Ǽȱȱȱȱȱȱȱǰȱȱà ȱȱȱȱ¡ȱƽȱŖȱ Ǽȱȱȱȱȱȱȱȱȱȱȱȱ¡ȱƽȱŗȱȱ àǱȱ Ǽ K( )= OLP K [ = ⎫ ⎪⎪ [ȱ→− ⎬ OLP K( [)= OLP K [ = ⎪ [ → ⎪⎭ [→ + ∴ȱ OLP K( [) = K() ȱȱȱȱǰȱȱȱȱȱȱǰȱȱ [ → àȱȱȱȱȱ¡ȱƽȱŖ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Ǽ K()= /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ = D− D+ − D = OLP K([)= OLP [ →ȱ− [→ − D[ − D[+ − D OLP K( [)= [→+ K ( [ ) ¡ȱ OLP [ →ȱ = ⇒ = − D ⇒ −= −D⇒ D = −D = = − (QHVWHFDVR K =ȱ OLP K [ [→ ȱȱȱȱȱŗȱ D = ȱ ȱȱȱàȱ ȱȱȱȱǱ ⎧ D[ȱ VL[< ⎪ ⎪⎪ [− K [ = ⎨ VL≤ [≤ ⎪ ⎪ E[ VL[> ⎩⎪ [ + ȱȱȱȱȱ¢ ȱȱȱȱȱȱȱȱȱ ǯȱ Ǳȱ ȱȱȱȱȱȱȱøȱǯȱ ȱȱȱȱ ȱ D ȱ =− ¢ E= ȱ ȱȱȱȱȱȱȱ¢ȱȱȱȱȱȱàȱȱ ȱȱȱ ǯ VL[ ≤ − ⎧ ⎪ I [ = ⎨D[+ E VL < [ < ⎪− VLȱ[ ≥ ⎩ Ǳȱ ȱȱȱȱȱȱȱøȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱȱ ȱ D ȱ =− ¢ E= ȱ ȱȱȱȱȱȱȱȱȱȱȱȱàȱȱ VL[≤ ⎧F[+ Jȱ [ = ⎨ ȱȱȱ ȱ VL > F[ [ ⎩ Ǳȱ ȱȱȱȱȱȱȱøȱǯȱ ȱȱȱ ȱ F = ȱ ⎧ [+ VL [ < ⎪ ȱȱà I [ = ⎨D[+ E VL≤ [ < ⎪[ VLȱ[ ≥ ⎩ Ƕ·ȱȱȱ ȱ¢ ȱȱȱȱȱȱàȱȱȱ ǵȱ Ǳȱ ȱȱȱȱȱȱȱøȱǯȱ ȱȱȱȱȱ ȱȱȱƽ Śȱ¢ȱȱƽȱȬŘ ȱ ȱȱȱȱȱȱ ȱ ¢ȱ ȱȱȱȱȱàȱȱ ȱȱȱ ȱ ǯ ⎧ [+ F VL [< − ⎪ I [ = ⎨F[+ N VL − ≤ [≤ ⎪[− N VL [> ȱ ⎩ Ǳȱ ȱȱȱȱȱȱȱøȱǯȱ ȱȱȱȱȱ ȱ Fȱ = ¢ N = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱà [ − [+ =ȱ ȱàȱȱȱ [ ] ⎤ ⎡ ȱ ȱȱȱ¡ȱȱ F ∈ ⎥ȱ ⎢ȱ F − F = ⎦ ⎣ ȱ J [ = [ − [ + [ +ȱ ǰȱȱȱ¡ȱ ȱ øȱ ȱȱ J F = − ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ I( [)=ȱH [ ¢ J ( [)=ȱ− [ ȱ ǯȱȱ àǱȱ ȱȱȱȱ¤ȱȱȱ I( [)=ȱH [ ¢ J ( [)=ȱ− [ ȱǰȱȱ ȱȱȱȱà Hȱ[ =− [ ȱàǯȱ Hȱ[ − + [ = I( [) ȱȱȱ [ȱ] ȱȱȱȱȱȱȱȱȱǯ Iȱ() =H − + = − ⎫⎪ ¢ Iȱ ⎬ ȱȱƽȱŖȱȱȱ Iȱ = H − + ≈ ⎪⎭ I ȱȱȱȱȱȱ¡ Fȱ∈() ȱ I(F)= ȱ ȱ FRVȱ[ = [ ȱàȱȱȱ [] ȱ ȱȱȱà H−[ = [ ȱàȱȱȱ ][ ȱ ȱ [ȱ + S[+ T= ȱȱÇ£ȱȱ Sȱ > ¢ T > ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ [ ] ȱ Iȱ [ = [ − ȱ ȱ ȱ ȱ [ȱ+N ]ǰȱ [ [ȱ] ȱȱȱȱȱ¡ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱȱàȱȱ OLP I [ ȱ= ǰȱǰȱȱȱȱȱȱ [→ ȱȱȱȱȱȱǱȱ Ǽȱ ǰ I =ȱ Ǽȱ ȱ ȱȱ ¡ȱƽȱśȱ Ǽȱ ǰ OLP I [ = OLP I [ ȱ= [→ + [→ − Ǽȱ ȱȱȱȱśǰȱȱȱȱÇȱȱ Ǽȱ ȱȱȱ I [ ȱ ȱȱȱȱȱȱȱŞǰȱȱȱȱ ¡ȱ·ȱ ȱà¡ȱȱśǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ Çȱ· ȱ Ǳȱ VHQ[ −ȱFRV[ WDQ[ ǯȱ OLP OLP [→ [→ [ [ [→ [ ȱȱÇȱ·ȱȱ ¤ȱȱÇȱȱ ȱÇȱ·ȱȱǱ OLP ȱ·ȱȱȱȱȱȱ ȱ ǯȱ ȱǱȱ ȱȱȱ¢ȱȱȱȱȱȱȱ ǰȱȱǱ OLPȱVHQ[= VHQD ¢ OLP FRV[ = FRVD ȱȱøȱȱǯ [→D [→ȱD Ǳȱ ȱȱȱȱȱ¢ȱȱǰȱȱȱÇȱǱ FRV [− FRV[+ [→ȱ + FRV[ • OLP • OLP Π [ȱ→ Ȧ VHQ[ [+ Ȧ VHQ[ [→ȱ [ • OLP π + Ȧ VHQ W ȱ ȱǰȱȱȱȱÇȱǰȱ VHQW ȱ ȱȱ¢ȱ ȱȱÇǰȱȱȱ ȱȱȱ ȱ¤ȱ •ȱ ȱȱȱ OLP W→ȱ ȱȱȱȱȱ ȱ ǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ OLP W→ȱ ¢ȱ àȱ I (W)=ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ W ȱ£ȱȱȱȱȱàȱ VHQW W ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ VHQW àǱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǻŗǰȱ ŖǼȱ ¢ȱȱ ȱ ȱ ∩ ȱ %'ȱ ȱȱȱȱǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ∩ %'ȱ ȱȱȱȱȱȱȱȱȱǯȱ ȱ ȱ ȱ ȃȄȱ ȱ ȱ ȱ Ǳ W VHQW OLP =ȱ= OLP ǯ W→ VHQW W→ W ⎛ ∩ȱ ⎞ P⎜⎜ %'⎟⎟ = W ⎝ ⎠ ( ) P(%&)= WDQ W P $' = VHQW ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ àǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à W ȱȱȱǰȱȱȱǰȱȱȱǯȱ I (W)=ȱ VHQW Wȱ > ǰȱȱȱȱȱȱȱȱȱǱ VHQW <ȱ W < WDQ W < Wȱ < π π ȱ ȱ ȱ Wȱ∈⎤⎥ ⎡⎢ VHQWȱ > ȱ ȱ ȱ ȱ ȱ ȱ ⎦ ⎣ ȱ >ǰȱ ȱ ȱ ȱ ȱ ȱ ¡àȱ ȱ ȱ ȱ VHQWȱ VHQW ȱȱȱȱȱȱǻȱȱȱȱȱȱ Ǽȱ ¢ȱ ȱ VHQW W WDQ W <ȱ < ⇒ VHQW VHQW VHQW W < < VHQW FRVW OLP+ = \ OLP+ Wȱ→ ȱ W→ = ǰȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ FRVW 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 OLP+ Wȱ→ W = VHQW ȱȱ Wȱ < ǰȱȱȱȱȱȱȱȱȱǱ πȱ WDQ W <W < VHQW SDUD − < W < ⎛ ∩ ⎞ W = − P⎜⎜ %'⎟⎟ ⎝ ⎠ ( ) WDQ W = − P(%&) VHQW = − P $' πȱ W ∈⎤⎥ − ⎡⎢ ⇒ VHQW< ¢ȱ ȱ ȱ ȱ ⎦ ⎣ ȱ <ǯȱȱȱȱȱȱ¡àȱǰȱ ȱȱ VHQWȱ VHQW ȱǯ WDQ Wȱ VHQW OLP− W→ > W VHQW W > ⇒ > > VHQW VHQW FRVW VHQW = \ OLP− = ǰȱȱȱȱȱà ȱȱȱ W→ FRVWȱ OLP− Wȱ→ ȱȱ OLP Wȱ→ + W = ǯȱ VHQW W W = \ OLP = ȱȱȱȱǱ W → VHQW VHQW − OLP W → W = VHQW 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ − FRV[ Ǳȱ OLP [ [→ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ·ȱ ȱ VHQ[ȱ −ȱFRV[ ȱȱ ǯȱ ȱȱ¡à ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ [ [ ȱ ( −ȱFRV[)(+ FRV[)= − FRV [ = VHQ[ǯȱ ȱ ȱ ȱ ȱȱ £ȱȱǱ − FRV[ ⎡− FRV[ + FRV[⎤ OLP = OLP⎢ ⋅ [→ [→⎣ [ [ + FRV[⎥⎦ − FRV [ VHQ[ = OLP [→ [(+ FRV[) [→ [(+ FRV[) = OLP ⎡ VHQ[ VHQ[ ⎤ = OLP⎢ ⋅ [→ȱ⎣ [ + FRV[⎥ ⎦ ȱȱȱȱȱȱȱȱȱȱȱȱÇȱ ¢ȱ − FRV[ OLP = ȱǱ Ǳȱ [ȱ→ OLP [→ȱ [ WDQ[ [ ȱ ȱ ȱ ȱ ȱ ȱ ¡àȱȱȱȱȱÇȱ·ȱǱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ VHQ[ − FRV[ = y OLP = [ȱ→ [→ [ [ OLP Ǳ VHQ[ WDQ[ VHQ[ = OLP⎡ ⋅ VHQ[⎤ OLP = OLP FRV[ = OLP ⎥ [→ȱ⎢ [→ [→ [→ [FRV[ ⎣ FRV[ [ ⎦ [ [ ȱȱȱȱÇȱȱǱ OLP [ȱ→ WDQ[ [ = ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ¤ȱ ȱȱÇȱǯȱ VHQ[ [→ȱ [ ŗǼ OLP àǱ X HQWRQFHV = [ Ǳ X =[ VHQX VHQȱ[ ⎛ VHQX⎞ = OLP = OLP⎜ ⎟ [→ [ X→ ⎛ X⎞ X→⎝ X ⎠ ⎜ ⎟ ⎝ ⎠ OLP VHQX =ȱ ⋅ = = OLP [→ X [ȱ→ Xȱ → FRVθ − θ→ȱ VHQθ ŘǼ OLP àǱ (FRVθ − )⋅ FRV θ − = OLP ȱ θ → VHQθ θ → OLP FRVθ − θ = θ OLP θ → VHQθ VHQθ ⋅ θ θ = VHQȱθ řǼ OLP θ→ θ + WDQθ àǱ VHQθ OLPθ + WDQθ = OLP VHQθ ⋅ θ VHQθ θ =ȱ = = θOLP WDQθ + θ → → (θ + WDQθ ) + θ→ θ θ W ŚǼ OLP W→ȱ WDQ W àǱ W ⎞⎟ ⎛ W ⎞ ⎛ W ⎞ ⎛⎜ = ⎟ = OLP⎜ ⎟ = OLP ⋅ OLP =OLP⎜ W→ WDQ W W→⎝ WDQW⎠ W→⎝ WDQW⎠ ⎜ W → WDQW⎟ ⎝ ⎠ W 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD śǼ OLP /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ VHQ [− [ȱ→ [ + [− àǱ VHQ [− VHQ [− VHQX = OLP = OLP OLP [→ȱ [ + [− [→( [+ )([− ) X→(X+ + )X X =[− ⎡ VHQX⎤ = OLP⎢ ⋅ X→ȱ⎣ X+ X ⎥⎦ [ȱ→ X → =ȱ ⋅ = − FRV[ [→ȱ [ ŜǼ OLP àǱ ⎡− FRV[ + FRV[⎤ − FRV[ − FRV[ = OLP⎢ ⋅ ⎥ = OLP [→ [→⎢⎣ [ + FRV[⎥⎦ [→ [(+ FRV[) [ OLP ⎡ VHQ[ = OLP⎢ ⎣ [ [ȱ→ ⎢ ⋅ ⎤⎥ = ⋅ = + FRV[⎥ ⎦ VHQ[− FRV[ FRV[ πȱ ŝǼ OLP [→ àǱ OLP π [→ VHQ[− FRV[ VHQ[− FRV[ VHQ[− FRV[ = OLP = OLP FRV[ π π (FRV[− VHQ[)(FRV[+ VHQ[) [→ FRV [− VHQ [ [→ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD − = πȱ FRV[+ VHQ[ = OLP [→ ŞǼ OLP [→ πȱ − − =− =− + WDQ [ àǱ OLP VHQ[ = πȱ − [→ VHQ[ OLP WDQ[= OLP FRV[ πȱ− π− [→ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 [→ = +∞ ȱ OLP− [→ π FRV [ = + ⎛ ⎛ ⎞ ⎞ şǼ OLPȱ [⎜⎜− FRV⎜ ⎟ ⎟⎟ [→∞ ⎝ ⎝ [⎠ ⎠ àǱȱ X =ȱ [ [ȱ→∞ Xȱ → ǯȱȱȱàȱȱȱ£ȱ ⎛ − FRVX ⎛⎞ ⎞ Ǳ OLP [⎜⎜− FRV⎜ ⎟ ⎟⎟ = OLP (− FRVX)= OLP = [ȱ→∞ ⎝ X→ X ⎝ [⎠ ⎠ X→ X − FRV[ πȱ π − [ ŗŖǼ OLP [→ àǱ π π⎞ ⎛ − ⎜ FRVXFRV − VHQXVHQ ⎟ ⎟ − FRV[ ⎠ ⎠ ⎝ ⎝ = OLP = OLP OLP π⎞ X→ X→ ⎛ π − X− π π π − [ π − ⎜ X+ ⎟ [→ ⎠ ⎝ ⎛ − FRV⎜ X+ π⎞ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Xȱ =[− ⇒X + π π = [ π⎞ ⎛ OLP⎜ [− ⎟ = [→ π ⎝ ⎠ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ − ⋅ FRVX+ − VHQX ƽ OLP X→ − X − FRVX+ VHQX = OLP X→ − X ⎡− FRVX VHQX⎤ = OLP⎢ − ⎥ X→⎣⎢ − X X ⎦⎥ =ȱ − =− [− VHQ[ [→ [+WDQȱ[ ŗŗǼ OLP ȦȱŖȱ ⋅ − VHQȱ[ ŘŖǼ OLP π [→ π [ − ȦȱŖȱ VHQ[ ŗŘǼ OLP [→ WDQȱ[ ȱ Ȧ [ − [+VHQȱ[ ŘŗǼ OLP Ȧ −ȱ [ [→ VHQ[+WDQȱ[ ŗřǼ OLP [→ [ ȦȱŘȱ ŘŘǼ Ȧȱŗȱ [− VHQ[ ŘřǼ OLP [→ [+VHQȱ[ W +W W→ WDQȱW ŗŚǼ OLP FRVθ + θ →π π − θ ȦȱŖȱ − FRV[ [→ȱ VHQ[ ȦȱŖȱ ŗśǼ OLPȱ ŗŜǼ OLP WDQπȱ[ [→− [+ OLP ŘŚǼ OLP Ȧ πȱ Ȧ −ȱ VHQȱ[ Ȧ [→ [ +[ ⎡ [→⎢⎣ WDQ[−VHQȱ[ ŗŝǼ OLP [→ [ ȦȱŖȱ FRV π ȱ + K + ŗŞǼ OLP K→ K ȦȱŖȱ VHQ[ ŗşǼ OLP [→VHQȱ[ ȱ Ȧ ⎞ ⎤ ⎟⎥ ⎟ ⎝ [ − ⎠ ⎥ ⎦ ⎛ ŘśǼ OLP⎢( [− ) VHQȱ⎜⎜ ⎧− FRV[ ⎪ [ ⎪ ŘŜǼȱ Iȱ [ =⎨ ⎪ VHQ[− WDQ[ ⎪ ⎩ VHQ[ OLP I [ [→ȱ ȱ ȦȱŖȱ VL[< VL[= VL[> ȦȱŖ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àȱ ȱÇȱȱȱȱ Ǳȱ ȱȱàȱȱȱÇȱȱȱà ȱȱȱǯȱ ǰȱȱȱàȱȱ OLPȱ I( [) = ¡ǰȱȱȱȱ [→D àȱǯ ȱȱȱȱàȱȱȱȱȱȱȱȱ£àȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ǰȱ ȱ ȱ ȱ Çǯȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱȱȱȱȱȱȱȱàǰȱȱȱ ȱ ȱ £ȱ ȱ ¢ȱ ȱ ȱ ¢ȱ Çȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱȱÇǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ àȱǰȱȱȱǰȱȱ¤ȱǶ·ȱȱ ȃȱ ȱ ȱ ȱ ȱ ȱ Iȱ( [) ȱ ǵȱ ȱ Ƕ·ȱ ȱ ¤ȱȱȃȱ¡ȱȱȱȱȱǵ ǯȱ ȱ ȱà ȱÇǱȱ ȃǱ OLPȱ I([) = [→F ¢ȱȱȃȱÇȱ Iȱ( [)ǰȱȱ¡ȱȱȱǰȱȱȱ ȱ Ȅǰȱȱȱȱȱȱȱ Iȱ( [) ȱ ȱ ǻȱȱǼȱ ȱ¡ȱȱȱȱȱǰȱȱȱȱȱǯȄȱ ȱ·ȱȱȱàȱȱȱ£ȱȱȱȱȱȱ àȱȱȱǻ [ȱ ȱȱȱȱȱȱȱȱȱȱ¡ȱȱ ǰȱ [ −ȱD ȱȱȱȱȱȱȱȱ¡ȱȱǼǯȱ Ƕàȱȱ ȱȱǱȱȃȱ ȱȱȱ Iȱ( [) ȱ Ȅȱǵȱ ȱȱȱȱøȱȱǰȱȱȱȱȱȱȱ ȱ ȱ ǻ¡Ǽȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ øǰȱ ȱ ȱ ȱ ȱ ȃÛȄȱ ǯȱ ȱ ǰȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ǻ¡Ǽȱ ·ȱ ȱ ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱǻ¡Ǽȱȱȱȱ ȱ ȱ ȱ ǰȱȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱøȱ ǯȱ ȱ ȱ ȱ ȱ ǻ¡Ǽȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱȱ εǯȱȱȱȱȱ ȱ ȱ ȱ Ǳȱ εȱ > ǻ¢ȱ ȱ ȱ ȱ ȱ ȱ Ǽǯȱ ȱ εȱ ȱ Iȱ( [) − < ε ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Iȱ( [) ȱ ȱ ȱ¤ȱÛȱ εȱ ǯȱ Ƕ·ȱȱȱ¡ȱȱȱȱȱǵȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǻ¡Ǽȱ ȱ ȱ ȱ ȱ ȱ ]O−ε ȱO+ ε [ ȱ εȱ ǰȱ ȱ ȱ ȱ ȱ øȱ ǰ δǰȱ ȱ ȱ ȱȱȱȱ¡ǰȱȱȱȱȱȱǻ¡Ǽǰȱ ȱȱȱȱ ȱȱȱȱȱȱøǯȱ ȱȱø δȱǁȱŖǰȱȱȱ [ȱ− F < δ ǰȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ¡ȱȱȱȱȱÛȱ δȱ ǯȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱ [ȱ− F ≠ ǰȱȱȱȱ [ȱ− F > ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0ȱ 71 àǱȱ ȱȱȱàȱȱȱȱȱȱȱȱȱøȱ ǰȱ¡ȱ£¤ȱȱǰȱȱȱȱȱÇȱ Iȱ( [) ȱ¡ȱ ȱȱȱ¡ȱȱȱȱ ȱ ¢ȱǱ OLPȱ I( [) = [→F ȱȱȱø εȱ > ¢ȱȱ δȱ > (6HD ε > ) (H[LVWHȱ δ > ) Ǳ Iȱ( [) − <∈ ȱ ȱ< [− F < δ ǯ ȱàȱ·ȱȱȱȱ·ȱȱȱȱȱȱ àǱȱ ȱȱ εȱ > ȱ£ȱȱ ȱ£ \ = + ∈ȱ \ = − ∈ ȱȱȱȱȱ ø δȱ > ȱȱȱ ȱ ¡ȱȱ ] F−δ ȱF+ δ [ [ ≠ȱFǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱ \ =ȱ I( [) ¤ȱ ȱȱ \ȱ= − ∈ ǰ \ȱ= + ∈ ǰȱ ȱȱȱȱ¡ȱ ] F−δ ȱF+ δ [ ǯȱ ȱǰȱȱ [∈] F− δ ȱF+ δ [ ȱȱ I([) ∈(O− ε ȱO+ ε ) ǯȱ ȱ ȱ ȱ δȱ ȱ ȱ ȱ ȱ ·ȱ ȱ ¤ȱ ȱȱȱ δȱǯȱȱ Ǳȱ Iȱ [ = − [+ ȱȱȱàȱ OLP I [ = [ȱ→− àǱȱ Ǽȱȱȱȱ δȱ εȱ > ǻȱøȱǼǯȱȱȱȱ δȱ ǁŖȱȱ Iȱ [ −− I [ −− <ε ȱ < [ȱ− − < δ [+ [+ = + = = ( [+ ) = [ȱ+ ȱȱ Iȱ([) −− <ε [ȱ+ < ε ȱȱȱȱȱ δ = ε [ȱ+ < ε 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǽȱàȱ εȱ > à δ =ȱ ε < [ȱ+ < δ ȱȱ [ + [+ ε = [ȱ+ < • δ = • = ε + = [ȱ+ + <ε ȱ < [ȱ+ < δ ȱȱǰȱȱȱàȱȱÇȱȱȱȱ OLP [ȱ→− [+ =− ¤ȱ ȱȱȱȱ ŗǼȱȱ OLP [− = [ȱ→ àǱȱ Ǽȱȱȱȱ δȱ εȱ > ǻȱøȱǼǯȱȱȱȱ δȱ ȱǱ Iȱ [ − < ε ȱ < [ȱ− < δ I [ − = ([− ) − = [− = ( [− ) = [ȱ− ȱȱȱ Iȱ [ − < ε ȱ [ȱ− < ε [ȱ− < ȱȱȱȱȱ δ =ȱ Ǽȱàȱ εȱ > Ç δ = ε ε ε 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ε < [ȱ− < δ [ȱ−− = [− = [− < ⋅ δ = ⎛⎜ ⎞⎟ = ε ⎝ ⎠ ȱȱǱ ( [ȱ−) − < ε ȱ < [ȱ− < δ ȱȱǰȱȱȱàȱȱÇȱȱȱ OLP [− = [ȱ→ ŘǼȱȱȱàȱȱȱÇǰȱȱ OLP −[+ = − [ȱ→ àǱȱ Ǽȱȱȱȱ δȱ εȱ > ǻȱøȱǼǯȱ ȱȱȱ δȱ ȱ Iȱ [ −− < ε ȱ < [ȱ− < δ I [ −− = − [+ + = − [+ = − ( [− ) = [ȱ− ȱȱ Iȱ [ −−< ε ȱ [ȱ−< ε [ȱ− < ȱȱȱȱȱ δ =ȱ ε ε Ǽȱàȱ εȱ > à δ =ȱ ε ε < [ȱ− < δ − [ + + = −[ + = [− < ⋅ δ =ȱ⋅ = ε ȱȱǱ − [ȱ+ + < ε ȱ < [ȱ− < δ ȱȱǰȱȱȱàȱȱÇȱȱȱȱ OLP −[+ = − [ȱ→ řǼȱȱ OLP ([+ )= [ȱ→ ŚǼȱȱ OLP (− [)= [ →− 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱȱȱ Ǳȱ ȱȱȱȱȱàȱȱȱøȱȱȱȱàȱ ¤ȱ¢ȱȱàǯȱ ȱȱȱȱȱàǰȱȱȱàǯ ȱȱȱȱȱȱ¤ȱȱȱȱȱȱȱȱ ȱȱ¤ȱǱȱȱȱȱȱȱȱàǯȱ ȱȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱ ȱ ¢ȱ ǯȱ ȱ Çǰȱ ȱ ȱȱȱ¡ǰȱȱ¤ȱȱȱǯȱ ȱȱȱȱ·ȱ¢ȱȱȱȱȱȱ ȱ £ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱȱȱȱȱȱ£ȱȱȱȱȱȱǯȱ ȱȱȱȱȱȱȱÇȱ¢ȱȱȱȱȱȱ ǰȱȱȱȱǰȱ ȱȱȱȱȱÇǯȱȱȱ ȱȱȱȱȱȱȱ ȱȱȱȱǯȱ àǱȱȱȱȱȱàȱȱȱȱøȱǰȱȱ I Cȱ(D)ǰȱ I( [) − I(D) I ′ (D) = OLPȱ ȱȱÇȱ¡ǯȱȱ [→D [− D ȱ ȱ ȱ ȱ ¡Ȭ ȱ ȱ ȱ ȱ ȱ [ =ȱD+ Kǯ I ( [) =ȱ I(D+ K) ¢ȱȱ¡ȱȱ¡ȱ ȱǰȱȱȱ¡¤ȱȱŖǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ¤Ǳȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱȱàȱȱȱȱøȱǰȱȱ I ′ȱ(D)ǰȱ ICȱ(D) = OLP K → I(D+ K) − I(D) ȱȱÇȱ¡ǯȱ K ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Ǳȱ Ǽȱȱȱȱȱȱȱȱȱ¤ȱȱȱà I ( [)=ȱ \ ȱȱ (D ȱ I(D)) ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ I ′(ȱD)ǯȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ øȱ ȱ ¢ȱ ȱ \ =ȱ I(W)ǰȱ ȱ ȱ ¤ȱȱȱȱȱȱ W =ȱD ȱȱȱȱȱȱȱȱàȱȱ ȱǯȱȱǰȱȱȱ¤ȱȱȱȱȱ I ′(ȱD)ǯȱ ȱȱȱȱ¢ȱȱȱȱȱȱȱȱȱ¡ǯȱ ȱȱ \ =ȱ I( [)ǯȱȱȱȱ [ȱ [ȱ ȱȱȱȱ¡ȱǱ Δȱ[ = [ − [ǰȱ ¢ȱȱȱȱȱ¢ȱǱ Δȱ\ = I( [) − I( [)ǯȱ ȱȱ£àȱȱȱȱȱ ¢ȱȱȱȱ¡ǰȱȱȱ Δȱ\ I( [) − I( [) [[ȱ [] ȱǱ = Δ[ [ − [ ȱ¤ȱ£àȱȱȱ¤ȱȱ¢ǰ ȱȱȱ¡ǰȱǱ I([) − I( [) Δ\ = OLP Δȱ[→ Δ[ [→ [ [ − [ OLP ȱ I ′ȱ(D) ȱȱ£àȱ¤ȱȱȱ \ =ȱ I( [) ȱȱ ȱ¡ȱ [ =ȱ Dǯ ȱȱȱȱàȱȱȱȱǱȱȱȱȱȱȱȱ ȱàȱȱȱȱȱ ȱȱǯȱ ȱ ǰȱ ¤ȱ ¢ȱ øȱ ȱ ȱ ȱ ȱ ȱ ȱ à \ =ȱ I( [) ȃȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ Ȅǯȱ ȱ ȱ ȱ ȱ ȱ ȱǱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 I([+ K) − I([) K K→ I′([) = OLPȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ Ǳȱ \ =ȱ I( [) ȱȱȱȱȱȱ \′ȱ= I ′ ([) = G\ GI G = = I([) = 'I( [) = '[ I( [)ǯȱ G[ G[ G[ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çǰȱ ȱ ȱ ȱ ȱ ¡ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ¡ȱȱȱȱȱàȱȱȱȱȱȱȱȱǯȱ ȱàȱȱȱ ȱȱȱ I ′ȱ(D) ¡ǯȱȱȱȱȱȱ ȱȱȱȱȱȱȱȱǯ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àǯȱȱȱȱ ·ȱàȱȱȱǯȱ ȱȱàȱȱȱȱȱàǰȱȱȱȱȱȱȱ ȱȱ OLPȱ I( [) = I(D)ǰȱȱàȱ£ȱȱȱȱȱȱ·ȱ [→D ȱȱ¤ȱȱȱàȱȱȱȃȄȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ¤ȱȱȱȱȱȱ£ȱȱȱȱȱȱ¤ȱ ȱȱàȱȱȱǯȱȱȱ£ȱȱȱȱȱȱ¤ȱȱȱ àȱȱǰȱȱȱȱȱȱȱȱȱ¤ȱȱȱȱȱȱȱǯȱ ȱȱȱȱȱȱȱȱǱȱȱ Ǳȱȱȱȱȱȱǰȱȱȱȱȱȱǯȱȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ I( [) − I(D) OLPȱ ¡ǯȱ [→D [− D 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 OLP ( I( [) − I(D)) [→D OLP ( I( [) − I(D)) = [→D ⎛ I([) − I(D) ⎞ OLP⎜ ⋅ ( [− D)⎟ [− D ⎠ I( [) − I(D) = OLP • OLP( [− D) [→D [→ D [− D = I′ (D)⋅ [→ D⎝ ¢3RUTXpVHSXHGHDSOLFDUSURSLHGDGHVGHOtPLWHV" = OLP I( [) = OLP( I([) − I(D) + I(D)) [→D [→ D = OLPȱ( I( [) − I(D)) + OLP I(D) [→D [→ D = + I(D) ∴ I HVFRQWLQXDHQD ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ·ȱ ǯȱ ȱ ǰȱȱȱȱ£ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ à I ( [)=ȱ [ ǰȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ Ŗǯȱ ȱ I ′ȱ() ȱ ¡ǯȱ ȱ ȱ ǰȱ ȱ ȱ ȱ [ −ȱ [ OLP = OLP ǰȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ [→ [− [→ [ ¡ǯȱȱȱȱ¤ȱȱȱà I ( [)=ȱ [ ȱ ȱ ȱ [ȱ = ¢ȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¤ȱ ȱȱàȱȱȱȱ I ′ȱ([) Ǳȱ ȱ ŗǼ I( [)=ȱ + [ à I([+ K) − I [ + [+ K− + [ = OLP OLP K K K→ȱ K→ ⎡ + [+ K− + [ + [+ K+ + [⎤⎥ = OLP ⎢ • ⎢ K + [+ K+ + [⎥⎦ K→ ⎣ + [+ K− (+ [) + [+ K−− [ = OLP = OLP K→ K⎛⎜⎝ + [+ K+ + [⎞⎟⎠ K→ K⎛⎜⎝ + [+ K+ + [⎞⎟⎠ I′( [) = K = OLP OLP ⎛ ⎞ = K→ K⎜⎝ + [+ K+ + [⎟⎠ K→ + [+ K+ + [ = + [+ + [ ŘǼ I [ =ȱ = + [ = + [ + [ [ řǼ I [ =ȱ [ ŚǼ Iȱ [ = [+ [+ śǼ Iȱ [ = [ [+ ȱ ŜǼ Iȱ [ = [ + ŝǼ I [ =ȱFRV[ ŞǼ Iȱ [ = [+ VL≤ [ < ⎧[ ⎪ Iȱ [ = ⎨ VL [ = ⎪− [+ VL< [ ≤ ⎩ ǼȱǶȱ ȱ ȱȱ¡ȱƽŗǵȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱǼȱǶȱ ȱ ȱȱ¡ȱƽŗǵ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱȱ Ǽ I = ⎫ OLP I [ = OLP [ = − − ⎪⎪ [ → [→ ⎬ ⇒ OLP I [ = OLP I [ = OLP −[+ = ⎪ [→ ⎪⎭ [→ + [→ + I (ȱ)≠ OLP I [ ⇒ I ȱȱȱȱ¡ȱƽȱŗȱ [→ Ǽ Iȱ ȱȱȱȱ¡ȱƽŗǰȱȱȱȱȱȱȱ¡ȱƽȱŗȱ ȱ ⎧⎪ [ − VL[< ǯȱȱȱ ȱȱȱȱ¡ȱƽȱŚȱȱ ⎩⎪[− VL[≥ Iȱ [ =⎨ àǱȱ ȱ I ′ȱ() ¡ǰȱȱ¡ OLP [ȱ→ I([) − I() = ǰȱȱȱ [− ȱ¡ȱȱȱÇȱǱ I([)− I [−− = OLP OLP [− [→ − [→ − [− [ − = OLP = OLP ( [+ )= [− − [→ [→ − I −′ () = I([)− I [− − = OLP OLP [− [→ + [→ + [− ( [− ) = OLP [− + [ȱ→ I +′ ( ) = Iȱ−′ () = I+ ′ () ȱȱȱȱ [ȱ= ⎧⎪ [ + VLȱ → [ ≤ ȱ I [ = ⎨ VL → [ > ⎪⎩− [ Ǽȱ Ƕȱȱȱȱ¡ȱƽȱřǵȱ Ǽ Ƕȱȱȱȱ¡ȱƽȱřǵ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àǱȱ Ǽ I = OLP I [ = ⎫ ⎪⎪ [ȱ→ − ⎬ ⇒ OLP I [ = OLP I [ = ⎪ [→ [→ + ⎭⎪ I ȱ = OLP I [ ⇒ I ȱȱȱ¡ȱƽȱřȱ [→ Ǽ I −′ () = I([)− I [+ − [− = = = OLP ( [+ )= OLP OLP OLP [ − [ − [ − − − − [→ [→ [→ [ → − I([)− I − [− − [ = = OLP OLP OLP [− [− [→ + [→ + [→ + [− = OLP − (+ [)= − [ → + I +′ () = Iȱ−′ () ≠ I+ ′ () ⇒ ȱȱȱȱȱ¡ȱƽȱřȱ ȱ ⎪⎧ [ + [+ I [ = ⎨ ⎪ ⎩− [− [ VL[< − VLȱ[≥ − ǼȱǶȱȱȱȱ¡ȱƽȱȮŘǵȱȱȱȱȱȱȱȱȱȱȱǼȱǶ ȱȱ¡ȱƽȱȮŘǵȱ ȱ ⎧ [ + VL[ ≤ I [ = ⎨ ⎩− [ VLȱ[ > ǼȱǶȱ ȱ ȱȱ¡ȱƽřǵȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱǼȱǶȱ ȱ ȱȱ¡ȱƽřǵȱ ȱ ⎧ VL[ < E ⎪⎪ [ I ȱ[ = ⎨ ⎪− [ VL[ ≥ E ⎪⎩ Ǽȱȱȱȱȱȱȱȱȱȱȱȱȱǯȱ ǼȱǶȱ ȱ ȱȱȱȱȱȱȱȱȱȱǵ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àȱ Ǽȱ ȱàȱ¤ȱȱȱǰȱȱ I E =ȱ− E ȱȱȱȱǯ OLP I [ = E [ → E− ⎫ȱ ⎪ ⎪ E ⎬ ⇒ OLP I [ ¡ ⇔ = − E E OLP I [ = − ⎪⎪ [→ E [→ E+ ⎭ − E ⇔ = ⇔ = E− E ⇔ E − E+ = ⇔ (E− ) = E ⇔ E = I ȱE = OLP I [ ⇔ E= ⇒ I ȱȱȱȱȱȱƽȱŘȱ [→ E Ǽȱ ȱȱ£¤ȱȱȱàȱȱȱȱȱƽȱŘǯ − I( [) − I − [ ′ I− ( ) = OLP = OLP [ = OLP [− [− [( [− ) [→ − [→ − [→ − =− = OLP − [ [ → − [ [ − − − ( ) I [ I − ′ = OLP I + ( ) = OLP = OLP [− [− [− [→ + [→ + [→ + − [ = OLP − = − = OLP ( [− ) [ → + [→ + I −′ (ȱ) = I+ ′ ()⇒ I ȱȱȱȱƽȱŘȱ ȱ ȱȱ¤ȱȱȱàȱȱȱȱȱȱ Ǽ Iȱ ȱȱǱ ]−ȱ∞−]]− [ ¢ ]ȱ+∞[ Ǽ Iȱ′(− )= ¢ I ′ȱ() ȱ¡ȱ Ǽ OLP I [ = ¢ OLP I [ = +∞ [ȱ→−∞ [ȱ→+∞ Ǽ OLP I [ = ¢ OLP I [ = − + [ȱ→− [ȱ→− Ǽ OLP I [ = −∞ ¢ OLP I [ = +∞ − + [ȱ→ [→ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱàȱ ȱ ¢ ¤ȱȱȱȱ Ǽȱ ȱȱȱȱàȱȱȱȱȱȱÇǰȱȱ¡ǰȱ¢ȱ ȱȱȱȱȱȱȱ¡ȱÇȱ¢ȱȱȱȱȱǯȱ 9 OtP I [ 9 OtP I [ [→ − 9 OtPI [ [→ 9 OtP I [ [→ȱ [→ − 9 OtP I [ [→ − Ǽȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱȱȱȱȱǻȱȱȱȱǼȱ 9 [=− 9 [ = Ǽȱ Ǽȱ 9 [= 9 [ȱ = − ȱȱȱȱȱȱȱȱ ȱȱȱ ¡ȱȱȱȱȱȱȱȱȱȱȱ àǱȱ Ǽȱ Ǽ OtP I [ = Ǽ OtP I [ = Ǽ OLP I [ = Ǽ OtP I [ = [ȱ→− [ȱ→− [ȱ→ [ȱ→ Ǽ OtP I [ = − [ȱ→− ȱȱȱǼIȱ ȱȱ [ȱ=− ȱ Iȱ − = ¢ OtP I [ = [→− 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ǼIȱ ȱȱ [ȱ= ȱ Iȱ = ¢ OtP I [ = [ȱ→ Ǽ Iȱ ȱ ȱ ȱ [ȱ= ȱ Iȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱȱ OtP I [ ȱ¡ǯȱ [→ȱ Ǽ Iȱ ȱ ȱ ȱ [ȱ=− ¢ȱ ȱ ȱ ȱ ȱ OtP I [ ȱ¡ǯȱ [→ȱ− Ǽ (−∞ȱ−) ¢ (−ȱ) ·ȱȱȱȱȱȱȱȱ ȱȱ ȱ ȱǯȱ Ǽ Iȱ ȱȱ [ȱ = Iȱ ȱȱȱ [ȱ = ǯȱ ȱ ȱȱà Iȱ ¢ȱȱȱȱȱ ȱȱȱȱàȱǰȱȱȱ¡àȱȱȱȱȱȱȱȱ ȱȱȱàȱǯȱ Ǽ OLP I [ = ȏȏȏȏȏȏȱ [ȱ→− Ǽ OLP I [ = ȏȏȏȏȏȏȱ [ȱ→− Ǽ OLP I [ = ȏȏȏȏȏȏ [ȱ→−∞ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Ǽ OLP I [ = ȏȏȏȏȏȏȱ [ȱ→∞ Ǽȱ ȱ ȱȱȱȱȱ¡ȱ ȱȱȏȏȏȏȏȏȏȱ Ǽȱ ȱ ȱȱȱȱ ¡ȱ ȱȱȏȏȏȏȏȏȏȱ Ǽȱȱȱȱȱȱȱȱȱȱȱ ǼȱȱÇȱǰȱȱȱ¤ȱ ȱǰȱȱȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽ Iȱ − ȱȱȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽȱ ȱÇȱ£ǰȱȱȱ¤ȱȱȱǰȱȱȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽȱȱȱȱ¡ȱȱȱȱȱȱȱȱȱȱǱȱȏȏȏȏȏȏȱ Ǽȱ ȱȱ Iȱ ȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱȱ àǱȱ Ǽ OLP I [ = − [ȱ→− Ǽ OLP I [ = [ȱ→− Ǽ OLP I [ = −∞ Ǽ OLP I [ = [ȱ→∞ Ǽȱ Ȭŗȱ Ǽȱ řȱȱȱȱȱȬŘȱ [ȱ→−∞ ǼȱȱȱǱ (−∞ȱ−)[− −)(− )(+∞ ) ǰ [ȱ+∞) ǰȱȱȱȱȱȱȱȱ ǯȱ Ǽ [ȱ=− Ǽȱśȱȱȱȱȱŝȱ ǼȱŖȱ Ǽȱȱ Ǽ \ȱ = ȱ ȱȱà Iȱ ¢ȱ¤ȱȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ǻǼȱȱȱȱȱ¤ȱȱȱȱȱÇǰȱȱ¡ǲȱȱ ȱȱȱȱ¡ȱÇȱ¢ȱȱȱ OLP I [ OLP I [ OLPI [ [→ OLPI [ ȱ [→− [→− [→ ǻǼ ȱȱ ȱ ȱȱȱȱȱȱȱȱȱȱàǰȱȱ ȱȱȱȱȱȱȱǯȱȱȱȱȱǯ [= − [= − [ =ȱ ǻǼȱ ȱȱȱȱ¡ǰȱȱ ȱ ȱǰȱȱȱ ȱ I [ [= − [= − [= [ =ȱ ȱ ȱȱà Iȱ ¢ȱȱȱȱȱ ȱȱȱȱ¤ȱǰȱȱȱ¡àȱȱȱȱȱȱȱ ȱȱȱȱȱàȱȱ Ǽ OLP I [ ȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ [→− Ǽ OLP I [ ȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ [→ȱ− Ǽ OLP I [ ȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ [→ȱ Ǽ OLP I [ ȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ [→ȱ Ǽ Iȱ ȱȱȱȱȱ¡ȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽ Iȱ ȱȱȱȱȱ¡ȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ ǼȱȱÇȱ£ǰȱȱȱȱ Iȱ ǰȱȱȱȱȏȏȏȏȏȏȏȏȏȱ ǼȱȱÇȱǰȱȱȱȱ Iȱ ǰȱȱȱȱȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽ I ′ȱ() ȱȱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽȱȱȱȱ¡ȱ Iȱ ȱȱȱȱȱȱȱȏȏȏȏȏȏȏȏȱ Ǽȱȱȱ Iȱ ȱȱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱàȱ Ǳȱ ȱ ȱȱȱàȱȱȱȱ ȱ ȱȱ ǯȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ àǰȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱǯȱȱ Ǳȱ ȱȱàȱȱȱȱȱàȱȱ ŗǼ 6L I [ =F HQWRQFHV I′ [ = ȱ ȱ àȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ·ǰȱ ȱ ȱǯȱȱȱȱȱȱȱ ȱàȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ǯȱ ȱàȱȱȱȱ¤ȱȱȱ£ȱ¢ȱȱȱȱ ǰȱȱȱȱȱȱȱȱ¤ȱ I ȱ[ =F ȱȱǯȱȱ Ǳȱ ȱȱàȱȱȱȱȱàȱȱ I ȱ[ = [Q Iȱ′ [ = Q[Q− Qȱ∈ ,1− {} Ǳȱȱȱ Ǽ ( [ + K) = [Q + Q[Q−K+ Q Q(Q− ȱ) Q− [ K + + Q[KQ− + KQ Ǽ [ȱQ −DQ = [− D [Q− + [Q− D+ + [DQ− + DQ− 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱȱȱàȱȱȱȱȱ Q ∈ȱ,5 ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ àȱ ȱ ¤ȱ Q ∈ȱ,5ǯȱȱȱȱǰȱȱȱȱȱȱ¤ȱȱ£ȱȱȱàȱȱ ȱȱȱ ¤ȱ ȱ ȱ ¤ȱ ȱ ǯȱ ȱ ǰȱ ȱ ȱ ¤ȱ ȱ ȱ £ȱȱȱǱȱȱ ŘǼȱ I ȱ[ = [Q Iȱ′ [ = Q[Q− Q ∈,5 Ǳȱ ȱȱàȱȱȱ¢ȱȱȱȱȱÇȱȱ ȱàȱǯȱ ȱȱȱȱȱȱ¢ȱ ȱȱȱǱȱ ′ȱ řǼ (FI ( [)) = FI ′( [) ′ ŚǼ ( I ([) +ȱJ([)) = I ′( [) + J′([) ′ śǼ ( I ([) −ȱJ([)) = I ′([) − J′([) ȱ àȱ ȱ ȱ ȱ ¤ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱȱȱȱȱ¤ǯȱ 6L I\ J VRQIXQFLRQHVGHULYDEOHVHQWRQFHV ( I( [)J( [))′ = I′([)J([) + I([)J′([) ′ ⎛ I( [) ⎞ I′( [)J([) − I( [)J′( [) ⎟⎟ = ⎜⎜ J( [) ⎝ J( [) ⎠ ȱ ȱ àȱ ŝǼȱ ȱ ȱ ȱ ȱ ·ȱ ȱ ȱ ȱ ȱ ȱ¢ȱȱȱȱȱǯȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ¤ȱ Ǽȱȱȱȱȱȱȱȱ ŗǼ Iȱ [ = Ȧ Iȱ′ [ = Iȱ [ =π Ȧ Iȱ′ [ = ŘǼ řǼ Iȱ [ = Ȧ Iȱ′ [ = ŚǼ Iȱ [ =−[+ Ȧ Iȱ′ [ = − śǼ Iȱ [ =[ − [+ Ȧ Iȱ′ [ = [− ŜǼ Iȱ [ =[ − [ + [− Ȧ Iȱ′ [ = [ − [ + ŝǼ I [ =ȱ [ + [ Ȧ I ′ȱ [ = Ȧ I′ȱ [ = ŞǼ I [ =ȱ [ [+ şǼ I [ =ȱ [ [ + [ − [ ŗŖǼ Jȱ [ = [ + [− −[ + [+ ŗŗǼ Jȱ [ = −[ − [+ − [ + [ + ([− ) ŗŘǼ \= [ȱ + −[ ŗřǼ I [ = ŗŚǼ \= [+ [ȱ − [+ − [ ȱ+ [ Ȧ I ′ȱ [ = + [ [ − [ [ − [ − [ [ [ + [ Ȧ Jȱ′ [ = −[ − [ + [ + [− ȦJȱ′ [ = [ + [ − [ + [ − [+ Ȧ \ =ȱ − [ + [ + − [ Ȧ Iȱ [ = Ȧ − [ − [+ [ − [+ − [ + [ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Ǽȱȱȱȱȱ ŗǯȱȱȱàȱȱȱȱȱȱ Iȱ( [)= [ − [ + [− ȱȱȱǻŗǰȬřǼǯȱȱ àǱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Iȱ( [)=[ − [ + [− ȱ ȱ ȱ ǻŗǰȬřǼȱ Iȱ′= −ǯȱ ȱ ȱȱ ȱ + E ȱȱȱàȱȱȱȱ¢ȱȱȱȱȱàǱ−ȱ= − Ǳȱȱàȱȱȱȱȱ ȱȱ Iȱ( [)=[ − [ + [− ȱȱȱǻŗǰȬřǼȱȱȱ¢ƽȬŚ¡Ƹŗȱ Řǯȱȱȱàȱȱȱȱ ȱ \ȱ = − [− ¢ȱȱȱȱȱȱ ȱ Iȱ( [)= [ − [+ ǯȱȱ àǱȱ ȱȱȱȱȱȱȱȱȱȱǯȱ ȱȱȱȱȱȱȱȱ Iȱ( [)= [ − [+ ȱȱ (D ȱ I(D)) Iȱ′ D = D− ¢ȱȱȱȱȱȱ \ȱ = − [− ȱȬŗǯȱȱȱȱȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ à ⎛ ȱ ⎞ − = D ȱ− ⇒ D= ǯȱ ȱȱȱȱȱ ⎜ − ⎟ ¢ȱȱȱȱ ⎝ ⎠ ȱàȱȱȱȱȱ \ȱ=− [+ ǯȱ Ǳȱȱàȱȱȱȱ ȱ \ȱ = − [− ¢ȱȱȱȱȱȱ ȱ Iȱ( [)= [ − [+ ȱ \ȱ =− [+ řǯȱ ȱ ·ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ[ + \ = E ȱ ȱ ȱ ȱ ¤ \ =ȱ D[ ȱ¡ȱƽȱŘǯȱȱ àǱ ȱ[ + \= E⇒ \ = −[+ Eǰȱ ȱ ȱ \ =−ȱ[+ E ȱȱȮŘȱ ¢ȱ ȱȱȱȱȱȱȱȱ I ( [)=ȱ D[ ȱȱ ( Iȱ()) ȱȱ I ′() 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱ Iȱ′( ) = −⇒ D()= −⇒ D= −⇒ D= − ȱȱȱ (−ȱ)ǯȱ ǰȱȱȱ ȱ ȱ (−ȱ) ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ àȱȱȱȱǰȱȱǱ ȱ() +(− )= E ⇒ = Eǯȱ Ǳȱȱ ȱ[ + \ = E ȱȱȱȱ¤ − D ȱ = ¢ Eȱ= \ =ȱD[ ȱ¡ȱƽȱŘǰȱ Śǯȱȱȱȱȱȱ¢ȱȱȱȱȱȱ¤ȱ I ȱ[ = [ + E[ ȱ ȱȱȱȱȱȱȱ \ =ȱ[+ F [ȱ=− àǱȱ I [ = [ + E[ Iȱ ′(−) ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ I ( [)= [ȱ + E[ ȱȱ (− Iȱ(− ))ǯȱ ȱ ȱ ȱ ȱ ȱ \ =ȱ[+ F ȱ Řȱ ȱ ȱ ȱ ȱ Iȱ′( −) = ⇒ (−)+ E= ⇒ E= + ⇒ E= ǯȱ (− Iȱ(− )) ȱȱȱȱȱȱȱȱȱȱ ȱȱ I( [)=ȱ [ + [ ¢ȱȱ ȱ Ǳ \ =ȱ[+ Fǯȱ ȱ ȱ ȱ £ȱ ȱ−= I (−ȱ) = (−)+ F ¢ Iȱ − = − + − −+ F ⇒ −= F ȱ Ǳȱȱ ȱ ȱ \ =ȱ[+ F ȱ ȱ ȱ ȱ I ( [)= [ + E[ [ȱ = − ȱȱȱ Eȱ = ¢ Fȱ = − śǯȱ ȱȱȱȱȱȱ ȱȱȱ ȱȱȱŗşŖȱȱ ȱǰȱ ȱȱȱȱȱŚŞ SLHVȱ ǰȱȱȱȱȱȱȱȱǯȱ V Ǽȱ Ƕȱȱȱȱȱǵȱ Ǽȱ Ƕȱȱȱȱȱǵȱ Ǳ V W =ȱ−W + YW+ V Y ȱǰ Vȱ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Ǳȱȱȱȱȱȱȱ ȱśȱȱ¢ȱȱȱȱȱ ȱ ȱ ȱȬŗŗŘȱȦȱǻȱ ȱȱȱȱ £ȱȱȱ àȱ ǰȱȱȱǰȱȱǼǯȱ ŜǯȱȱȱȱȱȱȱȱȱŜŖŖȱȱȱǯȱȱǶ¤ȱȱȱȱ ȱȱȱǵȱǻǱ VW = −W + YW+Vȱ Ǽȱ Ǳȱȱȱȱȱȱȱȱȱ¡ȱȱȬŗşśǰşŜȱ Ȧǯȱ ŝǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ⎛ ⎞ ⎜ ȱ ⎟ ȱȱȱÇǯȱȱ ⎝ ⎠ I ( [)=ȱ− [ ȱ ȱ àǱȱ ⎛ ⎞ ȱ Ǳ ⎜ ȱ ⎟ ȱ ȱȱ ȱ ȱ ȱ ȱ ȱ ȱȱ ȱ ȱ ⎝ ⎠ ǯȱ (D −ȱD ) ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱ I ′ȱ(D)= −Dǯȱ ( ⎛ ⎞ −ȱD ⎜ ȱ ⎟ ¢ D ⎝ ⎠ ) ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ −ȱD − ·ȱÇȱȱȱȱȱȱȱǱ D ȱȱ ȱȱȱȱȱøȱǱ −D ȱ − = −D⇒ −D − = −D ⇒ D = D ⇒ D = ⇒ D = ± ( ) ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ I [ =ȱ− [ ȱ ȱ ⎛ ⎞ ⎛ȱ ⎞ ȱ ȱ ⎜ ȱ ⎟ ǰȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱ ⎜ ⎟ ¢ȱ ⎝ ⎠ ⎝ ⎠ ⎛ ȱ ⎞ ⎜ − ⎟ ǯ ⎝ ⎠ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱȱ ȱȱȱȱ I ( [)=ȱ− [ ȱȱȱȱ Pȱ = − ¢ȱȱȱȱȱȱȱȱȱ I ( [)=ȱ− [ ȱȱȱȱ Pȱ =ǯȱ Pȱ ⋅ P = − ȱȱȱȱȱȱȱȱȱ ⎛ ⎞ I ( [)=ȱ− [ ȱȱ ⎜ ȱ ⎟ ȱǯȱ ⎝ ⎠ Şǯȱ ȱȱȱ¡ȱȱȱȱȱȱȱ (ȱ) ¢ȱȱȱȱ£ȱ ȱȱȱȱȱȱà \ =ȱ [ àǱȱ ȱȱ¡ȱ ȱȱȱȱȱ (ȱ)ǯȱ \ =ȱ [ ȱȱȱȱ (D ȱD ) ȱ ȱ ȱ ǰȱ Dȱ= DD−−ǰȱ ȱ ȱ ȱ D ȱ −D+ = ǯȱ ȱȱȱȱȱàȱȱȱǰȱȱȱȱ ȱȱ¡ȱȱøȱ ȱ ȱȱȱà D ȱ −D+ = ǰȱȱȱ ¡ȱȱȱȱȱȱȱ (ȱ) ¢ȱȱȱȱ£ȱȱȱȱȱȱȱ à şǯȱ \ =ȱ [ǯȱ ȱȱȱȱȱȱȱȱȱȱ (ȱ) ¢ȱȱ ȱȱȱȱȱȱ I ( [)=ȱ − [ ǯȱȱ àǱȱ ȱ (ȱ) ȱȱȱȱȱȱȱȱȱȱȱȱȱ ǯȱ ( ) D −ȱD ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱ I ′ȱ(D)= −Dǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ ( /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ) (ȱ) ¢ D −ȱD ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ·ȱÇȱȱȱȱȱȱȱǱ −D ȱ − D− ȱȱ ȱȱȱȱȱøȱǱ −D ȱ = −D⇒ − D = −D − D⇒ D + D= D− ⇒ D= − R D= ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ I ( [)=ȱ− [ ȱ ȱ ȱȱ (ȱ)ǰȱ¢ȱȱȱȱȱȱȱȱȱȱ (ȱ) ¢ȱ (−ȱ) ǯȱ ȱȱȱȱȱȱȱȱ I ( [)=ȱ − [ ȱȱȱȱ Pȱ = ǰȱȱȱàȱȱȱȱȱȱǱ\ȱ = ǯȱ ȱȱȱȱȱȱȱȱ I ( [)=ȱ− [ ȱȱȱȱ Pȱ =−ǰȱȱȱàȱȱȱȱȱ \ȱ =− [− ŗŖǯȱ ȱȱȱȱȱȱȱȱȱȱȱ \ =ȱ [ ¢ȱȱȱȱȱ (−ȱ)ǯȱ Ǳȱȱȱȱȱȱȱȱȱȱȱ \ =ȱ [ ¢ȱȱ ȱȱȱ (−ȱ) Ǳ \ȱ = [− ¢ \ȱ =− [− ǯȱ ŗŗǯȱ ȱȱȱ ǰȱȱ ¢ȱ ȱ ȱȱȱ \ = [ȱ + D[ +E ¢ \ = F[ − [ ȱ ȱȱȱÇȱȱȱȱǻŗǰŖǼǯȱ Ǳȱȱȱȱȱȱà \ = [ȱ + D[ +E ¢ \ = F[ − [ ȱ ȱ ȱȱÇȱȱȱȱǻŗǰŖǼǰȱȱȱȱȱȱ¤ȱȱȱȱ ǱȱƽȬřȱǰȱƽŘȱȱ¢ȱȱƽŗǯȱ ŗŘǯȱ ȱȱàȱȱȱȱȱȱȱ¤ȱȱȱà I [ = [ȱ + ȱȱȱȱȱȱ ř¡ȱƸȱ¢ȱƸȱŘȱƽȱŖǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǳȱȱàȱȱȱȱȱȱȱ¤ȱȱȱà I [ = [ȱ + ȱ ȱȱȱȱà [ȱ+\+ = [ȱ+\− = ŗřǯȱ ȱȱȱȱȱȱȱǻŚǰȱŗřǼȱȱ¢ȱȱȱȱȱȱȱ \ = [ −ȱ ȱȱȱȱàȱȱȱȱȱȱǯȱ ǱȱȱȱȱàȱȱȱȱȱȱȱȱǻŗǰŗǼȱ¢ȱ ǻŝǰşŝǼǯȱ ŗŚǯȱ ȱȱȱȱȱ ȱȱȱȱȱȱǻŘǰȮřǼȱ¢ ȱ ȱȱȱȱȱ \ = [ȱ + [ ǯȱ Ǳȱȱȱȱȱ ȱȱȱȱȱȱǻŘǰȮřǼȱ¢ȱȱ ȱȱȱȱȱà \= [ȱ +[ \ȱ−[+ = ¢ \ȱ+[+ = ŗśǯȱ ȱàȱȱȱ¤ \ = [ȱ + E[+ F ȱȱȱȱȱ à ¢ȱƽȱ¡ȱ ȱȱȱǻŗǰŗǼǯȱȱ àǱȱ ǻŗǰŗǼȱȱȱȱȱǰȱȱȱȱȱȱȱàȱ ȱ= + E + Fǯȱ ȱȱȱȱ \ =ȱ [ ȱȱȱȱȱàȱ + Eȱ= ǯȱ ȱȱȱȱȱ Eȱ=− ¢ Fȱ= ǯȱ Ǳȱȱ¤ ȱȱȱȱȱà ¡ȱƽ¢ȱ ȱȱȱǻŗǰŗǼȱ \ = [ − [+ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱȱȱàȱ¡ȱ Ǳȱ ȱȱȱȱȱà ¡ǯ ȱȱȱȱȱà I ( [)=ȱD[ǯȱ ȱȱȱȱȱȱȱȱ£ȱȱàǯȱ I([+ K) − I([) D[+ K − D[ = OLP D[ ⋅ DK − D[ D[ DK − K K = OLP = OLP K→R K→ ( I′([) = OLP K K→ȱ K→ ) K ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ǰȱ D ȱ[ ǰȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ¤ȱȱ I ( [)=ȱD[ DK − D DK − = OLP K→ȱ K− K→ K I ′ = OLP ( ) D[ DK − ȱ ȱ ȱ K K→ȱ Iȱ C() ¡ǰȱ ȱ ȱ ȱ I′ [ = OLP ⎛ DK − ⎞ D[(DK − ) ⎟⎟ = D[ OLP⎜⎜ K→ȱ K→ K K ⎝ ⎠ ȱÇȱ¢ ȱǱ OLP ȱȱȱȱȱŖȱȱȱȱȱȱȱȱ¢ ȱȱ I′ȱ([) = I ′()D [ǯȱȱȱȱȱȱȱȱȱȱȱȱȱ·ȱ ȱȱ¡ȱ IȱC()ǯȱ D ȱ= I([)=ȱ [ ¢ OLP K→ȱ K −ȱ Ŗǰŗȱ ŖǰŖŗȱ ŖǰŖŖŗȱ ŖǰŖŖŖŗȱ ŖǰŖŖŖŖŗȱ Ŗǰŝŗŝŝȱ ŖǰŜşśŜȱ ŖǰŜşřŚȱ ŖǰŜşřŘȱ ŖǰŜşřŗȱ ( ) = [ OLP⎛⎜ K − ⎞⎟ [ K − K K K→ ⎜ ⎝ K ⎟⎠ K −ȱ K ȬŖǰŗȱ ȬŖǰŖŗȱ ȬŖǰŖŖŗȱ ȬŖǰŖŖŖŗȱ ȬŖǰŖŖŖŖŗȱ ŖǰŜŜşŜȱ ŖǰŜşŖŝȱ ŖǰŜşŘşȱ ŖǰŜşřŗŘȱ ŖǰŜşřŗŚȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 'HODVWDEODVDQWHULRUHVVHSXHGHGHGXFLUTXH I′()= OLP K→ VL I( [) =ȱ ⇒ I C([) ≈ () [ K −ȱ ( ) = [ OLP⎛⎜ K − ⎞⎟ [ K − K→ K Ŗǰŗȱ ŖǰŖŗȱ ŖǰŖŖŗȱ ŖǰŖŖŖŗȱ ŖǰŖŖŖŖŗȱ K [ D ȱ= I([)=ȱ [ ¢ OLP K − ≈ SRUORWDQWR K K→ ⎜ ⎝ K ⎟⎠ K −ȱ K ŗǰŗŜŗŘȱ ŗǰŗŖŚŝȱ ŗǰŖşşŘȱ ŗǰŖşŞŝȱ ŗǰŖşŞŜȱ ȬŖǰŗȱ ȬŖǰŖŗȱ ȬŖǰŖŖŗȱ ȬŖǰŖŖŖŗȱ ȬŖǰŖŖŖŖŗȱ ŗǰŖŚŖŚȱ ŗǰŖşŘŜȱ ŗǰŖşŞŖȱ ŗǰŖşŞśȱ ŗǰŖşŞŜ 'HODVWDEODVDQWHULRUHVVHSXHGHGHGXFLUTXH I′()= OLP K→ K − ≈ SRUORWDQWR K VL I([) = ⇒ I′([) ≈ () [ [ ȱàȱ¤ȱȱȱàȱȱ¤ȱ Iȱ′( )= ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Iȱ ′()≈ VLD= ¢ Iȱ′()≈ VL D= ȱ ȱ £ȱ ȱ ȱ ȱ¡ȱȱøȱȱȱȱȱŘȱ¢ȱřȱȱȱ Iȱ ′( )= ǯȱ ȱøȱ¡ȱ¢ȱȱȱȱǯȱ 'HILQLFLyQ HK − = K → K H HVHOQ~PHURTXHVDWLVIDFHTXH OLP I ( [) =ȱ H[ ȱ I′([) = H[ ⋅ I′() =H [() =ȱH [ ȱà I ([)=ȱH[ ȱȱȱȱȱȱȱǯȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱȱȱȱ· Ǳȱ ȱȱȱȱȱȱ·ǯ •ȱ ȱȱàȱȱȱ¢ȱȱàȱǱ VHQ( [+ \) = VHQ [FRVȱ \+ VHQ \FRV [ FRV( [+ \)= FRV [FRV \− VHQ [ VHQ \ Ǳ 6L I ([)= VHQ[ HQWRQFHV I′( [)= FRV [ 6L I([)=FRV [ HQWRQFHV I ′([)= − VHQ[ •ȱ ȱȱȱȱàȱ¢ȱȱǱ WDQ [=ȱ VHQ[ FRV[ FRWDQ[= VHF[= FVF[= FRV[ VHQ[ FRV[ VHQ[ Ǳȱȱ 6L I([) = VHF [ ⇒ I C([) = VHF[WDQ[ 6L I([) = FVF[ ⇒ I C([) = − FVF[FRWDQ[ 6L I([) = WDQ[⇒ I C([) = VHF [ 6L I([) = FRWDQ[ ⇒ I C([)= − FVF [ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ¤ȱ ǼȱȱǻȱȱȱǼȱ ŗǯ \ =ȱ [ − [ + [ H[ Řǯ \= H[ȱ [ − řǯ Iȱ [ = H [+ [ Śǯ \ =ȱ[ VHQ[− H śǯ \ =ȱ [VHF[+ H[ WDQ[ Ŝǯ \= [ VHQȱ[ ŝǯ \ȱ = Şǯ \= H[ H[ − − VHQ[ +FRVȱ[ şǯ I [ =ȱ WDQ[ + FRV[ ŗŖǯ I [ =ȱ WDQ[− VHF[ Ȧ \ =ȱ [ + [ + [+ H[ Ȧ \ = H[ȱ [ − [− [ − Ȧ Iȱ [ = −H [+ Ȧ \′ȱ= [ VHQ[+ [ FRV[− H[ Ȧ\=ȱVHF[ + [WDQ[ + H[ WDQ[+ VHF [ Ȧ \=ȱ [ VHQ[− [FRV[ VHQ[ − H[ \ ȱ = Ȧ H[ − Ȧ \ =ȱ VHQ[− FRV[− Ȧ I ′ȱ [ = Ȧ I′ȱ [ = + FRV[ VHF[ VHF[+ + VHQ[ + FRV[ WDQ[+ VHF[ Ǽȱȱȱȱǯȱ ŗǯȱ ȱȱàȱȱȱȱȱȱȱ \ȱ= [H[ + ȱȱ ȱǯȱȱ àǱȱ ȱ ȱ ȱ ȱȱ ȱ ȱ ȱ ȱ ȱ \ȱ = [H[ + ȱ ȱ ȱ \ȱ′ = ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱȱȱȱȱȱàȱȱȱȱ¢ȱ ȱȱȱȱàǱȱ= + E [ Ǳȱȱàȱȱȱȱȱ ȱȱ \ȱ = [H + ȱ ȱ ȱȱ¢ƽŘ¡Ƹŗȱ ȱ Řǯȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ \ =ȱ H[ ȱ ȱ [ Hȱ ǯȱ Ǳȱȱàȱȱȱȱȱ ȱȱ \ =ȱ Hȱ ȱ \ =ȱH řǯȱ ȱȱȱȱȱ H[ ȱȱ [ \=ȱ + H[ − [ ȱȱȱȱ ȱȱȱȱ [ȱ−\ = àǱ \ȱ= + H[ − [⇒ \′ = H[ − ȱȱȱȱȱȱȱřǰȱȱǰȱȱȱȱȱ ȱ ȱ[ − = \ (D+ȱH ) − D ȱȱȱȱȱ I ′ȱ(D)= PWDQ ȱ D −ȱ= ⇒ HD = ⇒ HD = ⇒ D = Q ȱȱ Iȱ ′( D)= H D Ǳȱȱȱ ȱ Śǯȱ ( Q−ȱ Q)ǯȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ \ =ȱ [+ FRV [ ȱ ȱ ȱ ǯȱ Ǳȱȱàȱȱȱȱȱ ȱȱ \ =ȱ [+ FRV [ ȱ ȱ ȱ \ =[+ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD śǯȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱàȱȱȱȱȱȱȱ \ = FRV[+ [VHQȱ[ ȱȱ πȱ−ǯȱ Ǳȱȱàȱȱȱȱȱ \ = FRV[+ [VHQȱ[ ȱȱ πȱ− ȱ \ȱ = −π[+ π − Ŝǯȱ ȱȱȱȱ¡ȱȱȱȱȱ¤ȱ I [ =ȱ [+ VHQ[ ȱ ȱȱ£ǯȱȱ àǱȱ ȱȱȱȱȱȱ£ǰȱȱȱȱǯȱȱȱȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ à Iȱ′ [ = ǰȱȱ +FRV[ȱ= ǯȱ Ǳȱ ȱ ¤ȱ I [ =ȱ [+ VHQ[ ȱ ȱ ȱ £ȱ [=ȱ ± π + Nπ ȱȱȱøȱǯȱ ŝǯȱ ȱ ȱ ȱ ȱ ȱ \ = FRVȱ[ ȱ ȱ ȱ ȱ ȱ ȱ + VHQ[ £ǯȱ Ǳȱ ȱ ȱ ȱ ȱ \ = FRVȱ[ ȱ ȱ ȱ ȱ ȱ ȱ + VHQ[ ⎛ π ⎛ π ⎞ − ⎞⎟ ⎜ +Nπ ȱ ⎟ ȱ ȱ ȱ øȱ £ȱ ȱ ȱ ȱ ⎜⎜ +Nπȱ ¢ ⎜ ⎟ ⎟ ⎝ ǯ ⎠ ⎝ ⎠ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱȱȱ Ǳȱ ȱ ȱȱȱȱȱȱȱȱȱȱȱȱ ȱȱȱȱȱàȱȱȱȱ¤ȱǯ ȱ ȱ ȱ ·ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱ ȱȱ ȱ ȱȱǰȱȱ·ȱ ȱȱȱȱȱàǯȱ ȱȱȱȱȱȱȱàǰȱȱȱȱȱàǰȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ Ûȱ ȱ ȱȱȱȱȱ¢ȱǯȱ ȱàȱȱȱȱȱ¤ȱ¤ȱȱȱȱǰȱȱȱȱ ȱȱǯȱȱ ȱȱȱȱ ȱȱȱȱȱȱȱȱ¢ȱƽȱȱȱàȱȱ ȱ ) ( [) =ȱ I( J( [))ǰȱȱȱȱȱ¢ȱȽȱȱ¡ȱȱ ȱ ) ′ȱ( [) = I ′ (J( [))J′([) ¤ȱ ǼȱȱǻȱȱȱǼ \ = [ +[ WDQ [ [ \= VHQFRV [ + \ = ] + VHQ [ −WDQ [ \= [ − [+ [−VHF [ \ = \= \= \ = [ + [+ [ + FRV [ [ + [ − [+ [ − [ + [ \= [ȱ − [ + [ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD \= \= /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 [+ [ + [ + FRV [ + [ [ + \ = [ +FRV [ \ = [+ [ + \ =VHF FRV[ \= [ + VHF [ [ FRV [ + ( ) \= [ + VHF[ \= WDQ⎛⎜⎝ [ + ⎞⎟ ⎠ \= [ + VHF [ \= [+ VHQ [ + [ + [ − [ J [ = \ = [ +FRV[ [ \ = ( D+ [) D − [ \ = + WDQ[ \ = [ − [+ \= ⋅WDQ [ [− ( )⎥⎦ \=⎡ [ − [VHF [ + [⎤ \=WDQ⎛⎜ H [ ⎞⎟ \ = H([ +[) ⋅ FRV( [− ) I [ = (FRV[− )(− VHQ[) + WDQ⎜ ⎟ ⎝ [⎠ I [ = I [ = (WDQ[) VHF[+ ⎢⎣ ⎝ ⎠ ⎛ ⎞ VHQ([ + ) FRV([) I [ = [HVHQ[ [VHQ [ + \= [ + I [ =(WDQ [− [) \= [ WDQ [ + I [ = \ = [ + [ VHQ[+ FRV [ ( ) I [ =([ + ) VHQ [ + I [ =FRVHVHQ[ + [ ( ) I [ = πȱH[ + VHF[ ǼȱȱȱÇȱȱ VHQ[− VHQD [→D [− D ŗǼ OLP VHQ([ + ) FRV([)+ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱ VHQ[− VHQD OLPȱ [→D [− D I [ =VHQ[ I′ [ = FRV[ = OLP [→D I [ − I D [− D =ȱ I ′ D = FRVD VHQ(+ [) − VHQ [→ȱ [ ŘǼ OLP àǱ VHQ(+ [) − VHQ [→ȱ [ I([) − I() = OLP [→ [− OLP = Iȱ ′() ( ) = FRV(+ ) (+ ) = FRV() ⎛ [πȱ ⎞ FRV⎜ ⎟ ⎠ ⎝ řǼ OLP [→ − [ àǱ ⎛ [πȱ ⎞ FRV⎜ ⎟ ⎝ ⎠ OLP [→ − [ ƽ ⎞ ⎛ ⎛ [π ⎞ ⎟ ⎜ FRV⎜ ⎟ ⎜ ⎝ ⎠ + [ ⎟ ⋅ OLP⎜ ⎟ [→ ⎜⎜ − [ + [ ⎟⎟ ⎠ ⎝ I [ = VHQ(+ [) = VHQ(+ ) = VHQ I ( ) I′ [ = FRV(+ [) ((+ [)) 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ( /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ) ⎛ ⎛ [π ⎞ ⎞ ⎜⎜ FRV⎜ ⎟ ⎟⎟ + [ ⎝ ⎠ ⎠ ⎝ ƽ OLP [→ − [ ⎤ ⎡ ⎛ [πȱ ⎞ ⎟ − ⎥ ⎢ FRV⎜ ⎝ ⎠ = OLP⎢ ⋅ − + [ ⎥ ⎥ [− [→ ⎢ ⎥ ⎢ ⎦ ⎣ ( I [ = FRV ) Ǳ π = I I ′ [ = = Iȱ′() ⋅ − =− π[ −π ⎛ π [ ⎞ VHQ⎜ ⎟ ⎝ ⎠ ⋅ − =π ⎡ HVHQ[ − ⎤ ŚǼ OLPȱ⎢ ⎥ [→π ⎣ [− π ⎦ àǱ ⎡ HVHQ[ − ⎤ OLPȱ⎢ ⎥ [→π ⎣ [− π ⎦ I [ = HVHQ[ I π = HVHQπ = OLP [→π I [ − I π [− π = H = I′ [ = HVHQ[ • FRV[ = Iȱ ′(π ) = HVHQπ FRV π = − Ǽȱȱȱȱȱ ŗǯȱ ȱȱàȱ ȱȱȱȱȱȱ \ =ȱH− [ ȱ ȱ ȱȱȱà [ȱ−\= ǯȱ Řǯȱ ȱȱàȱȱȱȱȱȱȱ \=ȱH[FRV [ ȱȱ ⎛πȱ ⎞ ⎜ ⎟ ǯȱ ⎝ ⎠ π Ǳ \ = − [+ + π 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ řǯȱ ȱ J ( [)=ȱ − [ ǰ K([) =ȱ I(J([)) ¢ Iȱ ȱȱȱřǯȱ ȱ K′ȱ()= ȱ Iȱ ȱȱ¢ȱ¤ȱȱȱȱȱȱ Śǯȱ ( ( )) Ǳ Iȱ()=ǰ Iȱ()=ǰ Iȱ′()= ǰ Iȱ′()= ǰ Iȱ′()= ¢ J ([) = I [ȱ + I [ + I([) J ′ȱ ȱ Iȱ ȱȱȱŖǰ OLP śǯȱ [ȱ→ I( [) J( [) = ¢ OLP = [ȱ→ [ [ Ǳȱ Ǽ Iȱ() Ǽ I ′ȱ() Ǽ OLP [→ȱ Ŝǯȱ I( [) J( [) ȱ Iȱ ȱȱàȱȱȱȱ I ( J( [))=ȱ [ ¢ I ′ȱ([) = + ( I( [))ǯȱ ȱ J ′ȱ([)= + [ ŝǯȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ K [ =ȱ I J [ ǰȱ ȱȱȱȱ£ȱȱȱȱȱȱȱ ¡ȱ Ŗȱ ŗȱ Řȱ řȱ Şǯȱ I′ D ǻ¡Ǽȱ ′ǻ¡Ǽȱ ǻ¡Ǽȱ ŗȱ řȱ Řȱ śȱ Řȱ Śȱ Řȱ Ŗȱ ŗȱ ′ȱǻ¡Ǽȱ ǻ¡Ǽȱ Ȭśȱ Ŗȱ ′ǻ¡Ǽȱ śȱ ŗȱ Ŝȱ Řȱ ŗŘȱ ȱȱȱȱȱ ¡ȱƽȱǰȱȱȱȱǁȱŖȱ OLP [→D I [ − I D ȱ [ − D ȱ·ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD şǯȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ I ′ȱ [ ¡ǰȱȱ OLP [→ [ [I [ −ȱ[I [ [− [ = I [ − [I′ [ ŗŖǯȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ I [ − I D OLP [→D J [ − J D ȱ I′ D =ȱ ¢ J′ D = ǯȱ ŗŗǯȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ øȱǯȱ¤ȱȱǱȱ Ǽ J [ = [I [ + ȱ Ǽ J D+ E =ȱ J D J E ȱ ȱ¢ȱǯȱ Ǽ OLP I [ ȱ= [→ ȱ J′ [ = J [ ǯȱ ŗŘǯȱ ȱ ȱ ȱàȱ¢ȱȱȱȱȱȱȱȱøȱȱ ȱǱ I D + E = I D ⋅ȱ I E ȱȱ ȱ ¢ȱ ȱ ȱǯȱ ¤ I =ȱ \ I ¡ǯȱ ȱ I [ ȱ ¡ȱȱȱ ¡ȱ ¢ȱ I [ = I ⋅ I [ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àȱÇȱ Ǳȱ ȱȱȱȱȱÇǯȱ ǰȱȱȱàȱÇǰȱȱȱȱȱ ȱ·ȱǯȱ ǰȱ ȱ ȱ àȱ Çǰȱ ȱ ȱ ȱ ȱ à I([)=ȱOQ [ǯȱ ǰȱ ȱ ȱ ȱ ȱ J( [)=ȱORJ D [ǯȱ I ( [)=ȱD[ ¢ ȱȱàȱÇǰȱȱȱȱȱȱ ·ȱ ǰȱ ȱ ȱ ȱ ȱ I ( [) =OQȱ[ J( [) = ORJD [ \ K( [)= D[ ȱ ȱ àȱ ȱ ǯȱ £ȱ ȱ àȱ Çȱ ȱ ȱ ȱ ȱȱȱ ȱȱȱȱȱȱȱ¢ȱàȱȱȱȱȱ ¡ȱ ȱ ȱ ¡ǯȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡Çȱȱ·ȱȱȱǯȱȱǱ \ =[ȱ + [− \ = [ + \ = VHQ[+ H[ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ¡ȱ ǯȱȱǱ • [ +\ = • \ȱ + \+ [= ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ¢ȱ ȱ ·ȱ ȱ ¡ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱÇȱȱȱàǯȱ ȱǰȱȱà \ȱ −[ = ȱÇȱȱȱǻȱǼǰȱ ȱǱȱȱ ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ŗǯ Řǯ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 I [+∞[ → [+∞[ Iȱ( [)= [ I [+∞[ → ]− ∞] I ȱ( [)= − [ ȱàȱȱȱ¡Çȱȱȱȱȱȱ¢ȱȱ·ȱȱ ¡ǰȱ ȱ Çȱ ȱ ¡à \ =ȱ I( [)ǲȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ¡Çȱȱǯȱ ȱǰȱȱȱȱȱȱ ¡Çȱȱàȱȱ¤ȱȱȱȱÇǰȱȱȱȱȱà [ +ȱ \ = [\ ȱȱȱȱ¢ȱȱ·ȱȱ¡ǯȱ ȱ ȱ ȱ ȃȱ àȱ Çȱ ȱ ȱ ȱ ¡à [ȱ +\ = Ȅȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ \ =ȱ I( [) ȱ ȱ ȱ ¡àȱ ȱ ǲȱȱǱȱ ŗǼ I [−]→ ,5 I( [)=ȱ − [ řǼ I [−]→ ,5 ⎧⎪ − [ VL − ≤ [≤ Iȱ( [) =⎨ ⎪⎩− − [ VL < [≤ ŘǼ I [−]→ ,5 I ( [)=ȱ− − [ ŚǼ I [−]→ ,5 ⎧− − [ VL − ≤ [≤ ⎪⎩ − [ VL < [≤ ⎪ Iȱ( [) =⎨ ȱȱȱȱȱȱàǰȱȱȱȱȱȱàȱ ¡ \ =ȱ I( [)ǯȱȱȱ£ȱȱ¤ǰȱȱȱȱȱȱ £ȱ ȱ ȱ àȱ ȱ Çǰȱ ȱ ȱ ȱ ǯȱ ȱ ȱȱȱȱȱàȱȱȱȱȱÇȱȱ ȱȱȱ¢Çȱȱȱȱȱ ǯȱ ȱȱȱ ȱǰȱȱȱ ȱȱ ȱ ȱȱȱ ȱ ȱ ȱ àȱ ǰȱ ȱ ȱ ¡ȱ ȱ ȱ ǯȱ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ·ȱȱàȱȱȱȱ G\ G[ =ȱ I ′( [) ȱ ȱàȱÇǯȱȱȱȱȱȱàȱÇǯȱȱ ȱŗȱ ȱàȱÇȱ G\ȱ ȱȱȱ [ȱ +\ = G[ àǱȱ ȱȱ ȱȱȱȱȱȱȱȱ ([ȱ +\ )′ = ()′ ([ )′ + (\ )′ = ȱ ȱ ¢ȱ ȱ àȱ ȱ ¡ǰȱ ȱ ȱ \ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ Ǳ [ȱ+\• \′ = ȱȱ ȱȱ ȱ\\′= −[⇒ \′ = − G\ G[ = \′ [ [ ⇒ \′ = − \ \ ȱȱÇȱȱȱȱȱȱȱȱ \ =ȱ I( [)= − [ \ =ȱ I( [)= − − [ \′ȱ= − [ ȱǯȱ \ \ =ȱ − [ ¢ȱȱȱǯ \ ′= ȱ • −[= − [ \ ′= −ȱ − [ − [ =− − [ • −[= [ = − [ [ ¢ȱȱ \ [ − \ = \ =ȱ− − [ ¢ȱȱȱ − [ \ ȱȱǰȱȱȱȱȱȱȱȱàȱÇǯȱȱ ȱŘȱ Ǳ [ \+[− \ = ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àǱȱ ŗǼȱȱȱȱȱǰȱȱ¢ȱȱàȱȱ¡ǰȱȱ [ȱ\ ȱȱ ȱȱȱȱǯ [\ȱ+[ G\ G\ + − = G[ G[ ŘǼȱ [ G\ G[ ( ) G\ G\ G\ G\ȱ −[\− = − = −[\− ⇒ [ − = −[\− ⇒ G[ G[ G[ G[ [ − ȱřȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à [ +ȱ \ = [\ ȱ ȱ ȱǻřǰřǼȱȱ àǱȱ ŗǼȱ G\ȱ Ǳ G[ [ +ȱ\ G\ G\ = \+ [ G[ G[ ŘǼȱȱȱȱȱȱȱȱȱȱ \ =ȱ I( [) ¤ȱȱȱ ȱȱȱȱȱȱȱǯȱ ȱøȱȱȱǻřǰřǼȱȱȱàȱȱȱŗǼ + G\ȱ G\ G\ G\ = + ⇒ = −⇒ = − G[ G[ G[ G[ ȱȱȱȱȱȱ Pȱ =− řǼȱȱȱàȱȱȱȱ \ =− [+ E⇒ ȱ= −+ E⇒ = E Ǳȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à [ +ȱ \ = [\ ȱ ȱ ȱǻřǰřǼȱ \ =− [+ ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱȱ·ȱȱ Ǽȱ ȱȱȱ¤ȱǰȱȱȱȱȱȱȱȱàȱȱ Iȱ ⎡−Π Π ⎤ → [− ] ǲ I ([)=ȱVHQ[ ȱ ȱ ȱ Iȱ − [− ] → ⎡ − Π Π ⎤ ǲ ⎢ ⎣ I − ⎥⎦ ⎢⎣ ([)=ȱVHQ [ǯȱȱȱȱ ⎥⎦ − \ = VHQ− [ ⇔ [ =ȱ VHQ\ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ǰȱ ȱ ȱ ȱ VHQ \ =ȱ [ Çȱȱȱȱ¡Ǳ FRV \⋅ȱ G\ G\ = ⇒ = G[ G[ FRV\ ȱ FRV \ȱ+VHQ \ = FRV \ =ȱ± − VHQ\ ¢ȱ π [ =ȱ VHQ\ FRV\ =ȱ± − [ ǯȱȱȱ − ≤ \ȱ≤ π ȱȱ FRV \ȱ ≥ǰȱ ȱǰ FRV \ =ȱ − [ ¢ȱȱȱǱȱ \ =ȱ VHQ− [ G\ =ȱ G[ − [ ȱȱ¤ȱȱȱ \ =ȱFRV − [ Ǽȱ − G\ = G[ − [ ȱȱȱà I ⎤⎥ −πȱ π ⎡⎢ → ,5 ȱ I( [)=ȱ WDQ [ǯȱ ⎦ ȱ ȱ ȱ ȱ ¤ȱ ȱ ⎣ ⎤ π π ⎡ ǲ ⎢ ⎦ ⎣ Iȱ− ,5 → ⎥ − I −( [)=ȱWDQ − [ǯȱ ȱ ȱ Çȱ \ = WDQ − [ ⇔ [=ȱWDQ \ ȱȱȱ WDQȱ\ = [ Çȱȱȱȱ¡Ǳ VHF \ G\ = G[ ⇒ G\ =ȱ G[ VHF \ ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ +ȱWDQ \ = VHF \ ¢ȱ WDQȱ\ = [ G\ =ȱ ǯȱ ȱ G[ + [ ȱǱȱ \ =ȱ WDQ − [ǰȱ G\ =ȱ G[ + [ ȱȱ¤ȱȱǱȱ \ =ȱFRW − [ǰȱ Ǽȱ G\ =ȱ− G[ + [ ⎡ π ⎡ ⎡ π ⎡ Iȱ ⎢ ⎢ ∪ ⎢π ⎢ → ]− ∞−] ∪ [+∞[ ȱȱ ȱàȱȱ ⎣ ⎣ ⎣ ⎣ I ȱ[ VHF[ǯȱȱ ȱȱȱ¤ȱȱ Iȱ − ]− ∞−] ∪[+∞[ → ⎡Π ⎡ ∪ ⎡⎢Π Π ⎡⎢ ǲ I −ȱ [ VHF− [ ⎢ ⎢ ⎣ ⎣ ⎣ ⎣ ȱȱȱȱȱ \ = VHF − [ ⇔ [ =ȱVHF \ ȱȱȱ [ =ȱVHF \ Çȱȱȱȱ¡Ǳ VHF \ WDQ \ G\ G\ =ȱ = ⇒ G[ VHF\WDQ \ G[ ⎡ π ⎡ ⎡ π ⎡ \ȱ∈⎢ ⎢ ∪⎢π ⎢ ⎣ ⎣ ⎣ ⎣ ȱȱ WDQ( \ȱ)> ¢ WDQ \+ȱ= VHF \ǰȱȱȱ WDQ\ȱ= VHF \− ǯȱ ȱǱ G\ȱ = ǯȱ G[ [ [ − ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ǯȱ \ =ȱVHF − [ \ =ȱFVF − [ G\ȱ G[ G\ G[ = = [ [ − − [ [ − ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ¡ȱ¢ȱÇȱ ȱȱȱȱȱȱǱ • I [ = OQ[ K [ =D[ D> • J [ =ORJD ȱ [D> • Ǽȱ Iȱ ,5→]+∞[ ǲ I ȱ[ =H[ ȱȱȱȱȱàȱȱ I −ȱ ]+∞[→ ,5 I − [ = OQ[ǰȱȱȱǱ \ = OQ [ ⇔ [ =ȱH\ ȱȱȱ H\ =ȱ [ Çȱȱȱȱ¡ G\ G\ G\ G\ H\ ⋅ȱ = ⇒ = \ ⇒ = OQ [ ⇒ = G[ G[ H G[ H G[ [ \ =ȱOQ [ G\ = G[ [ Ǽȱ I [ =D[ȱ = H[OQ D D> I′ [ = (H[ D)′ =ȱH 6L OQ [OQ D • OQD = D[ • OQD I [ =D[ HQWRQFHV I C [ =ȱD[ • OQD Ǽȱ I [ =ȱORJD [ǯȱ ȱȱȱȱȱȱȱȱǱ I [ =ȱORJD [ = ′ OQ[ ⎛ OQ[⎞ • = ⎟ =ȱ ǰȱ I′ [ = (ORJD [)′ = ⎜ OQD [ [OQD ⎝ OQD⎠ OQD I [ =ȱORJD [ I C [ = [OQD ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àȱÇ ȱ ȱ·ȱȱàȱÇȱȱȱȱȱ ȱȱǰȱǰȱȱȱȱȱ¤ȱǯȱȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¡àȱȱȱ¤ȱǯȱȱ Ǳȱ \ȱ = ŗǯȱ [ [ + VHQ[+ àȱ ȱȱȱȱȱȱȱȱ OQ \ȱ =OQ [ [ + VHQ[+ ȱȱȱȱȱȱȱȱ¡à OQ \ȱ= OQ[+ OQ [ + − OQ VHQ[+ ȱȱÇȱȱȱȱ¡ G\ [ FRV[ = + − \ G[ [ [ + VHQ[+ G\ [ FRV[ = \ + − G[ [ [ + VHQ[+ G\ȱ [ [ + [ FRV[ − + ƽ G[ VHQ[+ [ȱ [ + VHQ[+ \ȱ = Řǯȱ [ [ + [− ([− ) àǱ \ = [ [ + [− ([− ) ⎤ ⎡ ⎢ [ [ + [− ⎥ ⇒ OQ \ =OQ⎢ ⎥ ⇒ OQ \ = OQ[+ OQ [ + [− − OQ([− ) ⎢ ⎥ ([− ) ⎣⎢ ⎦⎥ ( ) ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱÇȱȱȱȱ¡ G\ ⎡ [ + ⎤ G\ [ + • = + − ⇒ = \⎢ + − ⎥⇒ \ȱ G[ [ [ + [− [− G[ ⎢⎣ [ [ + [− [− ⎥⎦ ( ) ( ) G\ [ [ + [− ⎡ [ + ⎤ = + − ⎢ ⎥ G[ ([− ) ⎢⎣ [ [ + [− [− ⎥⎦ ( řǯȱ ) WDQ [ \=ȱ [ àȱ ȱȱȱȱȱȱȱȱ WDQ[ OQ \=ȱOQ[ ȱȱȱȱȱȱȱȱ¡à OQ \=ȱWDQ[OQ[ ȱȱÇȱȱȱȱ¡ G\ WDQ[ = VHF [OQ[+ \ G[ [ G\ WDQ[⎞ ⎛ = \⎜ VHF [OQ[+ ⎟ [ ⎠ G[ ⎝ ( G\ȱ WDQ [ ƽ [ G[ Śǯȱ )⎛⎜⎝ VHF [OQ[+ WDQ[[⎞⎟⎠ ȱ \ =ȱ [VHQ[ àǱ \= [VHQ[ ⇒ \= [VHQ[ ⇒ OQ \ = OQ[VHQ[ ⇒ OQ \= VHQ[OQ[⇒ G\ VHQ[ G\ ⎛ VHQ[⎞ • = FRV[OQ[+ ⇒ = \⎜ FRVOQ[+ ⎟⇒ \ G[ [ G[ ⎝ [ ⎠ G\ VHQ[⎛ VHQ[⎞ = [ ⎜ FRVOQ[+ ⎟ G[ [ ⎠ ⎝ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¤ȱ ȱǯȱȱȱȱȱȱ ŗǯȱ ȱȱȱȱȱȱȱȱȱà [ + \ + =ȱ [\ ȱ ȱȱȱ£ǯȱȱ à [ +ȱ \ + = [\ [ + \ ⋅ \′ = \ + [⋅ \ ⋅ \′ \\′ − [\\′ = \ − [ ( ) \′ \ − [\ = \ − [ \′ = ( ) \ − [ \ ( \− [) ȱ [ + \ + =ȱ [\ ȱ ȱ ȱ £ȱ ȱ ȱ G\ȱ ȱȱȱȱà = ǯȱ G[ ( \ −[) = ⇔ \ − [ = ⇔ȱ \ = [ \ ( \− [) G\ȱ ȱȱȱȱȱà = ȱȱȱȱǻ¡ǰ¡Ǽǯȱ G[ ȱȱȱȱȱǻ¡ǰȱ¡Ǽȱȱȱȱà [ + \ + = [\ [ȱ +[ + = [⋅ [⇔ [ − [ + = ⇔ ( ) ( ) [ − [ + = ⇔ [ − = ⇔ [ − = ⇔ [= ± Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à [ +ȱ \ + = [\ ȱ ȱȱ£ȱȱǻŗǰŗǼȱ¢ȱǻȬŗǰȬŗǼǯȱ Řǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ à [\+ =ȱ [\ ȱȱ£ǯ ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àǱ [ \+ȱ= [\ ⇔ [\+ [\′ = \ + [⋅ \\′ ⇔ ( ) [\′ − [\\′ = \ − [\⇔ \′ [ − [\ = \ − [\ ⇔ \′ = \ − [\ ⇔ \′ = $ \− [) ⇔ \′ = ⇔ [([− $ [ − [\ \( \− [)= ⇔ \ = ∨ \ = [ Ǽȱȱȱȱȱȱ ( [ȱ ) ȱ£ȱȱȱ [ \+ȱ= [\ \ȱ = ⇒ + = ⇔ = ǯȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ( [ȱ ) ȱ£ȱȱȱ [ \+ȱ= [\ ǯȱ ȱ [) ȱ£ȱȱȱ [ \+ȱ= [\ Ǽȱȱȱȱȱȱ ( [ \ = [⇒ [([) + = [([) ⇔[ȱ + = [([) ⇔ [ − [ = −⇔ −[ = −⇔ [ = ⇔ [ = Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à [ \+ȱ= [\ ȱ ȱ£ȱ (ȱ)ǯȱ řǯȱ ȱȱà [ + \ = [ −ȱ\ ǰȱ àǱ G\ ȱȱ G[ȱ ( ȱ )ǯȱȱ [ + \ ([+ \\′) = [− \\′ ⇔ ⎛⎛ ⎞ ⎛ ⎞ ⎞⎟⎛ ⎜ ⎜ ⎞⎟ ⎜ + \′ ⎟ = − ⋅ \′ ⇔ ⎜ ⎟ ⎜ + ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟⎠ ⎜ ⎝ ⎠ ⎝ ⎠ ⎠⎟⎝ ⎝ ⎞ ⎛ ⎞⎛ ⎜ + ⎟⎜ + − \′ ⎟ = \′ ⇔ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎛ ⎞⎟ + ⋅ ⎜ \′ = \′ ⇔ − ⎟ ⎜⎝ ⎠ ⎛ ⎞⎟ ⎜ − \′ = \′ ⇔ + ⎜ ⎟ ⎠ ⎝ \′ + \′ = ⇔ \′ = ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Śǯȱ ȱ ȱ ȱ ȱ ȱ ȱ à [− + \ − ȱ = ǰȱ ȱ ȱȱȱȱȱȱȱȱȱ ȱȱȱȱȱȱȱ ȱà \+ [ =ȱǯȱȱ àǱ \ +[ = ⇒ \ = − [+ ⇒ \ = − [+ ([ȱ−) + ( \− ) = ⇒ ([− ) + ( \− )\′ = − ([− ) ⇒ ( \− )\′ = −( [− ) ⇒ \′ = ⇒ ( \− ) − ([− ) \′ = ( \− ) ȱ ȱ ( [ȱ\) ȱ ȱ [− + \ − ȱ = ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à \+ [ =ȱ ( [ȱ\) ȱ ȱ G\ȱ à =− ǯ G[ G\ −( [ − ȱ) = − ⇔ = − ⇔ ([− ) = ( \− )⇔ [− = \− ⇔ \= [⇔ \= [ ( \− ) G[ ȱ ȱ ȱ Ǳ ([ȱ[) ȱ ȱ ȱ ȱ [− + \ − ȱ = [ − + \− = ⇔ ( [− ) = ⇔ ( [− ) = ⇔ [− = ±⇔ [= ± (ȱ) ȱȱȱȱȱȱȱȱǰȱȱȱȱȱȱ ȱȱȱǯȱȱ Ǳȱȱȱȱȱȱȱȱȱà [− + \ − ȱ = ȱȱ ȱȱȱȱȱ \+ [ =ȱ (−ȱ−)ǯȱ śǯȱ G\ ȱȱà VHF [+ \ = [ȱ +VHQ\ ǯȱȱ G[ȱ àǱ G\ ⎡ G\⎤ VHF([+ \) = [ + VHQ\⇒ VHF([+ \) WDQ( [+ \)⎢+ ⎥ = [+ FRV \⋅ ⇒ G[ ⎣ G[⎦ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD VHF ( [+ȱ \) WDQ( [+ \) + VHF([+ \) WDQ([+ \) /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 G\ G\ = [+ FRV \ ⇒ G[ G[ G\ G\ − FRV \ = [− VHF([+ \) WDQ([+ \) G[ G[ G\ (VHF([+ \) WDQ([+ \) − FRV\) = [− VHF([+ \) WDQ([+ \) G[ G\ [− VHF( [+ \) WDQ( [+ \) = G[ VHF([+ \) WDQ([+ \)− FRV \ VHF( [+ \) WDQ( [+ \) Ŝǯȱ ȱȱȱȱȱȱà [− + \ − ȱ =ǯȱ ȱȱȱȱȱȱȱȱȱȱȱȱȱ£ȱ¢ȱ ȱȱȱȱȱȱȱȱȱǯȱȱ àǱ [ − + \− = ⇔ ([− ) + ( \− )\′ = ⇔ ( \− ) \′ = −([− ) ⇔ \′ = − ([− ) − ([− ) ⇔ \′ = ( \− ) ( \− ) •ȱ ȱ£ \′= ⇔ [− = ⇔ [= (− ) + ( \− ) = ⇔ ( \− ) = ⇔ ( \− ) = ⇔ \− = ± ⇔ \ = ± ⇔ \ = \= •ȱ ȱ \′ȱ ȱ ⇔ \ − = ⇔ \ = ([ȱ−) + (− ) = ⇔ ([− ) = ⇔ [− = ±⇔ [= ± ⇔ [= [= Ǳȱ ȱ ȱ ȱ à [− + \ − ȱ = ȱ ȱ ȱ ⎛ ⎞ ⎛ ⎞ £ȱ ȱ ȱ ⎜ȱ ⎟ ¢ ⎜ȱ ⎟ ǰȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ⎝ ⎠ ⎝ ⎠ ()() 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ŝǯȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ ȱ à [− − \ − =ȱǯȱ ȱ ȱ ȱȱ ȱȱȱȱȱ ȱ ¢ȱȱȱ ȱȱ ȱ ȱȱ£ǯȱȱ àǱȱ [ȱ− − \− = ( [− ) − ( \− )\′ = ȱȱ¤ ([− ) = \′ ( \− ) G\ȱ ȱȱÇǱ G[ ([− ) ( \− ) G\ȱ ȱȱȱȱȱȱǻ¡ǰȱ¢Ǽȱȱ£ȱ = ¢ȱȱȱȱ G[ G\ȱ ȱȱȱǻ¡ǰȱ¢Ǽȱȱ ȱǯ G[ \′ = •ȱ ȱȱ£ \ȱ′= ( [− ) = ⇔ ( [− )= ⇔ [= ( \− ) ȱȱàȱȱȱȱ [− − \ − =ȱǰȱȱȱ¡ȱƽȱŗȱȱ ȱ − − \ − = ⇔ − \ − = ⇔ \ȱ− = − ȱȱ ȱȱȱ øȱȱȱ¢ǯ •ȱ ȱȱ \′= ( [− ) VHLQGHILQHVL ( \−)= ( \− ) ⇔ \ȱ = ǯȱ ȱȱàȱȱȱȱ [− − \ − =ȱǰȱȱ ¢ȱƽȱŘ [ − − − = ⇔ [ − = ⇔ [ − = ⇔ [ − = ± ⇔ [ = ± ⇔ [ȱ = à [ȱ= Ǳȱ ȱ ȱ ȱ à [− − \ − =ȱ ȱ ȱ ȱ ⎛ȱ ⎞ ⎛ȱ ⎞ ȱ ȱ ⎜ ⎟ ¢ ⎜ ⎟ ǯȱȱȱ¡ȱȱȱȱȱȱȱȱȱ ⎝ ⎠ ⎝ ⎠ ȱȱȱȱ£ǯ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Şǯȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱàȱȱȱȱȱȱȱ [− \ = àǱ [ − \ = [+ \ ⇒ − \′ = − \ȱ′ = [+ȱ \ ȱȱȱǻřǰŗǼǯȱȱ ( (+ \ )⇒ + \ ) ⇒ − \′ = [+ \ + (+ \′)⇒ − \′ = + \′ ⇒ − \′ = − ⇒ \′ = ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱȱȱȱȱȱȱȱ Pȱ = − ǯ \ =− ȱ [+ E⇔ = − ⋅ + E⇔ = E Ǳȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ [ −ȱ \ = (ȱ) \ȱ =− [+ ǯȱ şǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à [ + \ =ȱ ȱȱȱȱȱȱȱȱŗǯȱȱ [+ \ ȱ ȱ ȱ ȱ ȱ ȱ àǱ [+ȱ\\′ = ⇒ \\′ = −[⇒ \′ = − \′ = ⇔ − [ [ ⇒ \′ = − \ \ [ = ⇔ − [ = \ ⇔ [= −\ \ ȱ ȱ ȱ ȱ ȱ ȱ à [ + \ =ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱŗȱȱȱȱ (−ȱ \\)ǯ (− \ȱ) + \ = ⇔ \ + \ = ⇔ \ = ⇔ \ = ⇔ \ = ± − \= ± =± Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à [ȱ +\ = ȱ ȱ ȱ ȱ ⎛ ȱ ⎞ ⎛ ȱ ⎞ ȱȱȱȱȱȱȱȱŗȱ ⎜⎜ − ⎟⎟ ¢ ⎜⎜ − ⎟⎟ ǯ ⎠ ⎝ ⎠ ⎝ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ à [ + \ = [\−ȱ ȱȱȱȱȱȱȱȱ£ǯȱȱ ŗŖǯȱ àǱ [ȱ + \ = [\− ⇒ [ + \ ⋅ \′ = \ + [\′ ⇒ \\′ − [\′ = \− [ ⇒ ( ) \′ \ − [ = \− [ ⇒ \′ = \ − [ \′ = ⇔ \ − [ \− [ \ − [ = ⇔ \= [ ( ) ( ) [ + [ = [⋅ [ − ⇒ [ + [ = [ − ⇒ [ − [ + = ⇒ [ − = ⇒ [ − = ⇒ [ = ⇒ [= [ȱ = ¢ \ = \′ȱ ȱǯȱ ȱȱǰȱȱ¡ȱ ȱȱȱȱȱ ȱà [ + \ = [\−ȱ ȱ ȱȱȱȱȱ£ǯȱ ŗŗǯȱ G\ G[ ȱȱ [⋅ \ = ȱ ⋅ =ȱ ǯȱȱ G[ G\ àǱ G\ •ȱ ¤ȱ G[ G\ \ +[ = G[ ⇒ •ȱ ¤ȱ G[ \ +[= G\ ⇒ •ȱ ¤ȱ G\ \ =− G[ [ ⇒ G \ = G[ − [ − \ G\ + \ − [⋅ + \ \ G[ [ = = [ [ [ G[ G\ G[ [ =− G\ \ ⇒ G [ = G\ − \ [ G[ + [ − \⋅ − + [ [ \ G\ = = \ \ \ G \ G[ ⋅ G[ G\ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Gȱ \ G[ \ [ ([\) ⋅ = ⋅ = = = = G[ G\ [ \ ([\) [\ () ȱ ȱàȱ ȱȱ ȱȱȱȱȱȱȱ ȱà [ \ = [+ȱ ȱȱȱǻŘǰŗǼǯȱȱ ŗŘǯȱ àǱ [ \ = [+ ⇒ [\+[ \′ = ⇒ ()() + ()\′ = ⇒ \′ = − ⇒ \′ = − PȱWDQ =− ⇒ \= − [+ E⇒ = − ⋅ + E⇒ = − + E⇒ + = E⇒ E= Ǳȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à [\ = [+ȱ ȱȱȱǻŘǰŗǼȱ ŗřǯȱ \ȱ =− [+ ȱȱàȱȱȱȱȱȱȱ [ [ + \ =ȱ \ ȱȱ ( )ǯȱȱ ȱ àǱ [ [ +\ = \ ⇒ ( ) [ȱ [ + \ + [([+ \\′)= \\′ ⇒ ⋅ + ⎛ ⎞ ⎛⎜ ⎞⎟ \′ = ⋅ \′ + ⋅ ⎜ + ⎟ + ⎜ ⋅ ⎟ ⎝ ⎠ ⎝ ⎠ \′ = \′ ⇒ \′ ⇒ = \′ ⇒ PWDQ = + = \ = [+ E⇒ = + E⇒ E= − Ǳȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ [ [ + \ =ȱ \ ȱ ȱ ( ) \ȱ=[− ȱ ŗŚǯȱ ȱȱȱȱȱȱ [\ + [\ =ȱ ȱȱȱȱ ȱȱȱȱȱȮŗǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àǱ [\ +[\= ⇒ [\ + [ ⋅ \\′ + \+ [\′ = ⇒ ( ) [\\′ + [\′ = −[\ − \⇒ \′ [\+ [ = −[\ − \⇒ − \(ȱ[\+ ) \ \ \ ′= ⇒ \′ = − ⇒ \′ = −⇒ − = −⇒ \ = [ [([\+ ) [ [ ȱ \ =ȱ [ ȱȱà [\ + [\ =ȱ ȱǱ ( )( ) [ȱ [ +[[= ⇒ [ + [ − = ⇒ [ + [ − = ⇒ [ = ⇒ [ = ± ǱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȬŗȱ (ȱ) ¢ (−ȱ−)ǯȱ G\ ȱȱà \ + \ =ȱ [ ȱȱȱǻŘǰŗǼǯȱȱ ȱ G[ ȱȱȱ ŗśǯȱ àǱ ( ) \ +\ = [⇒ \\′ + \′ = ⇒ \′ \ + = ⇒ \′ = ∴ \ + G\ = G[ \ + G\ \ + )() − \⋅ ( G \ G[ ⇒ = G[ (\ + ) ( ⎞ ⎛ ) ⎟ G\ − \⎜ ( ) \ \ + − ⋅ ⎟ ⎜ G \ G\ ⎝ \ + ⎠ G[ ⇒ = ⇒ = G[ G[ \ + \ + ( ⇒ ⇒ ŗŜǯȱ Gȱ \ G[ = ( ⎛ ⎞ ⎟ − \⎜ ⎜ \ + ⎟ ⎠ ⎝ G\ ) G[ () (\ + ) = − (+ ) = ⇒ G\ G[ = ) − \ (\ + ) − − − = = [ + \ =ȱ ȱ G\ −ȱ[ G[ = \ ǯ ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àǱ [ + \ G\ G\ G\ [ [ = ⇒ =− ⇒ =− G[ G[ G[ \ \ [ G\ [\ [ \ − + • − \ • −[+ [ • \ G \ \ G[ ⇒ G \ = ⇒ = G[ \ G[ \ − [\ − [\ ⇒ ⇒ȱ G\ G[ \ = \ G \ − [\() G[ = ⇒ \ ⇒ ( ) G\ − [\ \ + [ G[ = \ G\ − [ G[ = \ ŗŝǯȱ ȱȱȱà [ȱ −[\+ \ = ȱȱȱȱȱ¢ȱȱȱǯȱ ȱȱȱȱȱȱȱȱȱȱȱǯȱȱ àǱȱ ȱȱȱàȱȱȱȱȱȱȱ¢ Ǳ (ȱ) ¢ (−ȱ) ȱ \ =⇔ \ = ± •ȱ G\ G[ G\⎞ G\ G\ G\ ⎛ ȱ[ −⎜ \+ [ ⎟ + \ = ⇒ [− \− [ + \ = ⇒ G[⎠ G[ G[ G[ ⎝ G\ (\− [)= −[+ \⇒ G\ = − [+ \ G[ G[ \− [ ȱȱȱȱȱȱȱȱ [ȱ −[\+ \ = ȱȱ (ȱ) Pȱ = = ǯȱ ȱȱȱȱȱȱȱȱ [ȱ −[\+ \ = ȱȱ (−) − Pȱ = = − P =ȱP ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à [ȱ −[\+ \ = ȱȱ (ȱ) ¢ (−ȱ) ȱǯ ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ŗŞǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à \ȱ +[ + [\+ \ = ǯȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱ (−ȱ)ǯȱȱ àǱ \ +[ + [\+ \ = ⇒ \\′ + [+ \+ [\′ + \ = \′(\+ [+ )= −[− \⇒ \′ = PWDQȱ = \ = − − [− \ \+ [+ −− () − + − = = = − + + − + − − [+ E⇒ −= − + E⇒ −+ = E⇒ = E Ǳȱȱàȱȱȱȱȱȱȱȱȱȱȱ (−ȱ) \ȱ+[+ = ȱǯȱȱȱǻȱȱȱǼǯȱ ŗǯ Řǯ ( ) I [ =DUFWDQȱ[ I [ =ȱ ( Ȧ I ′ȱ( [)= [ FRV([) + [ (WDQ [) ) Ȧ I ′([) = [ OQ + [ [ ] WDQ−ȱ([) [FRV([) − [VHQ([) + [ − (WDQ [) − ( ) řǯ \ = + [ WDQȱ− [ + Śǯ \ =( DUFVHQ[)OQȱ[ ( ) Ȧ \ȱ′ = [WDQ− [ + + Ȧ \′= OQ [ − [ + [ FRV([) + [ + [ ( ) + ([ + ) [+ [ DUFVHQ[ [ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ (DUFWDQ[) śǯ \ =( DUFWDQ[)ȱȱ Ŝǯ ⎛ [ [− ⎞ ⎟ I [ = OQ⎜⎜ ⎟ + ȱ FRV [ [ ( ) ⎝ ⎠ ŝǯ J [ = DUFVHQ [+ + ȱ−[ Ȧ J ′ȱ( [)= Şǯ I [ = [WDQ−ȱ [ Ȧ I ′([) = WDQ−ȱ [ + şǯ J [ = VHQ−([)+ OQ ŗŖǯ \ = WDQȱ⎛⎜⎝ H [ ⎞⎟ ⎠ ( Ȧ \′ȱ= ) [− [+ ȱ + [ Ȧ I ′ȱ([)= Ȧ Jȱ′( [) = Ȧ \′( [) = [ ⎛⎜ [ − ⎞⎟ ⎝ ⎠ − [+ − [ − [ + VHQ[ + [+ FRV[ + −− [ OQ [ (+ [) − ([− ) ([+ ) H [ ȱ [⎛⎜ + H [ ⎞⎟ ⎝ ⎠ ȱǯȱȱȱȱȱ ŗǯȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ \ =ȱH− [ ȱ ȱ ȱ [ȱ−\ = ǯȱȱ àǱ \ =ȱH− [ ⇒ \′ = −H− [ ¤ [ȱ−\= ⇒ − \= −[+ ⇒ \= [− ¢ȱȱȱȱȱȱȱȱ ȱ ȱȱà \ȱ = [− ȱȱȱ − H −[ = − ⇔ H− [ = PȱWDQ = − 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 − ⎛ ⎞ − [ȱ= OQ⎜ ⎟ ⇔ − [= − OQ⇒ [= OQ ⇒ \= H−OQ = HOQ = ⎝ ⎠ ⎞ ⎛ ȱȱȱȱȱ ⎜ OQ ⎟ ⎠ ⎝ \ =− ȱ [+ E⇒ = − OQ+ E⇒ + OQ= E Ǳȱȱàȱȱȱȱȱȱȱ \ =ȱH− [ ȱȱ ȱ [ȱ−\ = \ȱ =− [+ ⎛⎜ OQ+ ⎞⎟ Řǯȱ ⎝ ⎠ ȱ ȱàȱȱ ȱȱȱȱȱ \ =ȱH[FRV [ ȱȱ ⎛πȱ ⎞ ⎜ ⎟ ǯȱȱ ⎝ ⎠ àǱ \ =H[ FRV [ ⇒ \′ = H[FRV[(FRV[+ − [VHQ[)⇒ πȱ π FRV ⎛ π π π⎞ π⎞ π ⎛π ⎞ ⎛ PWDQ = \′⎜ ⎟ = H ⎜ FRV − VHQ ⎟ = H⎜ − ⎟ = − ⎠ ⎠ ⎝ ⎠ ⎝ ⎝ \ = − π [+ E⇒ = − π π • + E⇒ + π = E⇒ + π = E ⎛πȱ ⎞ Ǳȱȱàȱȱȱȱȱȱȱ \ =ȱH[FRV [ ȱȱ ⎜ ⎟ ⎝ ⎠ \ȱ =− π [+ + π ȱǯȱ ŗǯ I [ = ȱàȱÇȱȱ I ′ [ ȱ ȱȱǱȱ [ ([+ ) FRV [ [ + 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱ ⎛ ⎞ ⎜ ⎟ [ ([+ ) FRV[ [ ([+ ) FRV[⎟ ⎜ \ȱ = \ ⇒ OQ = OQ ⇒ ⎜ ⎟ [ + [ + ⎜ ⎟ ⎝ ⎠ OQ \ = OQ[+ OQ([+ ) + OQ(FRV[) − OQ [ + ⇒ [ − VHQ[ \′ = + + − ⇒ \ [ [+ FRV[ [ + ( \′ = Řǯ I [ = [ ([+ ) FRV[ ⎡ [ ⎤ − WDQ[− ⎥ ⎢ + [ + [ + ⎦ ⎣ [ [+ ( řǯI [ = ) )OQ [ Ǳ \′= [ + ȱ [ [ +ȱ [[ FRV [ àǱ ⎤ ⎡ ⎢ [ [ + ⎥ ⇒ OQ \ = OQ⎢ \= ⎥ [ [[ FRV[ ⎢ [ FRV[ ⎥ ⎣ ⎦ ⇒ OQ \ = OQ[+ OQ [ + − [OQ[− OQ(FRV[) ( [ȱ [ + ( ⇒ \ \′ = Śǯ ⋅ \′ = \ = ( + ( ) ) [ ) − OQ[− [ [ + ) [ [ + [ VHQ[ + [ FRV[ ⎡ ⎤ [ + − OQ[− + WDQ[⎥ ⎢ [[ FRV[ ⎢⎣ [ [ + ⎥⎦ ( H[ [ + ( [+ ) ( [ +) ) ( )OQ[ȱ⎡⎢ OQ( [ + ⎣⎢ ) ⎤ [ + OQ[ + ⎥ [ [ [ + ⎦⎥ ( ) 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱ ⎡ ⎤ H[ [ + ⎥ ⎢ ⇒ OQ \= OQ \= ⇒ ⎥ ⎢ ([+ ) [ + ( ) + + [ [ ⎣ ⎦ OQ \ = OQH[ + OQ [ + − OQ( [+ ) − OQ [ + ⇒ H[ȱ [ + ( \ ⋅ \′ = + \′ = śǯ ( ) ( [ ) ) ( [ + − ) ( ) [ − ⇒ [+ [ + ⎡ [ [ ⎤ + − − ⎢ ⎥ ([+ ) [ + ⎢⎣ [ + [+ [ + ⎥⎦ H[ [ + ( \ = (VHQ[) FRV( [ ( ) ) −ȱ) àǱ ) \ = (VHQ[)FRV([ȱ − ⇒ OQ \ = OQ⎛⎜ (VHQ[)FRV([ −) ⎞⎟ ⇒ OQ \ = FRV [ − OQ(VHQ[) ⇒ ⎝ \ ( ( ( )) ⎠ ⋅ \′ = − VHQ [ − ⋅ [OQ(VHQ[) + [ ) (FRV([ − ))FRV[ ⇒ VHQ[ )))] \′ = (VHQ[)FRV([ −) − [(OQ(VHQ[)) VHQ [ − + FRW[FRV [ − Ŝǯ ( ( ) ( ( [ [− \ȱ= ( [− ) àǱ \ȱ= \ ŝǯ [ [− ([− ) ⋅ \′ = \ = ⎛ [ [− ⎞ ⎟ = OQ[+ OQ([− ) − OQ( [− ) ⇒ ⇒ OQ \= OQ⎜ ⎜ ([− ) ⎟ ⎝ ⎠ [ [− ⎡ ⎤ ′ + − ⇒ \ = − ⎢ + ⎥ [ ([− ) [− ([− ) ⎣ [ ([− ) [− ⎦ [+ [+ [VHQ[ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱ \ = [+ [+ [VHQ[ ⇒ OQ \ = OQ([+ ) + \′ = Şǯ ([+ )ȱ I [ = OQ([+ ) − VHQ[OQ [⇒ [+ ⎡ VHQ[⎤ ⎢ [+ + [+ − FRV[OQ [− [ ⎥ ⎣ ⎦ VHQ[ [ [[ [ + ([− ȱ)VHQ −[ àǱ ( ) ( ) OQ [ + + − OQ([− ) − OQ VHQ−[ ⎤ [[ȱ [ + ⎡⎢ [ ⎥ \′ = OQ [+ + − − [ + [− VHQ−[ − [ ⎥⎦ ([− )VHQ−[ ⎢⎣ OQ \= [OQ[+ ( şǯ I [ = ( [− ) [+ȱ [FRV [ àǱ OQ([+ ) − OQ[− OQFRV[ ([− ) [+ ⎡ + − + WDQ[⎤ \′ = ⎥ [FRV[ ⎢⎣ [− ([+ ) [ ⎦ OQȱ \ =OQ( [− ) + ŗŖǯ I [ = ( ) [VHQ [ [ + VHQ− [ ) 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱ ( ) ( ) OQ\= VHQ[OQ[+ OQ[ + − OQ VHQ−[ ⎤ ⎡ ⎥ ⎢ [ [ + VHQ[ [ \′ = [ [ FRV OQ + + − ⎥ ⎢ VHQ−[ ⎢ [ [ + VHQ−[⎛⎜ − [ ⎞⎟ ⎥ ⎝ ⎠⎦ ⎣ ( VHQ[ȱ I [ = ŗŗǯ ) ( [ [+ ([−ȱ) VHQ [ àǱ \ = [ȱ [+ ([− )VHQ[ OQ \ = OQ[+ \′ = ŗŘǯ ⇒ OQ([+ ) − OQ([− ) − OQ(VHQ[) ⇒ [ [+ ⎡ ⎤ + − − FRW [⎥ ⎢ [ [ [ + − ⎦ ([− ) VHQ[ ⎣ [ I [ =(DUFWDQȱ[) àǱ \ =(DUFWDQ[) [ OQ \ = [OQ(DUFWDQ[) ⎡ \′ = (DUFWDQ[) [⎢OQ(DUFWDQ[) + ⎢⎣ ⎤ ⎥ (DUFWDQ[) + [ ⎥⎦ [ ( ) ) 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱ ȱȱ Ǳȱ ȱȱȱȱ ȱȱȱȱȱȱȱȃȱȱȄȱ Iȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ Çȱ ȱȱȱȱȱȱàȱȱȱǯȱȱȱȱȱȱȱ ȱȱȱà Iȱ ¡ǰȱȱÇȱȱȱ Iȱ ǰȱȱȱȱȱȱ ȱȱ Iȱ ǰȱȱȱȱÇ Iȱ ǯȱȱȱȱȱȱÇȱȱȱàȱȱȱ ȱǯȱǻ Iȱ ȱȱȱȱ Iȱ Ǽȱȱȱȱȱȱ¢ȱ Ǳȱ ȱȱȱ Iȱ ǰȱȱȱȱȱÇ Iȱ ȱȱȱ Iȱ ǰȱȱȱȱȱÇ IȱY ȱ·ȱȱ Iȱ ǰȱȱȱȱȱÇ Iȱ Q ȱȱȱȱȱȱȱȱ£àȱȱȱȱ ȱ £àȱȱǯȱ V =ȱV W ȱȱàȱȱàȱȱȱȱȱȱȱȱȱÇȱǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Yȱ W ȱ ȱ ȱ ȱ àȱȱǱ Y W =Vȱ W = GV GW ȱ £àȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à D ȱ W ȱ ǯȱ ȱ Çǰȱ ȱ àȱ àȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ¢ǰȱȱǰȱȱȱȱȱ ȱ ȱàȱ ȱàǱ D W =ȱY W = V W ȱȱȱàȱȱ£ D =ȱ Ǳȱ GY G V = GW GW I ′ȱ′() I ( [) = [ȱ − VHQ(π[) àǱȱ I ( [) = [ − VHQ(π[) ȱȱ ⇒ I ′( [) = [ − πFRV(π[) ⇒ I ′ȱ′( [) = [+ πVHQ(π[) Iȱ′′() = ⋅ + π VHQ(π )= + = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ £ȱ ȱ ǯȱ Ǳȱ ȱȱ£ȱȱȱ ȱȱȱȱȱȱȱȱȱ¤ȱ¤ȱȱǱȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ǰȱ [ȱ W ǰ\ȱ W ǰȱȱȱȱǰȱ¤ǰȱøǰȱǯȱ ȱ [ȱ W ǰ\ȱ W ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àǯȱ ȱȱȱȱȱȱȱȱ¤ȱȱȱȱȱȱ¢ȱȱ ȱȱȱ ȱȱ ȱȱǲȱȱȱȱ [ȱ W ¢ȱȱ ȱ \ȱ W ȱȱ·ǯȱ ȱȱȱǰȱȱȱȱȱàȱȱȱ ȱ [ȱ W ǰ\ȱ W ¢ȱȱȱȱȱȱǯȱ ȱ àȱ ȱ ȱ [ȱ W ǰ\ȱ W ȱ ȱ àȱ ȱ ȱ Çȱȱ [ȱ W ǰ\ȱ W ȱȱȱǰȱȱȱȱȱ ȱȱàȱÇǯȱȱ ¤ȱ ŗǯȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱřŖȱȱȱȱȱȱȱȱȱǯȱ ȱȱ¤ αȱ ȱȱȱ¢ȱȱÇȱȱàȱȱǰȱȱȱ£àȱ π ȱ ȱ ȱ ǰȱ ȱ ·ȱ £àȱ ȱ ȱ ȱ α =ȱ ǵȱȱ àǱȱ WDQα = [ Gα G[ = GW GW G[ π ⎛ ⎞ π VHF⎜ ⎟ ⋅ = ⋅ ⎝ ⎠ GW VHF α π 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 πȱ G[ = ⋅ GW G[ ⎛ π ⎞ = ⎜ ⎟ SLHVVHJ GW ⎝ ⎠ ⋅ ⎛π ⎞ Ǳȱȱȱȱȱȱȱ ⎜ ⎟ SLHV ȱVHJ ȱȱȱȱȱ ⎝⎠ ȱ¤ȱȱàȱȱȱȱȱ πȱ ǯȱ Řǯȱ ȱ ¤ȱ ȱ ȱ àȱ ·ǯȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ £àȱ ȱ ŖǰŘȱ ȱȱȱȱȱȱȱ ȱ ȱƽȱśȱǯȱȱǶȱ·ȱ £àȱ ȱȱȱ ȱȱàȱȱȱǵȱȱ àǱ 9 =ȱ π U ⇒ ⇒ GY GU GY SXOJVHJ = π U ⇒ = π ⋅ ⋅ GW GW GW GY = π SXOJVHJ GW Ǳȱȱȱȱàȱ·ȱȱȱȱȱ πȱ SXOJ VHJ ȱȱȱȱȱȱȱȱȱśȱǯȱ řǯȱ ȱȱȱȱ£àȱȱŗŖȱȱȱȱȱȱȱȱȱȱ ȱ ŗŖŖȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ £àȱ ȱ ȱ ȱ ¤ȱ ȱ àȱȱȱȱȱȱȱȱȱȱŗŖŖȱȱȱǯȱȱ àǱȱ G[ = SLHVVHJ GW G[ +DOODU FXDQGR[= GW [ Į Ǳȱ WDQα = [ G[ G[ VHF α = GW GW ⎛ π ⎞ G[ = ⋅ VHF⎜ ⎟ ⎝ ⎠ GW G[ = GW G[ = UDG VHJ GW ( ) ȱ ¤ȱ ȱ àȱ ȱ ȱ ȱ £àȱ ȦȱȱȱȱȱȱȱȱȱȱȱȱȱȱŗŖŖȱǯ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Śǯȱ ȱȱȱàȱȱÇȱȱ£àȱȱśȱȱřȱȦǯȱȱȱȱȱȱŗśȱ ȱ¢ȱȱȱȱȱȱȱȱŝǰśȱǯȱǶȱ·ȱ£ȱ¤ȱȱ ȱȱȱÇȱȱȱȱȱȱȱȱȱŝǰśȱǵȱȱ àǱ FP G9 = FP VHJ GW GK +DOODU VL K= FP GW U = ⇒ U = K K U 9 = π UK⇒ 9 = ⇒ 9 = π K ⇒ K π ⎛ ⎞ ⎜ K⎟ K ⎝ ⎠ GY π GK = K GW GW &XDQGRK FPVH WLHQHTXH GK GK GK ⋅ FPVHJ =ȱ ⇒ = ⋅ ⇒ = GW π () GW π GW π Ǳȱ ȱ ȱ ȱ Çȱ ȱ ȱ ȱ £ȱ ¡ȱ ȱ ŖǰŖŗŘŜ FP ȱVHJ ȱȱȱȱÇȱȱȱȱȱȱŝǰśȱǯȱ śǯȱ Ƕȱ·ȱ£àȱȱȱ¤ȱȱȱ¤ȱ¤ȱȱȱȱȱŗŖȱ ȱ¢ȱȱȱƙȱȱȱǵȱȱ àȱ $= GO = FPVHJ GW G$ = GW G$ = GW G$ = GW Ǳ ȱȱ¤ȱȱ¡ȱȱŚǰřřŖ FPȱ VHJǯȱ ⇒ GO ⇒ GW ⋅ ⇒ FP VHJ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ŝǯȱ ȱȱȱȱàȱ£ȱȱȱȱȱȱǰȱȱȱ ȱȱŞŖȱȱȱȱ¢ȱȱȱȱȱȱȱ ȱȱǯȱǶȱ·ȱ£ȱȱȱȱȱȱȱȱ Ŗǰśȱȱ·ǵȱȱ àǱȱ [ \ řȱ G[ = PLOODVPLQ GW G\ +DOODU VL [= PLOODV GW [ +ȱ= \ ⇒ G\ G[ [ = \ ⇒ GW GW [ G[ G\ G\ = ⇒ = ⋅ PLOODVPLQ \ GW GW GW Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ Ŗǰśȱ ǯȱ ·ȱ ȱ ȱ ȱ ȱ ȱ ·ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ŝşǰŝŝŜȱ Ȧǯȱ ŝǯȱ ȱȱȱȱȱȱȱȱȱȱǰȱȱȱȱȱ ȱ ·ȱ ȱ ¡ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ £àȱ ȱ ŗŖȱ Ȧǯȱ Ƕȱ ·ȱ £ȱ ȱ ȱ ¤ȱȱȱȱȱȱŚȱǵȱȱ àǱ U GU = FPVHJ GW G$ +DOODU VL U = FP GW 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD $=ȱπU ⇒ ⇒ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 G$ GU G$ = πU ⇒ = π ⋅ FP VHJ GW GW GW G$ = π FP VHJ GW Ǳȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ƽȱ Śȱ ȱ ȱ ȱ ȱ ¡ȱȱŘśŗǰřŘŝȱȱŘȱȦǯȱ Şǯȱ ȱ ȱ ȱ Ŝȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Şȱ Ȧǰȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ ŗŞȱ ǰȱ Ƕȱ ·ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱȱȱȱȱȱŗŖŖȱȱȱǵȱȱ àǱ G[ = SLHVVHJ GW G\ +DOODU VL [= SLHV GW \ =ȱ ⇒ \ − [= \⇒ \ = [ \ − [ G\ G\ G[ G\ = ⇒ = ⋅ ⇒ = SLHVVHJ GW GW GW GW ⇒ Ǳ ȱȱȱȱȱȱȱȱȱȱȱŗŘȱȦǯȱ şǯȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ¢ȱ ȱ ȱ ŗŜȱ àǰȱ ȱ ȱ ȱ ŚśŖȱ Ȧȱ ¢ȱ śŖȱ Ȧȱ ǯȱ Ƕȱ ·ȱ £ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱ¤ȱȱȱȱŗŘȱȱȱȱ¤ȱ¤ǵ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱȱ GD = NPK GW G] +DOODU VL D = NP GW ] Dȱ +D = ] G] G] D GD GD = ] ⇒ = GW GW ] GW GW G] G] ⇒ = ⋅ ⇒ = NPK GW GW D Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ NPȱ ŗŘȱǯȱȱȱ¤ȱ¤ȱȱȱȱȱȱȱŘŚŖ ǯȱ K ŗŖǯ ȱȱȱȱàȱȱȱȱ£àȱȱŘȱȱřȱȦǯȱȱȱȱȱ ȱȱŗŞȱȱ¢ȱȱȱȱȱȱȱȱȱȱȱŘŚȱǯȱǶȱ·ȱ ȱ¤ȱȱȱȱȱȱȱȱȱȱȱȱȱȱ Ŝȱǵȱȱ àǱȱ P GY = P PLQ GW GK +DOODU VL K= P GW P K 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 π UK 9 = $KRUD ȱ = U ȱȱ K ⇒ K= U ⇒ K= U ⇒ K= U π ⎛ ⎞ 9 =ȱ ⎜ K⎟ K ⎝ ⎠ GY GK ⇒ 9 = = π K π K ⇒ GW GW ⋅ GK ⇒ = π ⋅ GW GK ⇒ = PPLQ GW π ǱȱȱȱȱȱȱȱŜȱȱȱȱȱȱȱȱȱ ȱȱȱȱ¡ȱŖǰŖŖşşśȱȦǯȱ ŗŗǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ¤ȱ ¤ȱ π ¢ȱȱ£àȱ Ȧǯȱȱȱ ȱȱȱ ¢ȱȱŚŖǰȱ ȱ ȱȱ·ȱ£ȱȱȱ¤ȱȱȱȱȱ¤ȱȱ π ǯȱȱ ȱ àǱȱ [ \ Gα π = − UDGVHJ GW G$ π +DOODU VLα = UDG GW 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD VHQȱα = $= [\ [ \FRVα = /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 \ ⇒ $= VHQα FRVα G$ Gα Gα = FRV α − VHQα GW GW GW G$ ⎛π ⎞ π ⎛π ⎞ π − VHQ⎜ ⎟ ⋅ − ⇒ = FRV⎜ ⎟ ⋅ − GW ⎝ ⎠ ⎝ ⎠ ⇒ ⇒ π G$ π = ⋅ ⋅ − − ⋅ − GW Ǳȱ ȱ ȱ ¤ȱ ¡ȱȱřŚǰşŗ πȱ ȱ ȱ ¤ȱ ¢ȱ ȱ ȱ ȱ FPȱ ǯȱ V ŗŘǯ ȱ ȱ ȱ ŗŖȱřȱ ȱ àȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ·ȱȱȱȱǯȱȱȱȱȱȱȱȱȱśŖǰȱȱ ¤ȱ¤ȱȱȱ£àȱ ȱP Ƕ¤ȱ¤ȱ¢ȱȱȱȱȱ V ȱȱȱǵȱȱ àǱ U 9 =P 6L U = P⇒ K +DOODU GK GW G9 GU GK = π UK + π U GW GW GW GK GK ⋅ + π ⋅ ⇒ π =− = π ⋅ ⋅ GW GW π 9 =π U K⇒ GK GK − =− ⇒ = PVHJ GW GW π ⋅ π GU = PVHJ GW 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ śŖȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱȱȱȱ¡ȱȱŖǰŖŖŖŗŖŘȱȦǯȱ ŗřǯȱ Ƕȱȱ£ȱȱȱȱȱȱȱȱȱ ȱȱÇȱ ȱŜȱȱ¤ǰȱȱȱȱȱȱȱȱȱ£àȱ Pȱ ǵȱȱ PLQ àǱȱ U = P GK GY VL = −P PLQ GW GW 9 = πK GY GK = π GW GW GK − = π GW GK Ǳȱ ȱ £ȱ − = π GW +DOODU ȱ ȱȱȱȱ¡ȱŖǰŗŖŜŗȱȦǯ ȱ ȱ ȱ ȱ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱȱàȱ Ǳȱ ȱȱȱȱȱàǯȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱȱ¤ȱȱ¡ȱǯȱ ȱȱȱȱȱàȱȱȱȱ¡àȱȱȱ øȱǯ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ¤ȱȱȱ¤ȱ£ȱȱ ȱȱȱ¤ȱȱȱȱ ¤ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱȱȱȱȱ ȱȱȃȄȱȱȱ¤ǰȱȱȱ ȱ ȱ ȃȱ ȱ ÛȄǯȱ ȱ Çȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ·ǯȱ ȱ ȱ ȱ ¤ȱ Çȱ ȱ ȱ ȱ ȱ ȱ ¤ǰȱ ȱ ȱ ¢ȱ £ǰȱ ¢ȱ ȱ ȱ Çȱȱȱ ȱȱàȱȱȱȱȱȱȱȱ¢ȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱȱȱǯȱ Ƕȱȱȱȱȱȱȱàǵȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱȱȱǰȱ¡ǰ ȱȱȱȱȱǯȱȱ ȱŗǱȱ ȱȱà Iȱ [ =[ − [+ ǰȱȱ¤ȱȱȱȱȱȱ àȱȱȱȱȱ¡ǰȱȱȱŘȱȱřǯȱȱ àǱȱ ȱȱȱȱȱŘȱȱřǰȱȱȱȱȱȱȱ Δ[ȱ = ǯȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ à Iȱ [ =[ − [+ ǰȱ ȱ ȱ ȱ ȱȱȱ¤ȱȱȱàȱȱ¡ǯȱ ȱ ǰȱ ȱ Iȱ += I = ⋅ − ⋅ + = ǰ Iȱ =⋅ − ⋅ + = ǰȱ ȱ ȱȱȱàȱȱàȱȱàȱȱȱȱ Δ\ = Δ\ ȱ= I + − I = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱŘǱȱ ȱȱà Iȱ [ =[ − [+ ǰȱȱ¤ȱȱȱȱȱȱ àȱȱȱȱȱ¡ǰȱȱȱȬřȱȱȬŗǯȱȱ àǱȱ ȱ ȱ ȱ ȱ ȱ Ȭřȱ ȱ Ȭŗǰȱ ȱ ȱ ȱ ȱ ȱ ȱ Δ[ȱ = ǯȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ à Iȱ [ =[ − [+ ǰȱ ȱ ȱȱȱȱȱ¤ȱȱȱàȱ ¡ǯȱ ȱ ǰȱ ȱ ȱ Iȱ − = ⋅ − − ⋅ − + = ¢ Iȱ − = ⋅ − − ⋅ −+ = ǰȱȱȱȱ ȱ àȱȱàȱȱàȱȱ ȱàȱ Δ\ Δ\ ȱ= I −+ − I − = − ȱȱǰȱȱȱȱȱàȱ¢ȱƽȱǻ¡ǼǯȱȱÛȱà Δ¡ȱ ȱ ȱ ȱ ¡ǰȱ ȱ ȱ Ûȱ à Δȱ\ = I( [+ Δ[) − I( [) ȱ ȱ ¤ǯȱ ȱȱàȱȱȱ¤ȱȱ ȱ ȱ ȱ ȱ àȱ ȱ àȱ ȱ ȱ ¡ȱ ȱ ȱ ·ȱ Δȱ[ ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Ƕ·ȱȱȱȱȱ¢ǵȱ ȱȱǰȱȱȱȱȱàȱ¢ȱƽȱǻ¡Ǽǰȱȱȱȱȱàȱ ȱȱȱȱ ǯȱ ȱȱȱÛȱàȱȱȱȱ¡ǰȱȱȱÛȱ àȱȱȱ¤ǯȱ Δ¡Ǳȱ Δ¢Ǳȱ àȱ ¡ àȱȱȱ¤ȱ ȱȱ Δȱ\ = I( [+ Δ[) − I( [)ǯȱ ȱ ȱ ȱ ȱ I′([) = OLPȱ OLP ⎡ I( [+ Δ[) − I( [) ⎤ − I′([)⎥ = ǯȱ Δ[ ⎦ Δ[→ I( [+ Δ[) − I( [) ǰȱ ȱ ǰ Δ[ Δ[ ȱ→ ⎢ ⎣ I( [+ Δ[) − I( [) − I′( [)ǰȱ OLP[K(Δ[)]= ¢ȱ ȱ ȱ Δ[ ȱ→ Δ[ Δȱ[ K(Δ[) = I([+ Δ[) − I([) − Δ[ I′( [) ȱ I ([+ȱΔ[) − I([) = Δ[ I′([) + Δ[ K(Δ[)ǰȱ ȱ ȱ K(Δȱ[) = ȱȱ Δȱ\ = Δ[ I′( [) + Δ[K(Δ[) ¢ȱ Δ[ȱ→ ǰ Δȱ\ ≈ I′([)Δ[ǯȱ Çǰ G\ȱ ȱȱȱǲȱȱȱȱȱ¡ȱ¢ȱ G[ȱǯȱȱȱ G[ȱ ȱȱȱȱȱÇȱ¢ȱ¡ȱȱȱȱȱȱȱȱ Iȱ ȱ ȱȱȱ·ȱ G\ ǯȱ ȱȱ·ȱȱȱȱȱȱǱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ 3 ȱ [ I [ ¢ 4 [+ȱΔ[ I [+ Δ[ ȱȱȱ¤ȱ Iȱ ǲȱ¢ȱ G[ =ȱΔ[ǯȱȱȱ ȱȱȱ¢ȱ Δȱ\ = I [+ Δ[ − I [ ȱȱȱȱȱ 35ȱ ȱȱ I ′ȱ [ ǯȱȱÇǰȱȱȱ G\ Ǽȱȱȱǰ G\ȱ ȱȱ G[ àȱȱȱȱǰȱȱȱ Δȱ\ ȱȱàȱȱȱ ȱȱȱȱȱ I ′ ȱ[ G[= G\ ǻȱ I ′ ȱ[ = ȱȱ¡ȱȱȱȱ G[ȱǯȱ ȱȱȱȱȱ¡ȱ ȱȱȱȱȱ Δȱ\ ≈ G\ ¢ȱȱȱȱȱȱȱ Δȱ[ ȱ ȱ ¡à Δȱ\ ≈ G\ ǯȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ǰȱ Δȱ\ ȱȱǰȱȱȱȱȱ¡ȱȱ ȱȱȱȱȱǯȱ ȱȱàȱȱȱǰȱȱ¡àȱȱȱǱ I [+ Δ[ − I [ ≈ I′ [ G[ ⇒ I [+ Δ[ ≈ I [ + I′ [ G[ I ȱ[+Δ[ ≈ I [ + G\ ⇒ ŗǱȱ ȱȱȱ VHQȱ àǱȱ I [ =ȱVHQ [ ǰȱȱȱ¡ ƽ πȱ π ¢ G[=⎛⎜ − ⎞⎟ ⎝ I [+ Δ[ − I [ ≈ I′ [ G[ ⇒ ⎛ ⎞ ⎛ π ⎞ ⎛ π ⎞⎛ π ⎞ VHQ⎜ ⎟ − VHQ⎜ ⎟ ≈ FRV⎜ ⎟⎜ − ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎛ ⎞ ⎝ ⎠ ⎛ π ⎞ ⎛ π ⎞⎛ π ⎟ + FRV⎜ ⎟⎜ − ⎝ ⎠ ⎝ ⎠⎝ VHQ⎜ ⎟ ≈ VHQ⎜ ⎛ ⎞ VHQ⎜ ⎟ ≈ + ⎝ ⎠ ⎛ ⎞ VHQ⎜ ⎟ ≈ ⎝ ⎠ ⇒ ⎞ ⎟ ⎠ ⇒ ⇒ ⎠ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱŘǱȱ ȱ £àȱ ·ȱ ȱ ŗŖ SXȱOJ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱ¡ȱǯȱȱȱȱȱȱȱȱ£àȱ¤ȱȱȱȱàȱ Ǳ 9= πȱ [ − [ ȱȱȱȱȱȱȱȱ£àȱ¢ȱȱȱȱȱ ś SXȱOJ ȱȱȱ¤¡ȱȱȱȱ SXȱOJǯȱȱȱȱȱ¤¡ȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £àȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱȱ àǱȱ ȱ Δȱ9 ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Δȱ[ ǰȱ ȱ ȱ ȱ ¡ȱ ¡ȱ ȱ ȱ ȱ ȱ G9 = πȱ [− [ G[ = π [− [ G[ ȱȱ [ȱ= \ G[± ȱ G9 = πȱ − ± ≈ SXOJ ȱȱȱ¤¡ȱȱȱȱȱȱȱȱȱȱȱ £àȱȱ¡ǰȱŗś SXȱOJ ȱȱȱȱȱȱȱȱȱȱȱ HUURU HVWLPDGR HUURU UHODWLYR =ȱ YDORU H[DFWR ȱȱǰȱȱȱȱȱȱȱȱȱȱȱ¡ǰȱȱ G[ȱ = = = [ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ G9 = = = 9 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¤ȱ ȱȱȱȱ ŗǯȱ ȱȱȱȱȱȱ¡ȱȱȱǻŗŞŗķǼǯȱȱ àǱ \ȱ =VHQ[⇒ G\ = FRV [G[ π [= ⇒ G[ = = VHQ() ≈ VHQ() + FRV() • ⇒ π π ⇒ VHQ()≈ − VHQ() ≈ + (− ) • Řǯȱ ȱȱȱ¤ȱȱȱȱȱȱȱȱŚŖȱȱȱȱȱ ȱȱȱȱŖǰŜȱǯȱȱȱȱȱȱȱȱȱ ȱȱȱ¤ȱȱȱȱȱȱǯȱȱ àǱ U ȱ = FP GU = FP $= π U G$= π UGU ⇒ G$= π • • = π ⇒ G$ π = = $ π Ǳ ȱȱȱȱȱ¤ȱȱȱȱȱȱȱȱȱřƖȱ řǯȱ ȱȱȱȱȱȱŗŖȱȱȱȱ ȱȱȱȱȱŗǯȱ ȱȱȱȱȱ¤¡ȱȱȱȱȱȱ¤ǯȱȱ àǱ U =ȱFP = PP⇒ GU = PP $= π U ⇒ G$= π UGU⇒ G$= π • • PP = π PP Ǳ ȱ¤¡ȱȱȱȱȱȱ¤ȱȱ π ȱFP ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Śǯȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ŜŖȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ Ŗǰŗȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱ¤ȱȱȱǯȱȱ àǱ [ = FP G[= FP ⎛ π ⎛ [⎞ ⎞ $= ⎜ [ + ⎜ ⎟ ⎟FP ⎜ ⎝ ⎠ ⎟⎠ ⎝ [FP ⎛ [π ⎞ ⎟FP ⇒ G$= ⎛⎜ [+ [π ⎞⎟ G[⇒ $ȱ= ⎜⎜ [ + ⎟⎠ ⎠ ⎝ ⎝ π ⎞ + π ⎛ G$= ⎜ ⋅ + ⇒ ⎟⋅ = ⎠ ⎝ + π (+ π ) G$ = = $ + π (+ π ) + π + π = = ≈ + π + π Ǳ ȱȱȱȱȱȱ¤ȱȱȱȱȱȱŖǰřřƖȱ śǯȱ ȱȱȱȱȱȱȱȱȱȱȱ ȱƽȱřǰȱȱȱȱ¤¡ȱ ȱŖǰŖŗǰȱȱȱȱ¤¡ȱȱȱȱȱȱ ȱȱȱ ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱ G9 = π (UURU Pi[LPR SRVLEOH 9 =ȱ π U ⇒ G9 = π UGU⇒ G9 = π ⋅ ⋅ ȱǱ G9ȱ π = = Y π Ǳȱ ȱȱȱȱȱȱȱȱȱȱȱȱŗƖǯȱ Ŝǯȱ ȱǰȱȱȱȱ¡ȱ ȱǯȱȱ àǱ \ȱ = [ ⇒ G\ = G[ [ [= ⇒ G[= − ≈ + ŝǯȱ ⋅ −⇒ ≈ − = = ȱȱȱ¡ȱȱȱȱȱȱŗŚŝķǯȱȱ àǱȱ \=ȱFRV ([) ȱȱȱ G\ =ȱ−VHQ([)⋅ G[ǯȱ ȱ = π UDG ¢ ȱ = π UDG ȱ [ȱ= π ⎞ π ⎛ π ⎞ ⎛ π ⎞ ⎟⋅− ⎟ − VHQ⎜ȱ ⎟ ≈ FRV⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ȱȱ FRV⎛⎜ ǰȱȱǱ π − π π ⎛ π ⎞ FRV⎜ − ⋅− =− + = ⎟≈− ⎝ ⎠ π ¢ G[ȱ=− π ǯȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ¤¡ȱ¢ȱÇȱȱȱàǯȱ Ǳȱ ȱ ȱ ¤¡ȱ ¢ȱ Çȱ ȱ ¢ȱ ȱ ȱ ȱ àȱ £ȱȱ¤ǯȱ ȱ ȱ ¤¡ȱ ¢ȱ Çȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱǯȱ ȱȱȱȱ£ȱȱàȱȱȱȱȱȱ ȱȱȱǯ ȱȱȱ¤ȱȱȱȱȱȱàȱȱȱȱȱ ȱ ȱ ȱ ȱ ȱ Ǳȱ ȱ £àǰȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ǯȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤¡ȱ ȱȱȱȱȱȱȱȱÇȱȱȱȱ¢ȱ ȱǰȱȱǰȱǯȱ ȱàȱȱȱȱȱȱȱȱȱȱȱȱȱ àȱ£ȱȱ¢ȱȱȱȱǯȱȱȱȱȱȱȱ¤ȱȱȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ¢ȱǯȱȱ àǯ •ȱ ȱà Iȱ £ȱȱ¤¡ȱȱȱǰȱȱȱȱȱȱ ȱ¡ȱȱȱȱ Iȱ ǰ I F ≥ȱ I [ ǲȱȱȱ Iȱ F ȱȱȱȱ ¤¡ȱ Iȱ ǯ •ȱ ȱà Iȱ £ȱ ȱÇȱȱȱǰȱȱȱȱȱȱ ȱ¡ȱȱȱȱ Iȱ ǰ I F ≤ȱ I [ ǲȱȱȱ Iȱ F ȱȱȱȱ Çȱ Iȱ ǯȱȱ Ǳȱ Ǽȱ ȱȱȱȱȱȱȱȱ¤ȱǰȱȱȱȱȱ ¤ȱ ȱ ȱ à Iȱ ,5→ ,5 I [ = [ + [− ȱ ȱ ¤ȱ àȱ ȱǰȱ¢ȱ·ȱȱȱȱȱȱǻȬŗǰȬŚǼǯȱȱȱȱȱ ȱȱȬŚȱȱȱ¤¡ȱȱȱȱàǯȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǽȱ ȱà I ȱ ,5 → [− ] I [ = VHQ[ ȱȱ¤¡ȱȱȱŗȱ¢ȱȱ Çȱ ȱ ȱ Ȭŗȱ ¢ȱ £ȱ ȱ ¤¡ȱ ȱ ȱ ȱ øȱ ȱ ȱ π [=ȱ + Nπ Nȱ ȱøȱǲȱ£ȱȱÇȱȱȱȱ øȱȱȱ [=ȱ π + Nπ Nȱ ȱøȱǯȱ Ǽȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ǰȱ ȱ ȱ ȱȱȱàȱ£ȱȱ¤¡ȱȱȱȱ¢ȱȱȱȱǰȱȱ ȱȱȱÇȱȱȱ¢ȱȱ£ȱȱǯȱȱ àǯ •ȱ ȱà Iȱ £ȱ ȱ ¤¡ȱȱǻȱ¤¡ȱǼȱȱǰȱȱȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ øȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰ I F ≥ȱ I [ ǲȱȱȱ Iȱ F ȱȱȱȱ¤¡ȱȱ Iȱ ǯ •ȱ ȱ à Iȱ £ȱ ȱ Çȱ ȱ ǻȱ Çȱ Ǽȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ øȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰ I F ≤ȱ I [ ǲȱȱȱ Iȱ F ȱȱȱȱÇȱȱ Iȱ ǯ Ǳȱ Ǽȱ ȱ à I ȱ[∞[→ ,5 I [ = [ ȱ ȱ Çȱ ȱ ȱ Ŗǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ≤ȱ I [ ȱȱ¡ȱȱǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǽȱ ȱȱ¤ȱǰȱȱȱȱǰȱȱȱȱȱȱȱ àȱȱȱÇȱȱȱȱ¢ȱȱȱȱ£ȱȱÇȱǲȱ¤ȱȱ ȱȱȱ¤¡ȱȱȱ¢ȱ·ȱȱǯȱ Ǽȱ ȱ ȱ ¤ȱ ǰȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ àȱ£ȱȱÇȱȱȱȱ¢ȱȱȱ¢ȱ£ȱȱ¤¡ȱȱȱȱ ¢ȱȱŖǯȱǻȱȱȱȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱ¡ǯȱ Iȱ ȱ ȱ ȱ ȱ ȱ [D ȱE]ǰȱ Iȱ £ȱ ȱ ȱ ¤¡ȱ Iȱ F ¢ȱ ȱ ȱ Çȱ Iȱ G ȱ ȱ øȱȱ¢ȱȱ [D E] Ǳȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ǰȱ ȱ ȱ £ȱ ȱ àȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àǯȱ ȱ ȱ ȱ ȱ àȱ ȱȱȱȱ¡ȱȱȱǰȱȱȱȱ£ȱȱȱàȱ ȱ¡ȱǰȱǱȱ ȱȱȱ Iȱ ȱȱ¤¡ȱȱȱÇȱȱȱȱ¢ȱ I ′ȱ F ¡ǰȱ I ′F = Ǳȱȱȱȱȱȱȱȱȱ¤¡ȱ¢ȱÇȱȱȱ ȱ¢ȱƽȱǻ¡Ǽǰȱ ȱȱǰȱȱȱȱȱ£ǯȱȱ Ǳȱ ȱ ȱ ȱ ȱ à \ =ȱ I [ ȱ ȱ àǰȱ ȱ ȱ ȱ ȱȱȱȱȱǰȱǰȱȱ¢ȱȱȱȱȱȱ \ =ȱ I [ ȱŖǰȱǰȱȱ ¢ȱȱǯȱȱǰȱǰȱȱ¢ȱȱȱȱ£ȱȱȱȱȱ ȱȱǰȱȱȱǱ Iȱ′ = ǰ Iȱ′ D = ǰ Iȱ′F = ǰ Iȱ′E = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱÇȱȱȱȱǰȱȱȱǯȱ ȱǰȱ I [ =ȱ [ ȱȱȱȱȱȱȱ ȱȱ ǲȱȱ I [ =ȱ [ ȱȱȱ¡ȱȱǯȱ ȱ¡ȱȱȱ¡ȱȱ I ′ȱ F ȱ¡ǯȱ ȱ ǰȱ I ȱ[ = [ ȱ ȱ ȱ ¡ȱ ȱ ǲȱ ȱ I ȱ[ = [ ȱȱÇȱȱȱǯȱ ȱȱȱȱȱȱȱȱȱȱǯȱ àȱ ȱøȱÇȱȱȱà Iȱ ȱȱøȱȱȱȱȱȱȱàȱ Iȱ ȱ Iȱ′F = I ′ȱ F ȱ¡ǯ ȱ ȱ ȱ ȱ ȱ ȱ ¡ǰȱ ȱ ȱ ȱ ȱ àȱ ȱȱȱ¡ǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱàȱ£ȱȱ¤ȱȱȱȱȱȱȱ ȱ àȱ ȱ ȱ ȱ ȱ ¡ǯȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱȱȱȱȱȱǯȱ Ǽȱȱ¤ȱȱȱàȱ£ȱȱ¤¡ȱȱȱȱ¢ȱȱÇȱ ȱȱǯȱȱȱȱȱȱȱȱ¡ȱȱȱǯȱ Ǽȱȱ¤ȱȱȱàȱ£ȱȱ¤¡ȱȱȱȱ¢ȱȱÇȱ ȱȱǯȱȱȱȱȱȱȱȱ¡ȱȱȱǯȱ Ǽȱȱȱ¤ȱȱȱàȱ£ȱȱ¤¡ȱȱȱȱ¢ȱȱÇȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ¤¡ȱ ȱ £ȱ ȱ ȱ ¡ȱ ȱ ȱ ǲȱȱÇȱȱȱøȱÇȱǻ Iȱ′F = Ǽǯȱ Ǽȱȱ¤ȱȱȱàȱ£ȱ¡ȱȱȱȱ¢ȱȱǯȱȱȱ ȱȱȱ£ȱȱøȱÇȱǻ I′F = Iȱ′G = Ǽǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǽȱȱ¤ȱȱȱàȱ£ȱȱ¤¡ȱȱȱȱ¢ȱȱÇȱ ȱ ȱ ǯȱ ȱ ¤¡ȱ ȱ £ȱ ȱ ȱ ¡ȱ ȱ ȱ ǲȱ ȱ ÇȱȱȱøȱÇȱǻ I ′ȱ F ȱ¡Ǽǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱǯȱȱȱǰȱȱȱȱȱȱ Iȱ [ ¤ȱ ȱ¢ȱ¤ȱÛȱȱȱȱ¤ Iȱ [ ȱ¢ȱȱȱ¡ȱȱȱȱ ȱȱǯȱ ȱ¡ȱȱȱȱȱ¡ȱȱ£ȱȱȱȱ ȱ¡ȱȱȱȱȱ£ȱȱȱȱȱȱȱȱȱ ȱ¢ȱȱȱøȱȱȱȱȱøȱÇǯȱ ȱ ȱ ǰȱ ȱ àȱ ȱ ȱ ȱ àȱ ȱ ¡ȱ ȱȱȱàȱȱȱȱ [D ȱE] ȱ£ȱȱȱ ȱǱ •ȱ •ȱ •ȱ •ȱ ȱȱøȱÇȱ Iȱ ǯ ȱȱ¤ȱȱȱøȱÇǯ ȱȱ¤ȱȱȱ¡ȱȱǯ ȱȱȱ¤ȱǰȱȱ¢ȱȱȱ¤¡ȱȱ¢ȱȱ ȱȱȱÇȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ¤ȱ ŗǯȱ = [ +[ȱ Ƕȱ·ȱȱȱ I [ £ȱ ȱ¤¡ȱ¢ȱ ȱ ÇȱȱȱȱǽȬřǰŗǾǵȱȱȱȱ¤¡ȱ¢ȱȱÇǯȱȱ àǱȱ Ǽ Iȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ¡ȱȱȱȱȱȱ [−ȱ]ǯȱ Ǽ I ′([) = ȱ − − [ + [ = [ (+ [) I ′([) = ⇔ [ = − I ′([) VHLQGHILQH VL [ = I(− ) ≈ I() = I(− ) ≈ I()= Ǳ ȱ¤¡ȱȱ Ŝȱ¢ȱȱÇȱȱ Ŗǯȱ Řǯȱ Ƕȱ ·ȱ ȱ ȱ I [ = [ − [ȱ − [ £ȱ ȱ ¤¡ȱ ¢ȱ ȱÇȱȱȱȱǽȬŗŖǰŘǾǵȱȱȱȱ¤¡ȱ¢ȱȱÇǯȱȱ àǱ Iȱ( [) ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ ǯ Iȱ ′( [) = [ − [− ∨ [= − ⎛ ⎞ I⎜ − ⎟ = ⎝ ⎠ I ′([) = ⇔ [ − [− = ⇔ ([− )([+ ) = ⇔ [= ⎛ ⎞ ⎝ ⎠ I(− ) = − I()= I⎜ ⎟ = − Ǳ ȱ¤¡ȱȱȱřŚȱ¢ȱȱÇȱȱȱȬśŖşŖǯȱ řǯȱ ȱ ȱ ¤¡ȱ ¢ȱ ȱ Çȱ ȱ ȱ à [ −ȱ] ǯ I [ = − [ [ȱ + ȱ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àǱ Iȱ([) = − [ [ + ⇒ I′( [) = [ − [− ([ + ) I C([) = ⇔ [ − [− = ⇔ ([− )([+ ) = ⇔ [= ∨ [= − I(− ) = I() = − − I() = I(− )= Ǳ ȱ¤¡ȱȱ ȱ ¢ȱȱÇȱȱ −ȱ ǯȱ ȱ ȱ ȱ Çȱ ¢ȱ ȱ ȱ ¤¡ȱ ȱ ȱ à Śǯȱ I [ = [ȱ + [+ ȱȱ [−ȱ] ǯȱȱ àǱ I ′( [) = − [ I′( [)= ⇔ + − [ + = ⇔ − [ = −⇔ [ = −⇔ −= [ ⇔ [= − I ′ȱ( [) ȱȱ [ȱ = ǯ Iȱ(−) = − I(−) = I()= Ǳ ȱ¤¡ȱȱȱŘȱ¢ȱȱÇȱȱȬřǯȱ śǯȱ ȱ ȱ ȱ ¤¡ȱ ¢ȱ Çȱ ȱ I [ = −[ȱ ȱ ȱ [ −ȱ] ǯȱȱ àǱ Iȱ([) = − [ ⇒ I ′( [) = −[ − [ = − [ − [ I ′( [) = ⇔ [ = I ′([) VHLQGHILQH⇔ [= ±VLQ HPEDUJR ± ∉ [− ] I(− ) = I() = I()= Ǳ ȱ¤¡ȱȱȱřȱ¢ȱȱÇȱȱ ȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ¤¡ȱ ¢ȱ ȱ Çȱ ȱ ȱ à I [ = [ −[ȱ ȱ ȱ Ŝǯȱ ⎡ −ȱ ⎤ ǯȱȱ ⎢⎣ ⎥ ⎦ àǱ Iȱ′([) = [ − [ ( ) ⎡ ⎤ ⎣ ⎦ I′([) = ⇔ [ [ − = ⇔ [= [= ± ∉ ⎢− ⎥ ⎛ ⎞ ⎝ ⎠ I(− ) = − I⎜ ⎟ = − I() = I(− )= Ǳ ȱ¤¡ȱȱ Řȱ¢ȱȱÇȱȱ ȬśŜǯȱ ŝǯȱ I [ = [ − [+ȱ ȱ ȱ ȱ [ −ȱ] ǯȱ ȱ ȱ ȱ ·ȱ ȱȱȱȱ ȱȱ¤¡ȱ¢ȱȱÇȱ¢ȱøǯȱȱ àǱȱ Ǽ Iȱ( [)= [ − [+ ȱ ȱ ȱ Iȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ǯȱ Ǽ ( ) Iȱ′( [) = [ − ⇒ I′([) = ⇔ [ − = ⇔ [ = ± ∉ [− ] I(− ) = + I(− ) = I()= Ǳ ȱ¤¡ȱȱ +ȱ ¢ȱȱÇȱȱȱŖǯȱ ȱ ȱ ȱ ¤¡ȱ ¢ȱ Çȱ ȱ I [ = Şǯȱ [ [+ ȱ ȱ ȱ ȱǽŗǰŘǾǯȱȱ àǱ I ([) = [ ⇒ I′([) = [+ ([+ ) Iȱ ′( [)≠ ȱȱ¡ǰȱ¤ I ′ȱ( [) ȱȱȱȱ¡ȱȱȱ [ȱ]ǯ I () = I()= 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ Ǳ ȱ¤¡ȱȱ şǯȱ ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ¢ȱȱÇȱȱ ǯȱ I W =ȱVHQ Wǯȱȱȱȱȱ¡ȱȱȱ ȱ ȱȱ [−π ȱπ] ǯȱȱ àǱ I (W) = VHQW⇒ I ′(W) = FRVW I ′(W)= ⇔ FRVW = FRV Wȱ = ⇔ W = π \ W = − π I(− π ) = VHQ(− π ) = I(π ) = VHQ(π )= ⎛π ⎞ ⎛π ⎞ ⎛ π⎞ ⎛ π⎞ ⎟ = VHQ⎜ ⎟ = I⎜ − ⎟ = VHQ⎜ − ⎟ = − ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ I⎜ Ǳ ȱ¤¡ȱȱ ¢ȱȱÇȱȱ ȬŘǯȱ ŗŖǯȱ ȱȱȱÇȱ¢ȱȱȱ¤¡ȱȱȱà I [ ȱȱ [ −ȱ] ǯȱȱ = [ −[ȱ àǱ − − Iȱ ′([) = [ − [ = [ (− [) I ′([) = ⇔ [= I C([) = VHLQGHILQH ⇔ [= I(− ) = I() = I()= Ǳ ȱ¤¡ȱȱ ¢ȱȱÇȱȱȱŖǯȱ ŗŗǯȱ ȱȱȱÇȱ¢ȱȱȱ¤¡ȱȱȱà I [ = ȱȱ [ −ȱ] ǯȱȱ Ǳ ȱ¤¡ȱȱ +ȱ ¢ȱȱÇȱȱȱŚǯ [ +ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱǱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ¤ȱ ¤¡ȱ ȱ ȱ ȱ ȱȱ ȱ Çȱ ȱ U = FP ǰȱȱȱȱ ȱȱȱ ¤ǯ ȱ ȱ£ȱȱȱȱŘŖ FPȱ ȱǰȱȱȱȱȱǰȱ¢ȱȱȱȱȱ ȱȱȱȱȱǯȱǶàȱȱȱȱȱȱȱȱȱ¤ȱ ȱȱȱȱȱȱȱȱǵ ȱ ȱ £ȱ ȱ ȱ ȱ ŗŖȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ¤ȱ ¤ǯȱ Ƕàȱ ȱ ȱȱȱȱȱȱ¤ȱȱȱ¤¡ǵ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ŗŘȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱǯȱȱȱ ȱ ¤¡ȱȱȱ ȱȱ ȱ ȱÇǯ ȱ ȱȱ¤ȱȱȱȱȱȱȱȱȱȱȱȱÇȱȱȱřȱ ȱǰȱ¢ȱȱȱȱÇȱǯȱȱȱȱȱȱȱȱ ȱȱȱ£ȱȱȱȱȱȱȱȱŞȱȱȱ¢ȱȱȱȱ ¤ȱȱȱȱǯȱ ȱ£ȱȱȱȱȱȱǱȱ Ǽȱ ȱȱȱȱ¢ȱ£ȱȱȱÇȱȱȱȱȱȱ¤ȱ ȱàȱȱȱȱȱȱȱ¢ȱȱȱȱȱȱǯȱ Ǽȱ Çȱȱȱȱȱȱȱȱȱ Ǽȱ ȱøȱȱÇȱȱȱȱȱȱ£ȱȱȱ¢ȱ ǰȱ¢ȱȱȱȱǯȱ NPȱ NPȱ ȱ ȱ ȱ ȱ Ŝ ¢ȱ ȱ ȱ Ş ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ K K ȱȱ¢ȱȱȱȱȱȱȱȱȱȱȱ¤ȱȱ ǯ ȱ ȱ ȱ ¤ȱ ȱ ¤ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ¤ȱ ¢ȱ ȱ ȱ ȱ řȱ ¢ȱ ȱ Śǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ¤ȱȱȱǯ ȱ ȱȱȱȱ¤ȱȱ¢ȱ¤ȱȱȱȱ ȱȱ¤ȱàȱȱ E =ȱ ¢ȱȱ K = ǯ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ Çȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ŚŖȱ ȱ ŞŖȱ ǰȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ǞŞȱ ȱ ǯȱ ȱ ȱ ǰȱ ȱ ȱ ȱȱȱȱȱŞŖȱǰȱȱȱȱȱȱȱ ȱ ¤ȱ ȱ Śȱ ȱ ȱ ȱ øȱ ȱ ȱ ¡ǯȱ ȱ Ƕ¤ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ǵ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ£ȱȱȱȱȱȱȱȱȱȱ¢ȱȱ ȱ ȱȱȱǯȱȱȱȱȱȱȱȱ¤ȱÇȱȱ ȱȱȱ ȱȱȱȱ ȱȱȱȱȱ ȱȱ ȱ ǯ ȱ ȱȱȱȱȱȱȱȱȱȱȱ ¢ȱȱȱȱȱȱȱǯȱȱȱȱȱȱȱȱ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱŝŘȱǯ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £àȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱȱȱøȱȱȱȱȱȱȱȱ ¢ȱ ȱ øȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ¤¡ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱǯ ȱ ȱȱȱȱȱȱà [ȱ +\ = ǰȱǱ •ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ǻŚǰśǼȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯ •ȱ ȱȱ¤ȱȱȱȱǻŚǰśǼȱȱȱȱȱȱǯ ȱ ȱȱ ȱ ȱȱ ȱȱ¢ȱ ȱȱȱȱ ȱȱȱȱȱŜȱȱȱǯ ȱ ȱȱȱ¤ȱȱ¢ȱ¤ȱȱȱȱÇȱȱȱȱ ǯ ȱ ȱȱøȱȱȱ [ȱ] ȱȱȱȱȱȱøȱ¢ȱ ȱȱȱȱ¤¡ǯ ȱ ⎤ ȱ ȱ ȱ øȱ ȱ ȱ ⎡ ⎢ ⎥ ȱ ȱ ȱ ȱ ȱ øȱ ¢ȱ ȱ ⎣ ⎦ Çȱȱȱ¤¡ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱ¢ȱȱȱȱȱ ȱ ȱȱȱȱȱ¢ȱȱȱǯȱ ȱȱàȱȱȱȱȱȱȱȱ¢ȱȱ ȱȱȱȱȱȱǯȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱȱȱȱȱȱȱǯȱ ȱȱȱȱȱǯ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ 1ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ¤ȱ ȱ ȱ ǯȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱȱȱȱȱȱȱǰȱȱ¤ȱ ¤ȱ£ȱǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ǰȱ ȱ ȱ ȱ ȱ ȱȱ¡ȱȱȃȱȱȱȱȱȱȱȱ ȱȱȄǯȱȱ ȱȱȱȱ ȱȱǯȱ Iȱ ȱàȱȱȱȱȱàȱǱ • Iȱ ȱȱȱȱȱ [D ȱE]ǯ • Iȱ ȱȱȱȱȱ (D ȱE)ǯ • I D =ȱ I E ȱ¡ȱȱøȱǰȱ (D ȱE)ǰȱȱ Iȱ′F = ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ àȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱȱȱ·ȱȱǯȱ ȱȱ¡ȱ ȱȱȱȱȱǰȱȱȱȱ¡ȱ ȱȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ¡ȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £ǲȱ ȱ ȱ ¤ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ D ȱ I D ¢ E ȱ I E ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ·ȱȱȱ£ǯȱȱȱȱȱȱȱȱȱ£ȱȱȱ ¡ȱȱȱ ȱ ȱ ȱȱȱ ȱȱȱ ȱ ȱ¡ǰȱȱȱ ȱȱȱ¡ȱȱȱȱȱȱȱ¡ǯȱ ȱ ȱ ȱ ȱ ¤ȱ Ǽȱ ȱ ȱ øȱ ȱ ȱ ȱ [D ȱE] ȱ ȱ à Iȱ′F = ǰȱ øȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱǯȱȱȱȱ¤ȱǼȱ¢ȱǼȱȱ¡ȱȱȱȱȱ¤ȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ǯȱ ȱ ȱ øȱ ȱ ȱ ¤ȱǼȱ¡ȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ ȱ¡ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱŗǱȱ I [ =ȱ[ − [ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ⎤ ȱ ȱ ȱ ⎡ ⎢ ȱ⎥ ǯȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ⎦ ⎣ ǯȱȱ àǱ Iȱ ȱ ȱàȱ ǰȱȱȱȱȱȱ ǲȱȱȱȱȱ ⎤ ȱ ȱ ȱ ⎡ ⎢ ȱ⎥ ǯȱ ȱ ȱ ¤ǰ Iȱ ȱ ·ȱ ȱ ȱ ȱ ȱ ⎦ ⎣ ⎞ ǰȱȱȱȱȱȱȱȱȱȱȱ ⎛ ⎜ ȱ ⎟ ǯȱ ⎝ ⎠ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱȱǱ I ′ [ = [ − Iȱ ′ [ = ⇔ [ − = ⇔ [ = ⇔ ȱȱȱȱȱȱȱ Fȱ = [ = ⇔ [ = ± ȱŘǱȱ ȱȱȱà [ȱ + [− = ȱȱøȱàȱǯȱȱ àǱȱ ȱ ȱ ȱ ȱ ȱ øȱ àȱ ǰȱ ȱ ȱ ȱ ȱ ȱȱàǰȱȱȱȱ¤ȱȱȱȱȱǯȱ ȱȱà Iȱ [ = [ + [− ȱȱȱ [ȱ]ǯȱȱȱȱ Iȱ ȱȱàȱǰȱȱȱȱ£ȱȱȱȱ ȱȱǰȱȱȱ Iȱ ȱȱȱȱ [ȱ]ǯȱ¤ǰ Iȱ =− ¢ Iȱ =ǰȱ ȱȱȱȱȱȱȱȱŖȱȱȱ Iȱ ¢ Iȱ ǯȱȱȱ ǰȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ǯȱ ȱȱȱȱ¡ [ȱ ∈() ȱȱ¡Ŗȱ ȱàȱȱȱà [ȱ + [− = ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱȱàȱȱàȱøȱȱȱȱ£ȱȱàȱȱ ǰȱȱȱȱȱȱàȱȱȱȱǯȱȱȱȱ¢ȱȱ ȱȱȱȱ ȱȱȱàȱȱȱàǯ • Iȱ ȱȱȱȱȱȱ [D E] • Iȱ ȱȱȱȱȱȱ (D E) •ȱ ¤ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ Iȱ D = \ I E = ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ à [ȱ += ȱ ȱ ȱ ȱ àȱ ȱ ȱ àǰȱ ȱ ȱ ȱ ȱ ȱ àǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱȱǰȱȱȱ ȱ¡ȱ ȱ ȱȱ ¢ȱȱȱȱàȱȱ ȱ à [ȱ + [− = ǯȱ ȱȱȱȱà [ȱ + [− = ȱȱøȱàǯȱȱ ȱȱȱȱ ȱȱȱ ȱǰȱȱ·ȱ ȱȱȱǰȱ ȱ ȱ £àȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àǰȱȱȱȱȱȱȱǰȱȱȱȱȱȱȱàȱȱ ȱ¡ȱȱȱȱǯȱȱ ȱȱȱǯȱ Iȱ ȱàȱȱȱȱ àȱǱ • Iȱ ȱȱȱȱȱ [D ȱE]ǯ • Iȱ ȱȱ ȱȱ (D ȱE)ǯȱ ȱ¡ȱȱøȱǰȱ (D ȱE)ǰȱȱǱ I ′ ȱF = I E − I D E− D ȱǰȱȱȱȱȱ I E − I D = I′ F E − D 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ àȱ ȱ ǰȱ ȱ ȱ ȱ ¤ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ·ȱȱǯȱ ȱȱ¡ȱȱȱȱȱȱǰȱȱȱȱ¡ȱȱȱȱ ȱȱȱȱȱ¤ȱȱȱȱȱȱȱȱȱȱȱȱȱ ȱ ȱ ¡ȱ ȱ ǯȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ D ȱ I D ¢ E ȱ I E ȱ ȱ ȱ ¤ ȱ − I D I E ¢ȱ ȱ ȱ ȱ ȱ ȱ E− D ȱȱȱȱ I ′ȱ F ǯȱȱȱȱȱȱȱȱ¤ȱȱȱ ȱ I ′ ȱF = − I D I E E− D Ǳȱ I [ =ȱ [ − [ − [ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱȱȱȱȱ [ȱ]ǯȱȱȱǰȱȱȱȱȱȱȱȱ ȱǯȱȱ àǱ Iȱ ȱȱàȱǰȱȱȱȱȱȱȱǲȱȱȱ ȱȱȱȱȱ [ȱ]ǯȱȱȱ¤ Iȱ ȱ·ȱȱȱȱȱ ǰȱȱȱȱȱȱȱȱȱȱȱ (ȱ)ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱȱǱ I ′ [ = [ − [− Iȱ ′ [ = − I I − ⇔ [ − [− = −+ ⇔ [ − [− = − ⇔ ȱȱȱȱȱȱȱȱȱ F = [= [= 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¤ȱ ȱȱȱǯȱ ȱ ȱ ȱ àȱ ȱ I [ = [− ŗǯȱ ȱ [ [ȱ]ǰȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ǰȱ ȱ ȱ àǰȱ ȱ øȱ ¢ȱ¡ȱ¤ȱ£ȱȱȱǯȱȱ àǱ • • Iȱ ȱȱȱ [ȱ = ǰȱ ∉ȱ[] Iȱ ȱȱ ȱȱȱ [ȱ]ǯ I ′ȱ( [)= + [ ȱ ȱ ȱ [ȱ ≠ Iȱ ȱ ȱ ȱ ȱ [ȱ]ǯ • I [ = [− ȱ [ [ȱ] ȱ ȱ àȱ ȱ ȱ ȱ ȱ ǯȱ ȱ¡ Fȱ∈() WT I ′(F) = I() − I() − ǯ − + = ⇔ = − ⇔ = ⇔ F = ⇔ F= ± Fȱ F F Ǳ ȱȱȱȱȱȱȱȱȱȱȱ [ȱ] ǯȱ Řǯȱ ȱ I [ = [ − [+ȱ ȱȱàȱȱȱȱȱ ȱ ȱ ȱ [ −ȱ] ǯȱ ȱ ȱ ȱ ȱ ȱ øȱ ȃȄȱ ȱ ȱ ǯȱȱ àǱ I [ = [ − [+ȱ ȱȱȱȱȱ Iȱ ȱȱȱȱ [−ȱ] ¢ȱ ȱ ȱ ȱ ȱ ǻȬŘǰřǼȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱȱȱȱȱȱȱȱ [−ȱ]ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à I ′(F) = I ′(F) = I() − I() I() − I() − − − − ⇔ F − = − − ⇔ F = ⇔ F= ± 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ Ǳȱ ȱ ȱ F ȱ = − řǯȱ \ F = [−ȱ] /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ ǯȱ I [ = [ − [+ ǯȱȱǶȱ·ȱȱȱȱȱȱȱȱȱ ȱ ȱ ȱȱ [ȱ]ǵȱȱȱȱ F ∈]ȱ[ ȱȱȱ ǰȱȱ àȱȱȱȱǯȱȱ àǱ I [ = [ − [+ȱ ȱȱǰȱȱȱ Iȱ ȱȱȱȱ [ȱ] ¢ȱ ȱ ȱ ȱ ȱ ǻŖǰŗǼȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱȱȱȱȱȱȱȱ [ȱ]ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ I() − I() à I ′(F) = − Iȱ′( F) = I() − I() − ⇔ F − = (− + ) − ()⇔ F − = ⇔ F = ⇔ F= ± Ǳ ȱȱ [ȱ] ȱȱȱȱȱ Fȱ = Śǯȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ I [ = [ + [ +ȱ[ ȱȱ [ȱ]ǯȱȱ àǱ Iȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ [ȱ] ¢ȱ ȱ (ȱ)ǰȱ I() − I() ȱȱȱȱȱȱȱ¡ Fȱ∈() ȱ I′( F) = − ⇒Fȱ + F+ = − ± ⇒ F + F− = ⇒ F= − Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ [ȱ] F ȱ = − + ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD śǯȱ ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ = + ȱø F ∈ ] ȱ[ǯȱȱ Fȱ àǱȱ I ( [)=ȱ [ ȱȱȱ [ȱ]ǯ Iȱ ȱȱȱ [ȱ≥ ǰȱ Iȱ ȱȱ [] I′ȱ( [)= ȱ [ ȱ Iȱ′(F) = ȱ¡ȱȱ [ȱ > Iȱ ȱȱ (ȱ)ǯȱ ȱ I() − I() − ȱ ȱ = − ⇒ ȱ ¡ Fȱ∈() ȱ + = Ŝǯȱ àȱ ȱ ȱ ȱ ȱ ȱ ǽŘǰśǾȱ ȱ ¢ − I ȱ ≤ ȱ ≤ I ≤ I ′ [ ≤ ∀ [∈]ȱ[ǯȱ àǱȱ ȱ àȱ ȱ ȱ Iȱ ȱ ȱ ȱ ǽŘǰśǾǯȱ ≤ Iȱ′( [)≤ ȱ [ȱ∈[] ȱȱ£ȱ Iȱ ȱȱȱȱǽŘǰśǾǯȱȱȱȱȱ I() − I() I() − I() ȱȱ¡ Fȱ∈() ȱ I ′(F) = ⇒ I′(F) = − ⇒ ≤ Iȱ() − I() ≤ ⇒ ≤ I() − I()≤ ŝǯȱ ȱȱȱà [ + [ =ȱ ȱȱȱ¤ȱȱȱÇ£ȱ ǯȱȱ àǱȱ ¡ ȱ I ( [)= [ȱ + [ ȱȱàȱ ȱȱȱ ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 OLPȱ [ + [ = −∞ ¢ OLPȱ [ + [ = ∞ ȱȱ£ȱȱȱȱ [→−∞ [→−∞ ȱȱȱȱ¡ F ∈ȱ,5 ȱ Iȱ(F)= ǯȱȱȱǰ [ + [ =ȱ ȱ øȱȱǯȱ ȱ ȱȱ¡ D ∈ȱ,5 \ E∈ ,5 ȱ Iȱ(D) = \ I(E)= ǯȱ Ǽ Iȱ ȱȱ [D ȱE] ȱ Iȱ ǯȱ Ǽ Iȱ ȱȱ (D ȱE) ȱ Iȱ ǯȱ Ǽ I (D) =ȱ I(E)ǲȱȱȱȱȱ¤ȱǯȱ ȱȱȱȱȱȱ¡ Fȱ∈(D E) WT I′(F)= ǯȱȱȱ¡ Fȱ∈(DE) WT F + = Fȱ +≠ ȱF ∈ȱ,5ǯȱ ȱȱȱȱȱàǰȱȱȱȱȱȱȱȱ ȱàȱǰȱȱȱȱȱȱǯȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ¡ D ≠ȱE ȱ I (D) =ȱ= I(E) ȱ ȱ¡ȱȱøȱøȱȱȱ Iȱ(F)= ǯȱ Şǯȱ ȱȱȱà [ + [+ =ȱ ȱ¡ȱȱÇȱǯȱȱ àǱȱ ¡ Iȱ( [)= [ + [+ ȱȱȱ Iȱ ȱȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ⎞ ⎛ OLP [ + [+ = OLP [⎜+ + ⎟ = ∞ [→ −∞ [ [ ⎠ ⎝ ⎞ ⎛ OLP [ + [+ = OLP [⎜+ + ⎟ = ∞ [→ +∞ [→ +∞ [ [ ⎠ ⎝ I(− )= − + = − [ȱ→−∞ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ Fȱ ∈(−∞−) ¢ Fȱ ∈(− +∞) Iȱ(F )= ¢ Iȱ(F )=ǯȱ [ȱ +[+ = ȱȱȱȱȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à [ȱ +[+ = ȱ ȱ øȱ àȱ ȱ ȱ ȱ ȱ ȱ (−ȱ∞−) ¢ (ȱ+∞) ǯȱȱ ȱ¡ [ȱ \ [ ȱ ȱ (−ȱ∞−) ȱ I ([) =ȱ= I([ )ǯȱ Ǽ Iȱ ȱȱ [[ȱ [] ȱ Iȱ ǯȱ Ǽ Iȱ ȱȱ ([ȱ [) ȱ Iȱ ǯȱ Ǽ I ([) =ȱ I( [) ȱȱȱȱ¤ȱǯȱ ȱ ȱ ȱ ȱ ȱ ¡ F ∈ȱ( [ [) ȱ Iȱ′(F)= ǯȱ ȱ ȱ ¡ F ∈ȱ( [ [) ȱ Fȱ += Fȱ += ⇔ F = − ¢ −ȱ∉ ( [[) ȱȱȱȱ àǯȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ àȱ ȱ ȱ (−ȱ∞−) ȱøǯȱ ȱȱ£ȱ¤ȱȱ¢ȱȱȱàȱȱȱàȱȱ ȱ (−ȱ+∞) ȱøǯȱ ȱȱȱȱà [ȱ +[+ = ȱ¡ȱȱÇȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 şǯȱ ȱȱȱà [ + [ + [ − =ȱ ȱ¡ȱȱÇ£ȱ ȱȱȱȱȱ ]ȱ[ǯȱȱ àǱȱ ¡ ȱ Iȱ( [)= [ + [ + [− ȱȱȱ Iȱ ȱȱȱȱ [ȱ]ǯȱ¤ Iȱ()=− ¢ Iȱ()=ǯȱ ǰȱȱȱȱȱȱȱ¡ Fȱ∈() Iȱ(F)= ǯȱ ȱ ȱȱ¡ D ȱ∈() ¢ Eȱ∈() ȱ Iȱ(D) = I(E)= ǯ Iȱ ȱȱȱȱȱ [D ȱE] ¢ȱȱ (D ȱE)ǯȱ ȱȱȱȱȱ¡ F∈ȱ(D E) ȱ Iȱ′(F)= ǯȱ ǰ Fȱ +F + = Fȱ +F + ≠ ȱ F ∈ȱ,5ǯȱ ȱ Δȱ= − ()()= −ǰȱ àǰȱ [ȱ +[ + [− = ȱȱøȱàȱȱ ǯȱ ŗŖǯȱ ȱȱȱàȱ ȱ ȱȱ [DȱE] ¢ȱ I ′ [ =ȱ ȱȱ ¡ȱ ȱǻǰȱǼǯȱȱ I [ =ȱ [− D+ I D ȱȱ¡ [DȱE] ǯȱȱ àǱ Iȱ ȱȱ [D ȱE]⇒ I ȱȱ [D ȱ[] ȱ [∈ȱ(D E) ¢ȱ Iȱ′( [)= ȱȱ¡ȱȱǻǰȱǼȱȱȱȱȱ Iȱ ȱȱ ȱ ǻǰ Ǽȱ Iȱ ȱȱ (D ȱ[) ȱ [∈ȱ(D E)ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ F∈ȱ(D [) ȱ Iȱ( [) − I(D) I([) − I(D) I ′ȱ(F) = ǰȱ ȱ = ǯ [− D [− D ⇒ȱ I ( [) − I(D) = [− D⇒ I( [) = I(D)+ [− Dǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ŗŗǯȱ ȱȱȱà [ + [+ =ȱ ȱ¡ȱȱÇ£ȱǯȱȱ àǱȱ ¡ Iȱ( [)= [ + [+ ȱȱȱ Iȱ ȱȱȱ ǯ ( ) ( ) OLP [ + [+ = −∞ \ OLP [ + [+ = ∞ ǯȱ ȱ ȱ ȱ ȱ ȱ [ȱ→−∞ [→ −∞ ȱȱ¡ Fȱ∈,5 WT I(F)= ǯȱ ȱ ȱȱ¡ D ≠E WT I(D) = ȱ= I(E) Iȱ( [)= [ + [+ ȱ ȱ ȱ Iȱ ȱȱ [D ȱE] ¢ȱȱ (D ȱE)ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ Fȱ∈(D E) WT I′(F)= ǯȱ ȱ ȱ ¡ F∈(D E) WT Fȱ += Fȱ +≠ ȱ F ∈ȱ,5ǯȱ ȱȱȱȱȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ D ≠E WT I(D) = ȱ= I(E)ǰȱ ȱȱȱȱàȱȱȱà Iȱ( [)= ȱøǯȱ ŗŘǯȱ ȱȱ <[ȱ FRV [ > − [ ȱ ǯȱȱ àǱȱ I ( [) =FRV[+ [ȱ − ⇒ I′( [)= − VHQ[+ [ I ′( [)> ȱ⇔ − VHQ[+ [> ⇔ [> VHQ[ǯȱ ȱ £ȱ ȱ ¤ȱ Iȱ( [)= [ ¢ Iȱ( [)= VHQ[ ȱȱȱȱȱȱ [ȱ>VHQ[ ȱǰȱȱȱȱȱà [ >VHQ[ ∀[> ∀[> ǯȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ J ( [) =− VHQ[+ [⇒ J′( [) = − FRV [+ ⇒ J′([)> ⇔ − FRV[+ > ⇔− FRV [ȱ> −⇔ FRV[< FRV [ȱ< ȱ [ ∈ȱ,5ǯȱ ȱ ȱ ȱ ȱ Jȱ ȱȱàȱȱȱ [ ∈ȱ,5ǯȱȱȱàǰȱȱ [ȱ > ȱ Jȱ( [) > J( )⇒ − VHQ[+ [ > ȱȱȱ [ȱ> ȱ [ >ȱVHQ[ǯȱ [ȱ>VHQ[ ∀[> ȱȱ Iȱ ȱȱȱ [ȱ> ǰȱ ȱ àȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ [ȱ > ȱ I ( [) >ȱ I()ǯȱȱȱȱȱȱȱȱ [ȱ > ȱȱ FRV[ + [ − > FRV+ [ − ⇒ FRV[ȱ+ − > − ǯȱ ȱȱ [ȱ > ȱȱ FRV[ >ȱ− ȱȱ < [ < ŗřǯȱ πȱ [ ǯȱ ȱȱ [>ȱ [ ǯȱȱ àǱȱ I ( [) =WDQ[− [⇒ I′( [)= VHF [− Iȱ′( [) > ⇔ VHF [− > ⇔ (VHF[− )(VHF[+ )> ⎤ Π⎡ [ȱ∈⎥ ⎢ ȱ ȱ VHF [ȱ−> \ VHF[+ > VHF [ȱ−> ǲȱ ⎦ ⎣ Π ⎤ Π⎡ ȱȱȱȱ ⎥ ⎢ Iȱ′( [)> ǯȱȱȱȱ <[ȱ< ȱȱȱ ⎦ ⎣ I ( [)=WDQȱ[− [ ȱǯȱ ⎤ Π⎡ [ȱ∈⎥ − ⎢ ⎣ ⎦ [ȱ> ⇒WDQ [ȱ− [> ǯȱ ȱȱȱ <[ȱ< I( [) > I( )⇒ WDQ[− [> WDQ− Π WDQȱ[ > [ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 [= ⎧ ⎪ ŗŚǯȱ Ƕȱ·ȱȱȱ ǰȱ ȱ ¢ȱ ȱ ȱà I [ = ⎨− [ + [+ D < [< ǰȱ ⎪P[+ E ȱ ≤ [≤ ⎩ ȱȱàȱȱȱȱȱȱȱȱȱȱǽŖǰŘǾǵȱȱ àǱȱ ǰȱ£ȱȱȱȱȱàȱȱȱȱǯ Iȱ( [)=− [ + [+ D ȱȱàȱȱ Iȱ ǰȱȱȱ ]ȱ[ǯ Iȱ( [)=P[+ E ȱȱàȱȱ Iȱ ȱȱ ]ȱ[ǯȱ ȱ Iȱ ȱ ȱ ȱ ȱ ȱ ȱ Ŗȱ ȱ ȱ ȱ OLP+ I( [) = I() [ → ( ) OLP+ I( [) = OLP+ − [ + [+ D = Dǰȱ Iȱ ȱ ȱ ȱ ȱ ȱ ȱ Ŗȱ [→ȱ [→ D ȱ= ǯ D ȱ= ȱ Iȱ ȱȱ [ȱ = ȱȱȱ OLP I( [) = I() [→ OLP I( [) = P+ E⎫ ⎪ I( [)ǰȱ¡ǰȱȱ¢ȱȱ P + E= − ⎬ ⇒ OLP [→ ( ) OLP I [ = − ⎪⎭ [→− [→+ȱ I (ȱ)=P+ Eǰȱȱȱ Iȱ ȱȱ [ȱ= ȱȱ Pȱ+ E= − ǯȱ ȱ Iȱ ȱ ȱ ȱ ȱ £ȱ [ȱ = ȱ ȱ ȱ OLP I( [) = I() ȱ ȱP +E = P+ E ¢ȱȱȱȱȱȱ ȱȱ [ →ȱ− ȱ¢ȱǯȱ ǻŗǼȱȱȱ Iȱ ȱȱ [ȱ] Dȱ = \ P+ E = −ǯȱ ǰȱȱȱȱȱȱàȱ () Iȱ( [)=− [ + [+ D ȱȱàȱȱ Iȱ ȱȱ () 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 I ȱ( [)=P[+ E ȱȱàȱȱ Iȱ ȱȱ (ȱ) ȱ Iȱ ȱȱȱŗȱȱȱȱ OLP [ȱ→ I( [) − I() ¡ [− I( [) − I() − [ + [+ − (P+ E) − [ + [+ − P− E = OLP = OLP OLP [ → [→ [→ [− [− [− I( [) − I() P[+ E− (P+ E) P( [− ) = OLP = OLP = P OLP [→ [ → [ → [− [− [− − − + − + − − [ + [+ − (P+ E) I( [) − I() ǰȱ ¡ǰȱ ȱ ¢ȱ ȱ OLP = P ȱ ȱ ȱ [ȱ→ [→ [− [− ǲȱ OLP( [− )= ȱȱȱǱ OLP− − [ + [+ − P− E = OLP − ȱ [ȱ→ − ( ) [→ ( ) OLP − [ + [+ − P− E = ⇔ −+ + − P− E= ⇔ − P− E= ⇔ P+ E= [ȱ→ − Ǳ ( ) − [ + [+ − − [ + [− − [ − [+ = P ⇔ OLP = P⇔ OLP = P [− [− [− [ȱ→ − [→− [→− − ([− )( [− ) ⇔ OLP = P⇔ OLP − ([− )= P⇔ P= − [− [→ [→− OLP Pȱ= ⇒ E= ǻŘǼȱ ȱȱ Iȱ ȱȱ (ȱ) Pȱ= ¢ Eȱ= ȱǻŗǼȱ¢ȱǻŘǼȱȱȱȱ Iȱ ȱȱȱȱȱȱ ȱȱ (ȱ) D ȱ= ǰ Pȱ= ¢ Eȱ= ǯȱ ŗśǯȱ ȱ VHQE− VHQD ≤ E−ȱD ȱȱȱøȱ ȱ ¢ȱ ǯȱȱ àǱȱ I ( [)= VHQ[ Iȱ ȱȱ¢ȱȱȱ ǰȱȱȱȱ ȱ øȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ Iȱ ȱ ȱ [D ȱE] ¢ȱ ȱ (D ȱE)ǯȱ ȱȱȱȱȱȱȱȱȱȱ¡ F∈ȱ(D E) ȱ I(E) − I(D) VHQE− VHQD ȱȱ¡ F∈ȱ(D E) ȱ FRVȱF = ǯ I ′ȱ(F) = E− D E− D 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ àȱ ȱ [−ȱ] − ≤ FRVFȱ ≤ ǲȱ VHQEȱ− VHQD − ≤ ≤ ǯȱ E− D D ȱ<E⇒ E− D> ǰȱȱȱȱǱ −ȱ(E − D) ≤ VHQE− VHQD≤ (E− D)ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ¢ȱ VHQE −ȱVHQD ≤ E− Dǰȱ E −ȱD≤ E− D VHQE −ȱVHQD ≤ E− D ǯȱ ŗŜǯȱ Iȱ′ [ = ȱȱ¡ȱȱȱ (D ȱE) Iȱ ȱȱ (D ȱE)ǯȱ ŗŝǯȱ I ′ȱ [ = J′ [ ȱȱ¡ȱȱȱ (D ȱE) I ȱ[ =J [ + &ǰȱ &ȱ ȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱÇȱ¢ȱȱȱȱàȱ Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱàȱȱȱȱȱȱȱǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱȱǰȱȱȱȱȱȱ¤ȱȱȱàǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱȱǰȱȱȱȱȱȱ¤ȱȱȱàǯȱ ȱȱȱÇȱ¢ȱȱȱȱàȱȱȱ ȱȱȱȱ¢ȱȱǯȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱȱȱȱǯȱ ȱ ȱ ȱ ȱ ȱ ȱ øȱ Çȱ ¢ȱ øȱ ȱ ¡àǯȱ ȱȱȱȱ¡à ȱȱàǯ ȱȱÇȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à Iȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȃȄǰȱ ȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ȱ ȱȱȱǰȱȱȱȱǰȱȱȱ ȱȱàȱȱȱȱȱ ǯȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ à Iȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȃȄǰȱ ȱ ȱ ȱ ȱ ȱ ȱ£ȱȱ ȱȱȱ ȱǰȱȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ àȱ ȱȱȱȱǯ %ȱ $ $ % 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àǱȱ ȱȱȱȱà Iȱ ȱȱȱȱȱȱ ǰȱȱȱȱȱ I ( [) ≤ȱ I( [) [ȱ[ ∈, ¢ [ <ȱ [ǯȱ ȱȱȱȱà Iȱ ȱȱȱȱȱȱ ǰȱȱȱȱȱ I ( [) ≥ȱ I( [) [ȱ[ ∈, ¢ [ <ȱ [ǯ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ƽȱ ǻ¡Ǽǰȱ ȱ ȱ (FȱI(F))ǰȱ¤ȱȱ I ′ȱ F ȱȱȱȱ ǯȱ ȱȱȱȱàȱ ȱȱȱȱȱȱȱȱȱȱȱȱȱȱ àȱȱȱȱǰȱȱȱȱ£ȱȱ¤ȱǱȱ Ǽȱȱȱȱ¤ȱǰ Iȱ ȱ ȱ ȱ ȱ ȱȱ ȱ ȱ ȱ Iȱ′ D > ¢ Iȱ′E > ǲȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱ ȱȱȱ£ȱ Iȱ′F > ǻȱ ȱ ȱ ȱ ȱȱȱȱǼȱ Ǽȱȱȱȱ¤ȱǰ Iȱ ȱ ȱȱ ȱȱ ǯȱȱȱ Iȱ′ D < ¢ Iȱ′E < ǯȱ ȱ ¤ǰȱȱȱȱȱȱȱ ȱȱ ȱȱȱ£ȱ Iȱ′F < ǻȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Ǽȱȱ Ǳȱ ȱȱàȱȱȱȱȱ ǰȱȱ Ǽ Iȱ ȱȱȱȱȱȱ¢ȱàȱ Iȱ′ [ > ȱȱ¡ ȱ ǯȱ Ǽ Iȱ ȱȱȱȱȱȱ¢ȱàȱ Iȱ′ [ < ȱȱ¡ȱȱ ǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǳȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱȱȱȱàǱ •ȱ ȱȱȱȱ •ȱ ¢ȱȱȱàȱȱàȱȱȱǰȱøȱȱȱǯȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ àȱȱ ǯȱ ȱ ǰȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ǯȱ ȱ ¢ȱ ȱ ǻȱ ȱ ȱ ȱ ȱ àǼȱ ȱ ȱ ȱ àȱ ȱ ȱ àȱǻȱȱǼȱȱ ȱȱÇȱȱȱàǯȱȱ Ǳȱ ȱ ȱà I ,5→,5 I [ =ȱ [ ǰȱ¢ȱ¤ȱȱȱǱȱ ȱȱȱȱȱȱȱȱȱ [ȱ∞) ¢ȱȱ ȱȱ (−∞ȱ]ǯȱȱȱȱȱȱȱÇȱ [ȱ∞) ¢ (−∞ȱ]ǲȱȱ Iȱ′ = ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱ àǰȱȱȱȱ·ȱȱȱǰȱȱȱȱÇȱȱ ǻȱ ȱ àȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ øȱ Çȱ ȱ ȱ àǰȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ¡ǯȱȱ ȱ ȱà I ȱ,5→,5ǰ I [ =ȱ [ ȱȱ I ′ȱ [ = ǰȱȱȱ [ ȱȱ (−∞ȱ] ¢ȱȱ [ȱ∞) ǯȱȱȱÇǰȱȱàȱȱȱȱȱ [ȱ∞) ¢ȱ ȱȱȱ (−∞ȱ]ǰȱȱ øȱÇȱȱŖǰȱ ȱ I ′ȱ ȱ ¡ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱàȱȱ ȱȱȱ ȱȱ¡ƽȱȱ¢ȱȱȱȱȱ ȱȱȱȱȱȱȱȱȱȱàǰȱȱȱȱȱȱ ȱøȱȱÇȱȱȱàǰȱȱȱȱøȱÇȱȱȱàȱȱȱ øȱȱǯȱȱ ȱ ȱà Iȱ ,5−{} → ,5− {}ǰ I [ =ȱ [ ȱȱ I ′ȱ [ = − [ ǰȱȱ ȱ ȱ ȱ (−∞ȱ) ¢ȱ ȱ (ȱ∞) ǯȱ ȱ ȱ Çȱ ȱ àȱ ȱ ȱ ȱ ȱ (ȱ∞) ¢ȱȱȱȱ (−∞ȱ)ǰȱȱȱȱȱȱȱ ȱȱŖȱȱȱøȱÇǰȱȱȱŖȱȱȱȱȱȱȱàǯȱȱ ȱ ȱ ȱ ȱ ȱ àȱ ǰȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ (−∞ȱ D ]ǰ [Eȱ] ¢ [Fȱ∞) ǲȱ ¤ȱ ȱ ȱȱȱ [D ȱE] ¢ [ȱF]ǯȱȱȱȱøȱȱȱȱ¢ȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱȱ¤¡ǰȱȱ£ȱ¤¡ȱȱȱ¡ȱƽȱǰȱ¡ȱƽȱŖȱ¢ȱȱȱȱ ȱȱȱȱȱȱÇǰȱȱ£ȱÇȱȱȱ¡ȱƽȱȱ ¢ȱ¡ ƽȱǯȱ ȱȱȱȱȱȱȱàǯȱȱ ȱȱȱȱȱ ȱȱȱȱøȱÇȱȱȱàȱȱǯ •ȱ I ′ȱ ȱȱȱȱȱȱǰȱ Iȱ ȱȱ¤¡ȱ ȱȱǯ •ȱ I ′ȱ ȱȱȱȱȱȱǰȱ Iȱ ȱȱÇȱ ȱȱǯ •ȱ I ′ȱ ȱȱȱȱȱǰȱ Iȱ ȱȱ¡ȱȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱàȱ àǱȱ ȱ ȱ ¤ȱ Iȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ àȱȱȱȱ ǯȱȱȱȱȱȱȱȱȱ ǰȱȱȱàȱ ȱȱȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ £ȱ ȱ ǰȱ ȱ I ′ȱ D < I ′ E ǰȱ ȱ ȱȱȱȱȱȱȱ Iȱ ȱȱàȱȱ¢ȱȱȱȱ ȱȱȱȱȱǰ I ′ȱ′ ȱȱàȱǯȱ ȱȱȱ¤ȱȱȱȱȱȱ ȱȱ¢ȱ ȱ ȱ £ȱ ȱ £ȱ ȱ ǰȱ ȱ I ′ȱ D > I′ E ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Iȱ ȱ ȱ àȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱȱȱȱǰ I ′ȱ′ ȱȱàȱǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǳȱ ȱȱàȱȱȱȱȱ ǯȱ Ǽȱ I ′ȱ ȱȱȱ ȱ Iȱ ȱàȱȱȱȱ ǯȱȱǰȱ Iȱ′′ [ > ȱȱ¡ȱȱȱ ǰȱȱȱ¤ȱ Iȱ ȱàȱȱȱ ǯ E ȱ I ′ȱ ȱȱȱ ȱ Iȱ ȱàȱȱȱȱ ǯȱȱǰȱ Iȱ′′ [ < ȱ ȱ ¡ȱ ȱȱ ǰȱ ȱ ȱ ¤ȱ Iȱ ȱ àȱ ȱ ȱȱ ǯȱ àǱȱ ȱȱȱȱ ȱȱ¡ȱƽȱȱȱȱȱ¡àȱȱȱȱȱ ǯȱȱȱȱȱȱȱ¡àȱ (Fȱ I(F))ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡àȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ¡ȱȱȱȱȱȱǯȱȱȱȱȱȱȱȱȱ ȱ ¡ȱƽȱǰ ȱȱȱȱȱȱ¡àȱ ¡ȱƽȱǯȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ¡ȱ ƽȱ ǰȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ¡àǰȱ ȱ ȱȱ Iȱ′′(F)= ǯȱȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱȱȱȱȱ¡ȱȱȱȱàǱȱȱ ȱȱȱȱȱ I ′ȱ′ ȱȱȱȱȱȱȱǯȱ Ǽȱ Iȱ′F = ¢ Iȱ′′ F > ǰ Iȱ ȱȱÇȱȱȱǯȱ Ǽȱ Iȱ′F = ¢ Iȱ′′ F < ǰ Iȱ ȱȱ¤¡ȱȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ¤ȱ ȱȱȱǯȱȱ ŗǯȱ ȱ ȱ ȱ à I ( [)=D[ȱ + D[ + D[ + E[+ Fǰȱ ȱ ǰȱ ȱ ¢ȱ ȱ ȱ ȱǰȱȱȱȱȱȱ¡àǯȱȱ àǱȱ I ([)=D[ + D[ + D[ + E[+ F ⇒ Iȱ′([) = D[ + D[ + D[+ E⇒ I′′([) = D[ + D[+ D ( ) I ′′([) > ⇔ D[ + D[+ D> ⇔ D[ + [+ > ⇔ [ + [+ > ȱ Δȱ= − ȱ ȱ ȱ ȱ [ȱ +[+ > ȱ ȱ øȱǯȱȱǰȱȱȱàȱȱȱȱȱȱȱ¢ȱȱȱȱ ȱ ȱȱȱȱ¡àǯȱȱ Řǯȱ ȱ ȱ ¤¡ȱ ¢ȱ Çȱ ȱ ¢ȱ ȱ ȱ ȱ ¡àȱ ȱ ȱ à I ( [)=ȱ[ − [ǯȱȱ àǱȱ ¤ȱȱȱ¡ȱ I ([) =[ − [ ⇒ I′([) = [ − [ ⇒ I′([) > ⇔ [([− )> ⇔ [ȱ( [− )> ȱ Ŗ ¡ȱ Ȭȱ Ƹȱ Ƹȱ ř¡ȱȬȱŚȱ Ȭȱ Ȭȱ Ƹ I ′ȱ( [) Ƹȱ Ȭȱ Ƹ I ( [) 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱȱȱȱȱ [ȱ= ȱȱ¤¡ȱȱ Iȱ()=ǯȱȱȱȱȱȱȱȱ [ȱ= ȱȱÇȱ ⎛ ⎞ ȱ Iȱ⎜ ⎟ = − ǯȱ ⎝ ⎠ ¤ȱȱȱȱȱ¡àǱ Iȱ′′( [) = [ − [ ⇒ I′′( [) = [( [− ) I′′( [) > ⇔ [( [− )> ⇔ [− > ⇔ [> ¡ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ [ȱ = ¤ I ′ȱ() ȱȱ¡àȱ (−ȱ)ǯȱȱ řǯȱ ȱȱȱȱ¡àȱ I ( [)=ȱ [ − [ ǯȱȱ àǱ Iȱ( [) = [ − [ ⇒ I ′( [) = [ − [⇒ I ′′( [) = [ − I′′( [)> ⇔ [ − > ⇔ [ > ⇔ [ > I ( [)=ȱ [ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱȱǱ [ȱ > ⇔ [ > ⎛ ȱ ⎞ ȱȱȱȱ¡àȱ ⎜⎜ − ⎟⎟ ǯȱȱ ⎝ Śǯȱ I ([)=ȱ ⎠ D[+ E ǰȱȱǰȱǰȱȱ¢ȱȱȱ (Fȱ≠G ≠ )ǯȱȱ Iȱ F[+ G ȱȱȱ¤¡ȱȱÇȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱȱ I ([)= I ′( [) = D[+ E F[+ G (F[+ G)D− (D[+ E)F = DF[+ DG − DF[− EF = DG − EF (F[+ G) (F[+ G) (F[+ G) (F[ȱ+ G) > ǰȱȱȱ¡ȱǲȱ¤ (DG −ȱEF) ȱȱȱǯȱ I ′ȱ( [) ȱȱ ȱǰȱǻȱȱȱȱȱȬǼǰȱ Iȱ ȱȱ¡ȱǯȱȱ śǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à Iȱ( [)= ȱ àȱ ȱ [ − ȱ¢ȱȱȱȱȱàȱȱǯȱȱ àǱ ( ) − Iȱ([) = [ − ( ) − ⇒ I′([) = − [ − −[ ⋅ [= ⇒ ([ − ) − ([ − ) + [• ([ − )[ − ([ − )[[ − − [ ] − (− [ − ) [ + I′′([) = = = = ([ − ) ([ − ) ([ − ) ([ − ) I′′([) > ⇔ [ + ([ − ) ( ) ( ) > ⇔ [ − > ⇔ [ − > ⇔ [∈ (− ∞−) ∪ (+∞ ) ȱȱàȱȱ (−ȱ∞−) ¢ (ȱ+∞) ǯȱ ȱȱàȱȱ [−ȱ]ǯȱȱ Ŝǯȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ àȱ ȱ I ( [)= [ȱ + D[ + E[+ F ȱ ȱ ¤¡ȱ ȱ [ȱ=− ¢ȱ [ȱ= ȱ Çȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àǱ I ( [) = [ȱ + D[ + E[+ F⇒ I′( [)= [ + D[+ E ȱ ȱȱ¤¡ȱȱ [ȱ=− Iȱ′( − )= ǯȱ ȱ ȱȱÇȱȱ [ȱ= Iȱ′()= ǯȱ ȱȱ (−) + D (− )+ E= ⇒ − D+ E= () +D ȱ()+ E= ⇒ + D+ E= ⎧−D ȱ+ E= ȱȱȱȱ ⎨ ȱȱ D ȱ=− ¢ ⎩+ D+ E= Eȱ=− Ǳȱȱȱȱȱȱȱȱ D ȱ=− ¢ Eȱ=− ǯȱȱ ŝǯȱ I ( [)=D[ȱ + E[ + F[+ Gǰȱȱǰȱǰȱȱ¢ȱȱȱȱȱȱȱ ȱ ¡ȱȱ (ȱ) ¢ȱȱ¤ȱȱȱȱȱ¡àȱ (−ȱ)ǯȱȱ àǱ I([) =ȱD[ + E[ + F[+ G ⇒ I′( [) = D[ + E[+ F⇒ I′′([)= D[+ E ȱȱȱ¡ȱȱ (ȱ) Iȱ′( ) = \ I()= ǯȱ ȱȱȱȱȱ¡àȱ (−ȱ) Iȱ′′() = \ I()= −ǯȱ ȱȱǱ D ⋅+ E⋅ + F= ⇒ F= D ȱ⋅ + E⋅ + F⋅ + G = ⇒ G = D+ E= D+ E+ F+ G = −⇒ D+ E+ = − Ǳ D ȱ+E= ⎫ ⎬ ⇒ D = −⇒ D = − D+ E = −⎭ D =− ⇒ + E =− + ⇒ E= − =− 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ Ǳȱȱȱȱȱȱ D ȱ =− E = − F = \ G = ǯȱȱ Şǯȱ ȱȱȱǰȱǰȱȱ¢ȱȱȱȱ I ( [)= D[ȱ + E[ + F[+ Gǰȱȱȱ ¤¡ȱȱ (ȱ) ¢ȱȱÇȱȱ (−ȱ)ǯȱȱ àǱ I([) =ȱD[ + E[ + F[+ G ⇒ I ′([) = D[ + E[+ F⇒ I ′′([)= D[+ E ȱ ȱȱ¤¡ȱȱ (ȱ) Iȱ′( ) = \ I()= −ǯȱ ȱȱǱ D ⋅+ E⋅ + F= ⇒ F= D ȱ⋅ + E⋅ + F⋅ + G = ⇒ G = D+ E+ = ⇒ D+ E= D+ E+ + = −⇒ D+ E= − D +E ⎫ −D ȱ− E= − D ȱ = −⇒ D = ǯȱ ⎬⇒ D+ E= −⎭ D+ E= − Ǳ D ȱ= Eȱ= −D⇒ E= − ǯȱȱ Ǳȱȱȱȱȱȱ D ȱ = E= − F = \ G = ǯȱȱ şǯȱ I( [) =ȱ [ − (D+ E)[ + DE[ ȱǀǯȱ ȱȱ¤¡ȱ¢ȱÇȱ ȱȱǯȱȱ àǱ ( ) Iȱ′( [) = [ − (D+ E)[+ DE= [ − (D+ E) + DE = ( [− D)( [− E)> ȱ ȱ ¡Ȭȱ Ȭȱ Ƹȱ Ƹȱ ¡ȱȬȱȱ Ȭȱ Ȭȱ Ƹ I ′ȱ( [) Ƹȱ Ȭȱ Ƹ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Iȱ( [) ȱÇȱȱȱȱ I (E) =E − (D+ E)E + DE =ȱE − DE − E + DE = DE − E + DE ȱ¤¡ȱȱȱȱ I(D) = − (D+ E)D =ȱȱD − D − DE+ DE = DE− D ŗŖǯȱ I ( [)=D[ȱ + E[ + F[+ Gǰȱ D ≠ȱEǯȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱȱ¡àǯȱȱ àǱ I ′( [) = D[ȱ + E[+ F⇒ I′′( [) = D[+ E I′′([)> ⇔ D[+ E> ⇔ D[+ E> ⇔ D[> −E D ȱ> D[ >ȱ −E⇔ [ > − E ǯȱ D ⎛ E ⎛ E ⎞ ⎞ ȱȱȱȱȱȱ¡àȱ ⎜⎜ − I⎜ − ⎟ ⎟⎟ ǯȱ ⎝ ȱD ⎝ D⎠ ⎠ D ȱ< D[ >ȱ −E⇔ [< − E D ⎛ E ⎛ E ⎞ ⎞ ȱȱȱȱȱȱ¡àȱ ⎜⎜ − I⎜ − ⎟ ⎟⎟ ⎝ ȱD ⎝ D⎠ ⎠ ȱȱǰ Iȱ ȱ¡ȱȱȱȱ¡àǯȱȱ ȱȱȱȱ£àȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ \ = − [ ȱ ȱ ¤ȱ ¤ȱ à¡ȱ ȱ ȱ ǻŖǰŘǼǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱŚȱ¢ȱŜȱȱȱȱȱȱȱŝǰśȱǯȱȱȱȱȱȱ ȱ ȱ ǲȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ¢ȱȱȱǰȱȱȱȱȱȱȱǯȱǶȱ·ȱȱȱȱȱ ȱȱȱȱȱȱȱȱȱȱȱ£ȱȱÇǵ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ŜŘǰśȱ ȱřȱǯȱ ȱȱȱȱ£ȱȱȱȱȱȱ£ǯ π ȱ ȱȱȱÛȱȱȱȱȱȱȱȱȱȱ 9 = ǯȱ ȱ Ƕ¤ȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ£ȱȱ Çǵ ȱ ȱ ¤ȱȱ¤ȱŞȱȱŘȱǯȱȱȱ£ȱ ȱÇȱ ȱȱ ȱȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ǯȱ Ƕ¤ȱ ȱ ȱ ȱÇȱȱȱȱǵ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ŗŖŞȱ ȱ ȱ ȱ ¡ǯȱ ȱ ¤ȱȱȱȱȱŚȱȱȱȱȱȱ¢ȱȱ¤ȱȱ ȱ ȱ ȱ řȱ ȱ ȱ ȱ ȱ ǯȱ Ƕ·ȱ ȱ ȱ ȱ ¤ȱ £ȱȱȱȱȱǵ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ Çȱ ǰȱ ȱ ȱ ȱ ¤ȱ¤¡ȱȱȱǯ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ Pȱ ȱ ȱȱȱȱȱȱȱȱÇǯȱȱȱȱ ȱȱȱȱȱ¤¡ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¤ȱȱǯȱ Ǳȱ ȱȱ£ȱȱȱȱ¤ȱȱȱȱàǰȱ ȱȱǱȱȱȱȱȱǰȱ¤ȱȱȱȱ¢ȱȱ ǰȱ ȱ ¡ǰȱ ȱ ȱ ¡àǰȱ ¤ȱ ȱ ȱ ȱ ȱ ǰȱ ¤ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ¢ȱ à ȱȱȱàǯȱ ȱȱȱøȱȱȱȱàǯȱ £ȱ ȱ ¤ȱ ȱ ȱ àǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ¤¡ǰȱ¤ǰȱȱȱȱȱ¢ȱȱǯ ȱ ȱ àȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱȱ¤ȱȱǯȱȱ Ǳȱ ȱȱ¤ȱȱȱȱà I [ =ȱ [ − [ à •ȱ ȱȱȱ¤¡ȱ ȱȱȱàȱȱȱǰȱȱ¤¡ȱȱȱ •ȱ £ȱȱȱȱȱ ȱȱȱàȱȱȱǰȱȱȱȱȱ •ȱ £ȱȱ¡ȱȱ − [ = ∞ ǰȱ ¤ OLP [ − [ = −∞ ȱ ȱ ¢ȱ ȱ ȱ OLP [ [ȱ→−∞ [ȱ→∞ ¤ȱȱȱàȱȱȱÇȱ£ǯ •ȱ £ȱȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çǰȱ Çȱ ȱȱ¡ȱȱȱàǯ I ′ȱ [ = [− [ = [ − [ ȱȱȱȱȱÇȱȱȱ£ȱȱȱȱȱ ȱǰȱȱȱȱȱȱȱàǱ Iȱ′ [ > 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 I ′ [ > ⇔ [ȱ−[ > ¢ȱȱȱȱ¤ȱȱȱà I ′ȱ [ = [− [ ȱ ȱ ¤ȱ àȱ ȱ ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ȱ ]ȱ[ǯȱ Iȱ ȱ ȱ ȱ ȱ [ȱ] ¢ȱ ȱ ȱ ȱ ]−∞ȱ] ¢ [ȱ∞[ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Ŗȱ ¢ȱ ȱ Řǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ [ȱ= £ȱ ȱ Çȱ ǰȱ ȱ Iȱ = ¢ȱ [ȱ= £ȱ ȱ ¤¡ȱ ǰȱ ȱ Iȱ =ǯ •ȱȱ £ȱȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ àȱ ȱ ȱ ¢ȱ ȱ ȱ àȱ ȱ ǰȱ Çȱ ȱ ȱ ȱ ȱ ¡àȱȱȱȱàǯ I′ȱ′ [ = − [ ȱȱȱȱàǱ Iȱ′′ [ > I ′′ [ > ⇔ −[ > ⇔ [ȱ< ȱ ȱ ¤ȱ Iȱ ȱ àȱ ȱ ȱ ȱ ȱ ]−∞ȱ] ¢ȱ àȱȱȱȱȱ [ȱ∞[ ȱȱȱȱȱȱȱȱȱȱŗȱȱ¡ȱȱ ȱȱ¡àȱȱǻŗǰŘǼǯ •ȱ ȱȱà •ȱ ¤ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Ǳȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱ¤ȱȱȱȱà Iȱ [ = [ [ − à •ȱ ȱȱȱ¤¡ȱ ȱàȱȱȱǰȱȱȱȱȱ¤¡ȱȱȱȱȱȱ øȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱȱȱȱȱà Iȱ ,5 −{−} •ȱ £ȱȱȱȱȱ ȱȱȱàȱȱȱǰȱȱȱȱȱȱǯȱȱ ȱȱȱ ȱȱȱȱȱȱ¡ƽȬŗȱ¢ȱ¡ƽŗȱ OLP [ = +∞ [ȱ→+ [ − ¢ ȱȱȱȱȱàȱ¡ƽŗǯ OLP [ [ȱ→− [ − = −∞ ȱ ¡ȱ ȱ Çȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD OLP /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ [ [ȱ→−+ [ − = +∞ ¢ OLP [ [ȱ→−− [ − = −∞ ȱ ¡ȱ ȱ Çȱ ȱȱȱȱȱàȱ¡ƽȱȬŗǯ •ȱ £ȱȱ¡ȱȱȱ OLP [ [ȱ→−∞ [ − = ǰȱ ¤ [ OLP [ȱ→+∞ [ − = ȱ ¡ȱ ȱ Çȱ £ȱȱȱȱȱàȱ¢ƽŖ •ȱ £ȱȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çǰȱ Çȱ ȱȱ¡ȱȱȱàǱ Iȱ ′ [ = − [ + [ − ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ȱǰȱȱȱȱȱȱȱàǱ Iȱ′ [ > Iȱ′ [ < ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ àǰȱ ȱ ȱ ȱ ȱ ȱ ȱ Iȱ ȱȱȱȱȱǱ ]−∞ȱ−[ ǰ ]−ȱ[ ¢ ]ȱ+∞[ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ¡ȱǻȱ¤¡ǰȱȱÇǼǯ •ȱ £ȱȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ àȱ ȱ ȱ ¢ȱ ȱ ȱ àȱ ȱ ǰȱ Çȱ ȱ ȱ ȱ ȱ ¡àȱȱȱȱàǱ Iȱ ′′ [ = [ [ + [ − ȱȱȱȱàǱ Iȱ′′ [ > I ′′ [ > ⇔ [ [ + [ − > ⇔ [ȱ [ − > ǰȱȱ [ȱ + ȱȱ ȱȱȱȱ¡ǰȱ¤ [ȱ − ȱȱȱȱȱȱǰȱȱȱȱ ¡ȱȱǯȱ ȱȱȱà [ȱ > ǰȱȱȱȱȱȱ [− [+ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ǰȱȱȱȱȱȱ¤ȱȱȱàȱȱàȱȱȱ ȱ ȱ ȱ Ǳ ]−ȱ] ¢ ]ȱ+∞[ ¢ȱ àȱ ȱ ȱ ȱ ȱ Ǳ ]−∞ȱ−[ ¢ [ȱ[ǯȱ Ŗȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ¢ȱȱȱȱȱ¡àȱȱǻŖǰŖǼ •ȱȱ ȱȱà •ȱ ¤ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱ¤ȱȱȱȱà Iȱ [ = Ǳȱ [ [ + à •ȱ ȱȱȱ¤¡ȱ ȱàȱȱȱǰȱȱȱȱ¤¡ȱȱȱȱȱȱ øȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱȱȱȱȱà Iȱ ,5 •ȱ £ȱȱȱȱȱ ȱ ȱ ȱ àȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱȱ¤ȱȱȱàȱȱȱÇȱǯ •ȱ £ȱȱ¡ȱȱȱ OLP [ [ȱ→−∞ [ + = −∞ ¢ OLP [ [ȱ→+∞ [ + = +∞ ǰȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ¡ȱ ȱ ¢ȱ ȱ ȱ ȱ ǯȱȱȱȱ¤ȱȱȱàȱȱȱÇȱ£ǯȱ ¡ȱ ȱ ȱ ȱ Çȱ ȱȱ Çȱ ǰȱ ȱ ȱ ȱ \ȱ=P[+ E FRQ P≠ ǯȱȱ1ȱȱ ȱȱȱ ȱ ȱ£ȱ ȱ ǰȱȱȱ ȱȱ¤ȱȱȱà Iȱ [ ȱȃȱȄȱȱȱȱȱȱȱ ǯȱȱȱȱȱȱȱȱȱȱÇǰȱȱȱȱ Pȱ ¢ Eȱ ȱȱȱȱȱȱȱȱȱǱȱȱȱ Iȱ [ ¢ \ =ȱP[+ E ȱ ȱ £ȱ ¤ȱ Ûȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ¢ȱ ǯȱ ¤ȱȱȱȱǱ OLP I [ − P[+ E = ǯȱ [ȱ→∞ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ Çȱ ȱ àȱ ǯȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ǰȱ ȱ ȱ Iȱ [ = [ [ + = [− [ [ + ǯȱȱȱ I [ = [− [ [ + HQWRQFHV Iȱ [ −[= [ [ + ⎛ [ ⎞ ⎛ ⎞ ⎜ ⎟ ƽ ⎜ − [ ⎟ = ǯȱ ¢ȱȱȱ OLP I [ − [ ƽ OLPȱ − [ OLP [ →ȱ∞ [→∞ ⎜ [+ ⎟ [ȱ→∞ ⎜⎝ [+ ⎟⎠ ⎝ ⎠ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ Çȱ ȱȱà \ = [ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 •ȱȱ £ȱȱȱȱ ȱȱȱ ȱȱȱȱȱȱȱÇǰȱÇȱ ȱȱ¡ȱȱȱàǯ Iȱ ′ [ = [ [ + [ + ȱȱȱȱȱÇȱȱȱ£ȱȱȱȱȱ ȱǯȱȱ Iȱ′ [ > ȱȱ¡ȱȱȱȱȱàǰȱȱ ȱȱȱȱȱ Iȱ ȱȱȱȱȱǱ ]−∞ȱ−∞[ ȱȱȱȱȱȱȱȱȱȱàȱȱȱ ȱ¡ȱǻȱ¤¡ǰȱȱÇǼǯ •ȱ £ȱȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ àȱ ȱ ȱ ¢ȱ ȱ ȱ àȱ ȱ ǰȱ Çȱ ȱ ȱ ȱ ȱ ¡àȱȱȱȱàǱ Iȱ ′′ [ = ȱ [ − [ [ + ȱ ȱ ȱ ȱ ȱ ȱ ¡ǰȱ [ȱ + [ȱ + ȱȱȱȱȱȱǰȱȱȱȱ¡ȱȱǯȱ ȱ £ȱ ȱ ȱ [ −ȱ[ + [ ǰȱ ȱ ȱ ȱ ȱ ȱ Ǳ Ŗ −ȱ Ȭȱ Ƹȱ Ƹȱ Ƹ ȱ+[ ( ȱ−[ ¡ȱ )( ) [ ȱ−[ − [ Ƹȱ Ƹȱ Ƹȱ Ȭȱ Ȭȱ Ƹȱ Ȭȱ Ȭȱ Ƹȱ Ƹȱ Ƹ Ȭȱ ȱȱȱȱ ȱ¤ȱȱȱàȱȱàȱ ȱȱȱȱ ȱ Ǳ −ȱ ¢ ȱ∞ ¢ȱ àȱ ȱ ȱ ȱ ȱ ] ] [ [ ] ] [ [ Ǳ −ȱ∞− ¢ ȱ ǯȱ ȱ ȱ ȱ ȱ ¡àȱ −ȱ − ǰ ȱ ¢ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD •ȱȱ ȱȱà •ȱ ¤ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱȱàȱȱȱȱǰȱ¢ȱȱ¤ȱȱȱ àȱȱȱȱȱȱȱ àǱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ I [ = /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 [ ǯȱ ȱ ȱ ȱ ȱ ȱ ǻǰȱ ȱ ȱ ȱ [−ȱ ǰȱ ȱ ȱ ǰȱ ȱ ¡ǰȱ ǰȱ ȱ ȱ ¡àǰȱÇǼȱ¢ȱȱȱ¤ȱȱ I [ = [+ + [− I′ [ = [ − [ ( [− ) ȱ I′′ [ = ( [−) àǱȱ Ǳ ,5ȱ−{} ȱȱȱǱȱǻŖǰŖǼȱ ȱȱÇȱ¢ȱȱ¡ Iȱ′( [) > ⇔ [ − [ > ⇔ [( [− ) > ⇔ [∈ (− ∞) ∪ (+∞ ) ([− ) ȱȱǱ ]−ȱ∞] ¢ [ȱ+∞[ ȱȱǱ [ȱ[ ¢ ]ȱ] ȱȱ¤¡ȱȱȱŖȱ¢ȱȱ£ȱ [ȱ = Ǳ (ȱ) ȱȱÇȱȱȱŚȱ¢ȱȱ£ȱ [ȱ = Ǳ (ȱ) ȱ¢ȱȱȱ¡à Iȱ′′( [) > ⇔ > ⇔ ( [− ) > ⇔ [− > ⇔ [ > ([− ) ȱȱàȱȱ ]ȱ+∞[ ȱàȱȱ ]−ȱ∞[ ȱ¢ȱȱȱ¡àǯȱ ¤ȱȱȱȱȱȱ ȱ ȱȱȱ [ = OLP [→+ [ =∞ [− OLP [ȱ→− [ = −∞ [− ȱ¢ȱÇȱȱȱȱȱȱà [ = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¤ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ¡ȱ ȱ ǯ [ OLP =∞ [ȱ→∞ [− [ OLP = −∞ [→ −∞ [− ȱ¢ȱÇȱ£ȱ Ǳ [ȱ = + + [− [− [ȱ→ ±∞ Iȱ( [) ȱ¡ȱȱ \ȱ =[+ ȱ¢ȱȱÇȱȱȱȱȱȱà \ȱ =[+ ȱȱà 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱȱ¢ȱȱȱ¤ȱ I [ = [ [ − [ I ′ [ = [ − ( [ − ) I ′′ [ = [ + [ ( [ −ȱ) àǱȱ Ǳ ,5ȱ−{− } ȱȱȱǱȱǻŖǰŖǼȱ ȱȱÇȱ¢ȱȱ¡ Iȱ′( [) > ⇔ [ − [ ([ − ) > ⇔ [ − [> ⇔ [([ − ) > ( ) ( ⇔ [ − > ⇔ [∈ − ∞ ∪ ] ] [ ȱȱǱ −ȱ∞− ¢ +∞ [ ) [ [ ] ] ȱǱ −ȱ − ǰ ]−ȱ[ ¢ ȱ ȱ+∞ ȱȱ¤¡ȱȱ ⎛ −ȱ − ⎞ ⎟ ¢ȱȱ£ȱ [ȱ =− Ǳ ⎜⎜ − ȱ ⎟ ⎝ ⎠ ȱȱÇȱȱ ⎛ ⎞ ȱ ⎟ ¢ȱȱ£ȱ [ȱ = Ǳ ⎜⎜ ȱ ⎟ ⎝ ⎠ ȱ¢ȱȱȱ¡à Iȱ′′( [) > ⇔ [ + [ ([ ) − > ⇔ ( ) > ⇔ [ [ + ([ ) − ([ [ )> − ȱ ȱ àȱ ȱ ]−ȱ] ¢ ]ȱ+∞[ ȱȱàȱȱ ]−ȱ∞[ ¢ [ȱ[ ȱȱ¡à (ȱ) ¤ȱȱȱȱȱ OLP [→+ [ = +∞ [ − [ = +∞ OLP [→− [ − + OLP [→− [ = −∞ [− [ = −∞ OLP [ȱ→− [ − − ȱ¢ȱÇȱȱȱȱȱȱàǱ [ȱ = − ¢ [ = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ¤ȱȱȱȱȱàȱȱȱ¡ȱ ǯ [ OLP = +∞ [ȱ→+∞ [ − [ OLP = −∞ [→ −∞ [ − ȱȱ¢ȱÇȱ£ȱ Ǳ [ȱ [ = [+ [ − [ − [ȱ→ ±∞ [ȱ → ¢ Iȱ( [) ȱ¡ȱȱȱȱȱà \ =ȱ [ǯȱ [ − ¢ȱÇȱȱȱȱȱȱà \ =ȱ [ǯȱ ȱȱà 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱ¤ȱȱȱàȱȱȱȱȱǱȱ ¡ȱ Ȯ∞ȱ ∞ ȮŗȱȱȱȱȱȱȱȱȱȱȱŖȱȱȱȱȱȱȱȱȱȱȱȱŗ I ′ [ ȱ Ȯȱ Ȯȱ Ƹȱ Ƹ I ′′ [ ȱ Ȯȱ Ƹȱ Ƹȱ Ȯ OLP I [ = − OLP I [ = − OLP I [ = −∞ [→ −∞ [→ ∞ [→ −− [→ − [→ −+ [→ − OLP I [ = +∞ àǱ OLP I [ = +∞ OLP I [ ȱ= −∞ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱ¤ȱȱȱà ȱȱȱàȱ¢ȱǱȱ Ǽȱ ȱȱ¤¡ȱȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽ OLPȱ I( [) ƽȱȱȏȏȏȏȏȏȏȏȏȏȏȱ [→−+ Ǽ OLPȱ I( [)ƽȱ ȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽ OLPȱ I( [) ƽȱ ȏȏȏȏȏȏȏȏȏȏȏȏȱ OLPȱ I( [)ƽȱ ȏȏȏȏȏȏȏȏȏȏȏȏȱ [→− [→−∞ Ǽ [→+∞ Ǽȱ ȱȱàȱȱȱà ǻ¡Ǽÿ ÿŖ Ǳȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽȱ ȱȱàȱȱȱà I ′(ÿ [)ÿ ÿŖ Ǳȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽȱ ȱȱàȱȱȱà I ′′ÿ( [)ÿ ÿŖȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱȱ àǱȱ Ǽ ,5ȱ−{− } Ǽ OLPȱ I( [) = +∞ [→−+ Ǽ Ǽ Ǽ OLP I( [)= Ǽ Ǽ OLP I( [)= − Ǽ [ȱ→ − [ȱ→−∞ OLPȱ I( [) = ∞ [→+∞ ]−ȱ∞−[ ∪ ][ ]ȱ+∞[ ]−∞−[ ∪ ]+∞[ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ I [ = /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 [ + [+ȱ ǯȱȱ ȱȱȱȱ¢ȱȱȱ¤ȱȱ ǯ [ [ − ȱ I ′′ [ = I ′ [ = [ [ àǱȱ Ǳ ,5ȱ−{} ȱȱȱǱȱ ȱȱȱ¢ǲȱȱ ȱ∉' I ǯȱ ȱȱȱ¡ǲȱȱ Iȱ([)≠ ȱ [ ∈ȱ'I ǯȱ ȱȱÇȱ¢ȱȱ¡ Iȱ′( [) > ⇔ [ − > ⇔ [ − > ⇔ [∈ (− ∞−) ∪ (+∞ ) [ ȱȱǱ ]−ȱ∞−] ¢ [ȱ+∞[ ȱȱǱ [−ȱ[ ¢ ]ȱ] ȱȱ¤¡ȱȱȱȬŗȱ¢ȱȱ£ȱ [ȱ = − Ǳ (−ȱ−) ȱȱÇȱȱȱřȱ¢ȱȱ£ȱ [ȱ = Ǳ (ȱ) ȱ¢ȱȱȱ¡à Iȱ′′( [)> ⇔ [ > ⇔ [ > ȱȱàȱȱ ]ȱ+∞[ ȱàȱȱ ]−ȱ∞[ ȱ¢ȱȱȱ¡àǯȱ ¤ȱȱȱȱȱȱ ȱ ȱȱȱ [ = OLP [→ + [ + [+ = +∞ [ OLP [→ȱ − [ + [+ = −∞ [ ¢ȱȱÇȱȱȱȱȱȱà [ȱ = ¤ȱȱȱȱȱàȱȱȱ¡ȱȱǯ [ + [+ =∞ [→∞ [ OLP [ + [+ = −∞ [→ −∞ [ OLP 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ¢ȱÇȱ£ȱ [ +ȱ[+ Ǳ = [+ + [ [ [ȱ→ ±∞ ⇒ [ → Iȱ( [) ȱ¡ȱ \ȱ =[+ ȱ¢ȱȱÇȱȱȱȱȱȱà \ =[+ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ I [ = + [ −[ȱ Ǳ I ′ [ = /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ǯȱȱ ȱȱȱȱ¢ȱȱȱ¤ȱȱ ǯȱ [ (− [) I ′′ [ = [ + (−[) [ + ǯȱȱȱ ȱȱȱǯȱ [ [ − [ + ȱǱ I ′ [ = I ′′ [ = [ [ ȱ I [ =ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ àȱ ȱ ȱ ǰȱ ǰȱ ǰȱ ȱ Çǰȱ Çȱ ¢ȱ ¢ȱ ȱ ¤ȱ ȱ ȱ ȱ àǱ I [ = [ − [ + [ [ &RQVLGHUH I ′ [ = I ′′ [ = − [ ȱ ȱ ȱ ȱ ȱ ȱ àȱ I [ ǰȱ ȱ ȱ ¤ȱ Ǳȱ Ȯ∞ȱ I ′ȱ [ Ƹȱ Ƹȱ Ȯȱ Ȯ I ′ȱ′ [ Ƹȱ Ȯȱ Ȯȱ Ƹ Iȱ [ ȱ ¢ƽŗȱ Ƹ∞ȱ Ȯ∞ȱ Ȭŗȱ ȱȱȱȱ I [ = I ′ [ = ȱ ∞ ȮŗȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱŖȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱŗ [( [+ ) (+ [) Ȯ∞ȱ Ƹ∞ȱ [ (ȱ+[) I ′′ [ = ¢ȱÇ [ (+[) ȱȱȱȱ¢ȱȱȱ¤ȱ I [ = I′ [ = − [ ( [− ) I ′′ [ = ¢ƽŗ ( [ − ) ( [−) [ − [− ( [−) 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱàȱ ȱ ȱȱȱȱàǱȱ ȱ ȱ I − = − I = − I − = I = I ′ − = I ′ = I ′ [ =ȱ [ < − I ′ [ <ȱ ȱȱ ]−ȱ−[ ¢ ][ I ′ [ >ȱ ȱȱ ]−ȱ[ ¢ ]+∞[ I′′ [ >ȱ ȱȱ ]−ȱ[ ¢ ][ I′′ [ <ȱ ȱȱ ]+∞ȱ[ ¢ ][ OLP I [ =ȱ [→ ∞ ȱȱ¤ȱȱȱȱȱà ȱ ȱȱ¤ȱ \ = [ −ȱ[ǰȱǱȱǰȱȱȱàǰȱ ȱ¢ȱȱ¡ǯ ȱ I [ =ȱ [− DUFWDQ[ǯȱȱ ȱȱȱȱ¢ȱȱȱ¤ȱȱ ǯȱ Ǳ I ′ [ = [ − I ′′ [ = [ + [ ([ + ) ȱ I [ =ȱH− [ ǯȱȱ ȱȱȱȱ¢ȱȱȱ¤ȱȱ ǯȱ Ǳ I ′ [ = −[H−[ ȱ ȱǱ I [ = I ′′ [ = H− [ [ − [ + [− ȱ ǰ I [ = −[ ȱ ¢ I [ = − [ − [ +[ [ȱ Ǽȱǰȱȱ¡ǰȱȱÇȱȱȱ¤ȱȱȱà ǯȱ ȱȱ ȱ ȱ Çȱ ¢ȱ ȱ ¡ǯȱ ǻàȱ ǰȱ ǰȱ ȱ ¤¡ȱ¢ȱÇȱǼȱ Ǽȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ¢ȱ ȱȱ¡àǰȱÇǯȱ Ǽȱȱȱȱȱàȱȱ¢ȱȱ¤ȱȱȱ ȱȱ¤¡ȱ ǯ [ − [ + ȱàȱȱȱȱ¤¡ȱȱǯȱ [−ȱ [ − [ + I [ = ¢ I [ = ( [−ȱ) ( [−)ȱ ȱ I [ = ȱȱǰȱÇǰȱȱȱÇǰȱȱÇǰȱ ȱ¢ȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ ȱǱ Iȱ [ = /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 [ [ − ǰ Iȱ [ = −[ [ − ¢ Iȱ [ = [ + [ − Ǽȱ ǰȱȱ¤¡ȱȱȱȱàǯȱ Ǽȱ ǰȱȱ¡ǰȱȱÇȱȱȱ¤ȱȱȱà ǯȱ Ǽȱ ȱȱȱȱÇȱ¢ȱȱ¡ǯȱǻàȱǰȱ àȱȱǰȱȱȱ¤¡ȱ¢ȱÇȱȱǼȱ Ǽȱȱȱȱ¢ȱȱȱȱǯȱ ȱ¢ ȱȱ¡àǰȱÇǯȱ Ǽȱ ȱȱȱȱàȱȱ¢ȱȱ¤ȱȱ ȱ ȱǱ Iȱ [ = [ [+ ǰ Iȱ [ = [+ [ [+ ¢ Iȱ [ = [ + [+ [+ Ǽȱ ǰȱȱ¤¡ȱȱȱȱàǯȱ Ǽȱ ǰȱȱ¡ǰȱȱÇȱȱȱ¤ȱȱȱà ǯȱ Ǽȱ ȱȱȱȱÇȱ¢ȱȱ¡ǯȱǻàȱ ǰȱàȱȱǰȱȱȱ¤¡ȱ¢ȱÇȱȱǼȱ Ǽȱ ȱȱȱ¢ȱȱȱȱǯȱ ȱ¢ ȱȱ¡àǰȱÇǯȱ Ǽȱ ȱȱȱȱàȱȱ¢ȱȱ¤ȱȱȱǯ ȱȱȱȱȱȱà I ȱ[π[→ ,5ǰ I [ =ȱFRV [ + VHQ [ ¢ ȱ¤ȱȱàǯ ȱ ȱ I ȱ[ = [H[ ǯȱȱ ȱȱȱȱ¢ȱȱȱ¤ȱȱ ǯȱ Ǳ I ′ [ = [+ H[ I′′ [ = [+ H[ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Ǳȱ ȱȱȱȱȱàȱǯȱ ȱȱȱȱȱȱàȱǯȱ ȱȱȱȱ ȱǰȱȱȱȱȱȱȱȱàȱǯȱ ȱ ǰȱ ¡ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ¤ȱ ȱȱȱȱǯȱȱǱȱ ȱȱÇȱȱȱȱȱȱÇǰȱȱǰȱȱȱȱ ȱ ȱ àȱ ȱ ȱ ǲȱ ¢ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ǲȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱǻȱ Ǽǯȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àǯȱȱȱȱȱȱȱà ) ȱ ȱȱȱ Iȱ ǰȱ ȱ ) ȱ ȱȱȱ Iȱ ǯȱȱ àǱȱ ȱ à ) ȱ ȱ ȱ ȱ ȱ ȱ Iȱ ȱ ȱ ȱ ȱ ) ′ȱ [ = I [ ȱ [ ∈ȱ, ǯ ȱ ) ȱ [ =−[ + [ − [− ) ȱ′ [ = −[ + [− ǯȱ ȱ ȱ ȱ ȱ ȱ Iȱ ȱȱàȱȱ Iȱ [ =−[ + [− ǰȱȱȱ ) ȱ ȱȱȱ Iȱ ǯȱ *ȱ ȱ ȱ àȱ ȱ *ȱ [ =−[ + [ − [+ *ȱ ·ȱȱȱȱ Iȱ ǯȱ +ȱ ȱ ȱ àȱ ȱ +ȱ [ =−[ + [ − [+ +ȱ ·ȱȱȱȱ Iȱ ǯȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ǰȱȱȱȱøȱȱȱǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱǰȱȱȱȱȱȱȱȱȱȱȱ ǰȱȱ ǰȱȱȱȱ ǰȱ ·ȱȱȱȱȱȱ ǯȱ ȱƽȱȱƸȱ Ǳȱ ) ȱ ȱ ȱȱȱ Iȱ ȱ ȱ ȱ ǰȱȱȱ ȱȱ Iȱ ȱ ȱ¤ȱȱǱ ) ȱ[ +& ǰȱ &ȱ ȱȱǯȱȱ àǱȱ *ȱ ȱȱȱ Iȱ ȱ ǰȱ * ′ȱ [ = I [ ȱȱ¡ȱ ȱ ǯȱ ȱ ) ȱ ȱ·ȱȱȱ Iȱ ȱ ǰȱȱȱ ) ′ȱ [ = I [ ȱȱ¡ȱȱ ǯȱ ȱȱǰȱȱȱȱ * ′ȱ [ = )′ [ ȱȱ¡ȱȱ ǯȱ ȱ ȱ ǰȱ ¡ȱ ȱ &ȱ ȱ * ȱ[ = ) [ + & ȱ ȱ ¡ȱ ȱ ǯȱ ǻȱȱȱŗŝȱȱȱȱȱǼǯȱ *ȱ ȱ ȱ ȱ Iȱ ȱ ǰȱ ȱ ¢ȱ ȱ ȱ ȱ Iȱ ȱ ȱ ȱ ȱ ȱ ) ȱ[ +& ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ¢ȱȱȱȱȱǯ \ =ȱ) [ ȱȱȱ Iȱ ǰȱȱȱȱ ) ȱ [ ȱȱàȱ G\ ȱȱà =ȱ I [ ǯȱ G[ ȱȱȱȱàȱȱȱǰȱȱȱȱȱȱ ȱȱ G\=ȱ I [ G[ǯȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ǻȱ ȱ ȱ Iȱ Ǽȱ ȱ ȱ àȱ ȱ ȱ àȱ ¢ȱ ȱ àȱ ȱ ȱ ȱ ∫ \ = I ȱ[ G[ = ) [ + & ǯȱ £ȱ àȱ ȱ àȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ àȱ ǻ I ȱ[ G[Ǽȱ ·ȱ ȱ Çȱ ȱ àȱ ǻ ∫ȱ Ǽǯȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ àǰȱ Iȱ [ ǰȱ ¢ȱ &ȱ ȱ ȱ àǯȱ ȱ ȱ ȱ Ç ȱ ȱ ȱ ȱ ȱ ȱ ∫I ȱ[ G[ Iȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ǯȱ ȱ ȱ ȱ ¡ȱ ǻ¡Ǽȱ ȱ ȱ ȱȱ¡ȱ ȱȱȱȱàǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ·ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ·ȱȱȱȱàȱȱǯ ȱ £ȱ ȱ ȱ ȱ àȱ ¢ȱ ȱ àȱ ȱ ȱ àȱȱàȱȱȱȱȱȱàȱȱàǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱǱȱȱ ȱ¤ȱȱ à ∫ NI [ G[=N∫ I [ G[± ∫ J [ G[ ∫ I [ ± J [ G[= ∫ I [ G[± ∫ J [ G[ ∫ NG[ =N[+ & [Q Q ∫ [ G[= Q+ + & Q≠ − ∫ FRV [G[=VHQ[+ & ∫ VHQ[G[=− FRV [+ & + ∫ VHF [G[=WDQ[+ & FVF ∫ [G[=− FRWDQ[+ & ∫ VHF [WDQ[G[=VHF[+ & ∫ FRV HF[FRWDQ[G[=− FRVHF[+ & ∫ H G[=H + & ∫ [G[=OQ[ + & G[ ∫ − [ = VHQ [+ & [ [ − G[ ∫+ [ G[ ∫[ [ − ȱ = VHF = WDQ−[+ & − [+ & 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ¤ȱ ȱȱ ȱ ŗǯ Řǯ řǯ Śǯ śǯ Ŝǯ ∫ȱ ∫( ∫ȱ \ + \ − G\ \ ∫ ∫ ∫ ) FVF [− VHF[WDQȱ[ G[ Ş [ − [ + [ G[ [ şǯ (+ [) ∫ ∫ ŝǯ ([ ȱ ȱ ŗŖǯ − [ + VHQ[− H[ G[ ( ) [ [ −[+ [ G[ ŗŗǯ G[ WDQȱ [+FRW [− G[ VHQ[ FRVȱ [ G[ȱ ) VHQ[ ∫ [ ȱFRW[−VHQ[ ∫ ∫ G[ ȱ [−[ G[ WDQȱ \+G\ ȱ ȱ ȱ ȱ ȱ àȱ \′ȱ′ = [ǰȱ ȱ ȱ ȱ \ȱ′= ¢ \ȱ =ǯ ȱȱà \ =ȱ I [ ȱȱȱȱǱ I′′ [ = H[ + VHQ[ I′ = I = ȱ [ +[) ( I [ ȱ I ′ [ = ¢ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ [ȱ ȱǻŗǰŚǼǯ ȱ ȱȱǻȮŗǰřǼȱ¢ȱǻŖǰŘǼȱ¤ȱȱȱȱ¢ȱȱȱȱǻ¡ǰȱ¢Ǽȱȱ ȱ I ′′ [ = −ȱ[ǯȱȱȱȱàȱȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 G\ = [ [ − \ = − G[ [ ȱ ȱ ¢ȱ ȱȱȱǱ ȱ ȱ ǻ¡Ǽȱ ȱȱȱǱȱ [ I ′′ [ = ȱ ȱ¤ȱȱ ȱ ȱȱȱȱǻŚǰřǼǯȱ ȱ ȱȱȱȱȱ¤ȱȱȱȱȱǻŚǰřǼȱȱȱŘǯ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Ǳ 4 = 4′ = 4′′ = ȱ 4′′′ = ǯ ȱ ȱàȱȱȱȱȱȱȱȱȱȱȱǻŗǰřǼȱ \ = [ + ǯȱȱȱ ȱ ȱ ȱ ȱ ǻ¡ǰȱ ¢Ǽȱ ȱ ȱ ȱ ȱ ȱ G\ G[ =ȱ ǰȱ ȱ ȱ àȱȱȱǯ ȱ ȱÇǰȱȱȱȱȱÇȱȱ¢ȱȱàȱ¤ȱ¡ȱ FP ¢ȱ ȱ £ȱ ȱ V D W =ȱ+ W− W ǯȱ ȱ ȱ ȱ ȱ Y =ȱ− ȱŘȱȱȱȱŗŖȱǯȱȱȱȱȱà ȱàǯ ȱ ȱ Çǰȱ ȱ ȱ ȱ ȱ ȱ ȱ D W =ȱVHQW+ FRVW ǰ Vȱ = ¢Vȱ π = ǯȱȱȱȱȱà ȱàǯ ȱ ȱȱȱȱȱȱàȱȱȱȱȱȱȱŗŞŖŖȱ àǲȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ G9ȱ =W + W+ ǰȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à GW ȱȱȱȱȱȱÛȱ·ȱȱȱǯȱȱȱȱȱàȱ ȱ ¤ȱ ȱ ȱ ŗŖȱ Ûȱ ǰȱ Ƕ¤ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱȱŚȱÛȱȱȱȱȱǵ ⎧ VL[< ⎪⎪− [ ȱ K ȱ[ = [ ¢ +ȱ ȱǱ +ȱ [ =⎨ ⎪ [ VL[≥ ⎪⎩ ȱ Iȱ ȱȱȱ Iȱ (−∞ȱ∞ ) ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱàȱȱ Ǳȱ ȱȱ ȱȱ£ȱȱàȱȱǯȱ ȱȱȱȱȱǯȱ ȱ¢ȱȱȱȱǱ •ȱ ȱȱȱȱȱHQWHURVȱǯ •ȱ ȱȱȱȱȱȱȱȱȱȱǯȱ ȱȱÇȱȱȱǯ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ǰȱ ȱ ȱ ȱ £¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱǯȱ ȱȱȱȱȃȄȱȱȱȱǯȱȱ ȱ Ǳ 6ȱ = + + + + ǰȱȱȱȱȱȱȱȱȱ ȱȱǰȱȱ ȃȱȱȱȱȱǰȱ¤ȱǰȱȱǰȱ¤ȱǰȱ¤ȱ Ȅǰȱ ȱ ·ȱ ȱ ȱ ȱ ȱ ȱ ¡Çȱ ȱ ȱ ǰȱ ȱ ǰȱȱȱȱ ȱȃȱȱȱȱȱøȱ ǰȱȱȱ ȱȱȱȱŗȱ ȱȱȱȱȱśȄǯȱȱȱøȱȱȱȱȱȱȱȱȱȱ Ǳ 6ȱ = ∑ Lǯȱ L= 1ȱ ȱ ȱ ȱ ȃàȱ Ȅȱ ȱ ȱ ȱ ǻȱ ȱ Ȭ ∑ȱ Ȭȱ ȱ ¢øǼǲȱ ȱ ȱ ·ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱǰȱȱȱȱȱȱƽȱŗȱƸȱŘȱƸȱřȱƸȱǳȱƸȱśŘŞǰȱȱ ȱ ¤ȱàȱȱȱȱȱȱȱøȱ ǰȱȱȱȱøȱȱȱȱ ŗǰȱȱȱȱȱȱȱǰȱ¢ȱ 6ȱ = ∑ Lǯȱȱ L= 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àȱ D PDP+ȱ DP+ DQ ȱ øȱ ȱ Pȱ ¢ Qȱ ȱ ȱ P ≤ȱQǰȱ Q ∑ D ȱL ƽD P + DP+ + DP+ + + DQ L=P D ȱL ȱȱȱ·ȱȱȱȱǯ ȱȱȱȱàȱȱȱȱǯȱȱ ȱȱȱàȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Q ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ D L ǰȱ ∑ L =ȱ D ȱL ȱ ȱ ȱǰȱȱ DL =ȱF ȱ Lȱ ǯȱȱȱȱȱ ȱ àȱ ȱ ¤ȱ Qȱ ȱ ȱ ø F ǰȱ ȱ ȱ Ǳ Q ∑ F=F+ F+ F+ ȱ+ F= QFǯȱȱ L= 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Q ȱ ȱ ȱ ȱ ȱ ȱ ȱ D ȱL ȱ ȱ ∑ D L L=ȱ Q ∑ FD L ǯȱ ȱ ȱ ȱ ȱ ǰ F ǯȱ ȱ ȱ ȱ ǰȱ ȱ ȱ L=ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ø F ǰȱ ȱ ȱ Q ∑ L Q DL ǯȱ ∑ L FD L =FD + FD + FD + ȱ+ FDQ− + FDQ = F D + D + D + + DQ− + DQ = F = = ÇǰȱȱȱǱ Q Q L=ȱ L= ∑ FD L =F∑ DL ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ·ȱ ȱȱȱǱ Q ȱD L +EL ƽ D + E + D + E ∑ L + D + E + + DQ−ȱ + EQ− + DQ + EQ ǯȱ = ȱ£ȱȱȱȱȱ¢ȱȱȱȱȱ ȱ ȱ ȱ ȱ ȱ øȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Ǳ D + E + D + E + D + E + + DQ−ȱ + EQ− + DQ + EQ ƽ D + D + D + + DQ−ȱ+ DQ Ƹ E + E + E + + EQ−ȱ + EQ ƽ Q Q =ȱ L=ȱ D L Ƹ ∑ EL ∑ L ÇǰȱȱȱǱ ∑ Q Q Q ȱD L +EL = L= ȱȱ¤ǰȱȱǱ ∑ L= DL + ∑ EL L= Q Q Q = = = D L −EL = ∑ DL − ∑ EL ∑ L L L 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱǰȱȱȱȱȱǯȱȱ Ǳȱ ȱȱȱȱǰȱ Q L= Q Q Q D L +EL = L= Q = = FD L =F∑ DL ∑ L L ∑ F=QF ∑ Q ∑ L= DL + ∑ Q EL L= ∑ L Q D L −EL = = ∑ L = Q DL − EL ∑ L = ȱȱȱ ȱȱȱȱȱȱȱȱ¤ȱȱ àǱȱ ŗǯȱ ȱ £¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Qȱ ȱ Q L ∑ L ǰȱȱǱ 6 = =ȱ ȱ¤ȱȱ·ȱȱȱȱ¤ȱ ȱȱ ȱǻŗŝŝŝȬȱ ŗŞśśǼȱȱȱȱȱ£ȱÛǯȱ ȱ 6 =ȱ+ + + + + + Q− + Q− + Q ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ¢ȱ ǰȱ ȱ ȱȱȱȱ 6ȱ =Q+ Q− + Q− + + + + + + ȱȱǱȱ ȱ ȱ ȱ ȱ ȱ ¡ǰȱ ȱ ȱ £ȱ ȱ ȱ Ř ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱȱȱ Qȱ ȱȱȱ Qȱ+ ǯȱ Q Q+ 6ȱ =Q Q+ ȱȱ ǰ 6ȱ = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Q Q Q+ L= ∑ L ȱȱǱ ȱ= Řǯȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Qȱ ȱ Q L ∑ L ȱ ǰȱȱǱ 6 = = ȱȱǰ ȱ £ȱȱȱ ∑ [(ȱ+L) − L] ƽ ( −ȱ) + ( − ) + + ((Q + ) − (Q+ )) + ((Q+ ) − (Q)) Q L= ȱ£ȱȱǰȱȱȱȱȱȱȱȱȱȱ ǯȱ ȱ ȱ ȱ ·ȱ ȱ ȱ ȱ ȱ ȱ ·ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ·ȱ ǰȱ ȱ ȱ ȱ ȱ · Qȱ+ ǰȱ ȱ ȱ ȱ ȱȱøȱ·ǰȱ −ȱ = −ǰȱȱȱȱȱȱȱȱ ·ǯȱ Q ȱȱǰȱ ∑ L ȱ +L − L = Q+ − ǯȱȱȱȱȱȱ·ǰȱȱȱ = ȱȱȱȱȱ·ǰȱ¡ȱȱȱȱǻȱȱ¢ȱȱǼǰȱȱ ȱȱȱȱàǯȱ Q ǰ ∑ L ȱ +L − L ƽ = Q ∑ L ȱ +L + L+ L − L ƽ = Q ȱ +L ∑ L + L ƽ = ⎛ Q ⎞ Qȱ Q+ ȱ Ƹ ȱ LƸ ȱ L ƽ ⎜ ȱ L⎟ Ƹ Ƹ Qȱ ⎜ ⎟ L= L= L= ⎝ L= ⎠ Q Q ∑ Q ∑ ∑ ȱȱǰ ∑ Q ⎛ Q ⎞ ⎜ ȱ L⎟ ƸQȱ Q+ Ƹ Qȱ ƽ +L − L = Q+ − ⎜ ⎟ L= ⎝ L= ⎠ ∑ ∑ ⇒ ⎛ Q ⎞ ⎜ ȱ L⎟ ƸQȱ Q+ Ƹ Qȱ ƽ Q + − ⎜ ⎟ ⎝ L= ⎠ ⇒ ⎛ Q ⎞ ⎜ ȱ L⎟ ƽȱȬ Qȱ Q+ Ȭ Qȱ Ƹ Q + − ⎜ ⎟ ⎝ L= ⎠ ∑ ∑ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Q ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ L ∑ L ȱ ¢ȱȱȱȱȱ = Q ∑ L = Q Q+ Q+ Lȱ= ȱǰȱȱȱȱȱǯȱȱ Q ∑ L L= Q ∑ Q Q+ = L = Q Q+ Q+ L = ȱ¤ȱȱÇȱȱȱȱ ȱȱȱ¤ȱȱȱÇȱȱȱȱ ȱǱ Q ⎛⎜ ⎛ L ⎞ ⎞⎟ OLPȱ ∑ ⎜ ⎜⎜ ⎟⎟ +⎟ ǯȱ Q→ ∞ ⎟ L= Q⎜ ⎝ Q⎠ ⎝ ⎠ Q ȱ⎛⎜ ⎛ L ⎞ ⎞⎟ ȱȱȱ ȱ ∑ ⎜ ⎜⎜ ⎟⎟ + ⎟ Ǳ ⎟ L= Q⎜ ⎝ Q⎠ ⎝ ⎠ Q ȱ⎛ ⎛ L ⎞ ⎞ ȱ⎡ Q ⎛ L ⎞ Q ⎤ Q ⎜ ⎜ ⎟ + ⎟ ƽ ⎢ ∑ ⎜ ⎟ + ∑ ⎥ ƽ ∑ ⎛ L ⎞ + ⋅ Q ƽ Qȱ Q+ Q+ + ⎜ ⎟ ȱ ⎜ ⎝ Q⎠ ⎟ Q⎢ ⎜ ⎟ ⎥ Q Q Q L= ⎝ ⎠ Q L= ⎝ ⎠ ⎣L=⎝ Q ⎠ L= ⎦ ∑ Q+ ⎞⎟ + ⎟ Ǳ ⎟ Q ⎛ Q Q+ ȱ ȱÇ OLP⎜⎜ Q→ ȱ ∞ ⎜⎝ ⎠ ⎛ Q Q+ Q+ ⎞ Q ⎛⎜ ⎛ L ⎞ ⎞⎟ ȱ ⎜ ⎟ ⎟ ⎜ + ƽ + ƽ ⎟ OLP⎜⎜ OLPȱ ∑ ⎜ ⎜ ⎟ ⎟ ⎟ Q→ ȱ ∞⎝ Q→ ∞ ⎟ Q ⎠ L= Q⎜ ⎝ Q⎠ ⎝ ⎠ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ¤ȱ Ǽȱ ȱ¡Çȱȱȱȱ ŗǯ +Lȱ ∑ L řǯ = Řǯ ∑ L Lȱ = Lȱ Śǯ = Ǽȱ ∑ L+ L −L ∑ L + Lȱ+ = ȱ£ȱȱàȱȱȱȱȱȱǯȱ ŗǯȱ ŗȬŗƸŗȬŗƸŗȬŗƸŗȬŗƸŗȬŗȱ Řǯȱ ŗƸśƸşƸŗřƸŗŝȱ +ȱ + + + + + řǯ Śǯ OQ +ȱOQ+ OQ+ OQ+ OQ +ȱ + + + śǯ Ǽȱ ȱ = = = D ȱL = ǰ ∑ ELȱ = − ¢ ∑ FLȱ = ¢ȱ ȱ ȱ ȱ ȱ ∑ L L L Ǳ −D L + EL − FL + ∑ L = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Ǽȱ ȱȱȱȱȱȱȱ ŗǯ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ −Lȱ+ L ∑ L N Śǯ = Řǯ ∑ L = P řǯ ∑ L + L− ∑ L Q +Lȱ śǯ = ȱ L− ∑ L = Q ȱL−L + Ŝǯ = Ǽȱ = +Lȱ Ƹ − ȱ L ∑ L Lȱ ∑ L = ȱȱÇȱȱ ŗǯ Q ⎛ L ⎞ OLPȱ ∑ ⎜⎜ ⎟⎟ Q→ ∞ L= Q⎝ Q⎠ Řǯ ⎞ Q ⎛⎜ ⎛ L ⎞ L ⎟ ⎟ ⎜ OLP ∑ ⎜ ⎜ ⎟ + −⎟ Q→ ∞ Q ⎟ L= Q⎜ ⎝ Q⎠ ⎝ ⎠ +L− 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Ǳȱ ȱȱ¤ȱȱȱȱȱȱàǯ ȱ Çȱ ǰȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ǰȱ ȱ ¤ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱ¢ȱȱǲȱȱȱȱƽȱǯȱ ȱȱȱȱàȱȱ ȱ ȱ àȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱ ȱ ¤ȱ ǰȱ ȱ ȱ ¤ȱ ȱ¤ȱÇǯȱ ȱȱȱȱȱȱȱàȱȱȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ·ȱ ȱ ¡àǯȱ ȱ àȱ¤ȱȱȱȱ·ȱȱȱȱÇȱǻŘŞŝȱȬȱŘŗŘȱ ǯǯǼǯȱ ǰȱȱ·ȱȱȱȱȱÇȱȱȱȱȱȱȱ ¤ȱ ȱ ȱ Çǰȱ ȱ ȱ ȱ ȱ àȱ ¢ȱ ȱ ȱ ȱ ȱ àǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¢ȱ ȱ £¤ȱ ȱ Çȱȱȱȱ£ȱȱȱȱȱȱȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱȱàȱȱÇȱȱȱ¡ǯȱȱ ȱȱ¤ȱȱȱàȱȱȱȱ¤ȱ \ =ȱ [ǰȱȱ ȱ ȱ¡ȱ¢ȱȱȱȱà [ȱ= ¢ [ȱ= ǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ Ŗȱ ¢ȱ Şȱ ȱ ȱ ¤ȱ ȱȱȱ¤ȱ¢ȱȱȱŘȱ¢ȱȱȱ Śǯȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱȱ ȱȱȱȱŘȱ 6ȱ ¢ 6ȱ ȱ£ȱȱȱ [ȱ = ȱ ȱ ȱ ȱ ¡àȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ¤ȱ ¢ȱȱȱȱȱȱȱȱȱàȱ¢ȱ¢ȱȱȱȱȱȱȱȱ ȱȱȱȱȱàǲȱȱȱȱȱȱȱ¤ȱ¤ Iȱ()= = ¢ Iȱ( )== ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱ 5ȱ ȱȱȱȱ¤ȱȱȱ¤ȱȱ ¡àǰȱȱȱ 5ȱ =⋅ + ⋅ = + = ǯȱȱȱ¤ȱȱȱ ȱ ȱȱ 5ȱǰȱȱȱ $ȱ< ǯȱ ȱ ·ȱ ȱ ȱ ¡ȱ ȱ ¤ȱ ȱ ȱ ȱ 6ȱ ¢ 6ȱ ȱ¤ȱ¢ȱȱȱȱȱȱȱȱȱȱ¢ȱ¢ȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ǲȱ ȱ ȱ ȱ ȱ ȱ ȱ¤ȱ¤ Iȱ( )= ¢ Iȱ()=ǯȱ ȱ ȱ ȱ /ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ¤ȱ ȱ ¡àǰȱ ȱ ȱ /ȱ =⋅ + ⋅ = ǯȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ¢ȱ /ȱ ǰȱȱȱȱŗǀǯȱȱȱ¤ȱȱȱȱȱŗȱ¢ȱ śǲȱȱȱŗǀǀśǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ øȱ ¢ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Śȱ Ǳ 6ȱ ǰ 6ȱ ǰ 6ȱ ¢ 6ȱ £ȱ ȱ ȱ [ȱ = ǰ [ȱ= ǰ [ȱ = ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱȱȱàǰȱȱȱ¢ȱȱ¢Ǳȱ ȱ¢ȱȱȱȱ£ȱ¤ȱ¢ȱȱȱȱȱȱȱ ȱ ȱ àȱ ¢ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àǲȱȱȱȱȱȱȱ¤ȱ¤Ǳ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ⎛ ⎞ Iȱ⎜ ⎟ = /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ⎛ ⎞ \ I()= ȱ ȱ ȱ 5ȱ ȱ ⎝ ⎠ ⎝ ⎠ ȱȱȱ¤ȱȱȱ¤ȱȱ¡àȱȱǱ I() = I⎜ ⎟ = ⋅ + ⋅ + ⋅ = = ǯȱ 5ȱ = ⋅ + ȱȱȱȱȱ£ȱ¤ȱ¢ȱȱȱȱȱȱȱ ȱȱ ȱ ¢ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ⎛ ⎞ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ¤Ǳ Iȱ( ) = ǰ Iȱ⎜ ⎟ = ǰ ⎝ ⎠ ⎛ ⎞ Iȱ()=ǰ Iȱ⎜ ⎟ = ǯȱ ⎝ ⎠ ȱ ȱ ȱ /ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ¤ȱ ȱ ¡àȱȱ /ȱ = ⋅ + ⋅ + ⋅+ ⋅ = = ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ¤ȱȱȱȱ / =ȱ< $< = 5 ǯȱ ȱȱȱȱȱȱŞȱǱ 6ȱ ǰ 6ȱ ǰ 6ȱ ǰ 6ȱ ǰ 6ȱ ǰ 6ȱ ǰ 6ȱ ¢ 6ȱǰȱ ȱ ȱ ȱ ȱ ȱ £ȱ ȱ ȱ [ȱ= [= [= [= [= [= [= ǯȱ 5ȱ ȱȱȱǰȱǱ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ = ≈ 5ȱ = ⋅ /ȱ ȱȱȱȱǱ + ⋅ + ⋅ + ⋅+ ⋅ + ⋅ + ⋅ = ≈ /ȱ = ⋅ + ⋅ ȱ ȱ¤ȱȱȱȱ/ ≈ȱ< $< = 5 ǯȱ ȱȱȱȱȱȱȱ¤ȱȱȱȱȱŘǰŗŞŝśȱ¢ȱ řǰŗŞŝśǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱȱȱȱȱ¡àȱȱ¤ȱȱȱàǰȱ ȱ ȱ ȱ ȱ øȱ ȱ ȃȄǯȱ ȱ ȱ ȱ øȱ ȱ ¤ȱ ǰȱ¢ȱ /ȱQ ȱȱȱ¢ 5ȱQ ȱȱ ȱȱȱȱ ¤ǯȱ ȱàȱȱȱȱȱ /ȱQ ¢ 5ȱQ ȱȱȱ ÇȱȱǱȱ /Q 5ȱQ ȱ ȱ ȱ ȱ ȱ ȱ ŗŜȱ řŘȱ ŜŚȱ ŗŘŞȱ ŘśŜȱ śŗŘȱ ȱȱȱȱȱȱȱȱȱȱȱ¤ȱȱȱ ȱ ŘǰŜśŗŖŜŘŖŗȱ ¢ȱ ŘǰŜŞŘřŗŘŖŗȱ ȱ ȱ ŘśŜȱ ǲȱ ȱ ȱ śŗŘȱ ȱ ȱ ȱ øȱ ¤ȱ ¢ȱ ȱ ȱ ȱ ȱ ŘǰŜśŞŞśşŘśȱ ¢ȱ ŘǰŜŝŚŚŞŚŘśǯȱ ȱ ȱ Çȱȱȱȱ≈ȱǯȱ =ȱ ȱȱȱ /Q ȱ = ǯ ȱȱȱ ȱȱȱǱ ȱȱȱȱȱȱȱȱ 5ȱQ ȱ ȱǰȱȱ OLP Q→∞ Q Q→∞ Q ∑ ∑ ⎛ ⎞ ⋅ ⋅ L ⎜ L ⎟ = OLP Q→ ∞ Q→∞ Q⎝ Q⎠ Q Q L= L= = OLP OLP 5Q = OLP Q→ ∞ = OLP Q→ ∞ Q Q ∑ L L = OLP = Q→∞ Q(Q+ )(Q+ ) ⋅ Q ⎞ ⎛ ⎞⎛ Q⎜+ ⎟⎜ + ⎟ Q⎠ ⎝ Q⎠⎝ Q ⎞ ⎛ ⎞⎛ ⎜+ ⎟⎜ + ⎟ Q⎠ ⎝ Q⎠⎝ = OLP = = Q→∞ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ·ȱ ȱ ȱ ȱ ȱ ȱ ȱ /Qȱ ȱ =ȱ ȱ ȱȱȱǰȱȱ OLP /Q = ǯȱ ȱȱȱȱȱȱȱ Q→ ȱ ∞ Ǳ ⎛ (L− ) ⎞ OLP /Q = OLP ⋅ ⋅ L − L+ ⎜ ⎟ = OLP Q→ ∞ Q→ ∞ Q→ ∞ Q Q Q ⎝ ⎠ Q L= L= Q ∑ ⎡ = OLP⎢ ȱ ∞ ⎢ Q Q→ ⎣ Q ∑ L − L= Q Q Q ( ∑ ⎤ ) ⎡ Q(Q+ )(Q+ ) Q(Q+ ) ⎤ − ⋅ + ⋅ Q⎥ Q Q ⎦ ∑ L+ ∑⎥ = OLP⎢ ⋅ Q L= Q L= ⎥⎦ Q→∞ ⎣ Q ⎞ ⎛ ⎞⎛ ⎛ ⎜+ Q⎜+ ⎟⎜ + ⎟ Q⎠ ⎝ Q⎠⎝ = OLP + + = OLP ⎝ Q→∞ Q→∞ Q ⎞⎛ ⎞ ⎟⎜ + ⎟ Q⎠⎝ Q⎠ = = ȱ ȱ ȱ ȱ ȱ /Qȱ 5ȱQ ȱ ȱ ȱ ¡ȱȱȱ¤ȱȱǰȱȱȱȱȱȱȱ¤ȱȱȱȱÇȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ¤ȱ ȱ ¡àǲȱ ȱ Ǳ $ȱ= OLP 5Q= OLP /Q= ǰȱȱȱȱÇȱ¡ȱ¢ȱǯȱ Q→∞ Q→∞ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ǯȱ ȱ ȱ ȱ ȱȱȱ ȱȱàȱȱȱǯȱ ȱȱ [D ȱE]ǰȱȱ¤ȱȱàȱ [D ȱE]ǰȱȱȱȱ ȱ ȱ [D ȱE]ǰȱ 3 =ȱ{[ [[Q}ǰȱ ȱ [L ∈ȱ[D E] ȱ L=ȱQ ¢ D= [ < [ < [ < ȱ< [Q− < [Q = Eǯȱ ȱ ǰȱ ȱ àȱ [D ȱE]ǰȱ ȱ ȱȱȱȱȱȱǰȱȱȱȱȱȱȱȱ¢ȱ ȱøȱǯȱȱ ȱàȱ Qȱ+ ǰȱȱȱȱ ȱȱ ȱ [[ȱ []ǰ [[ [][[Q−ȱ[Q]ǯȱ [[L−ȱ [L] ȱ ȱ ¤ȱ ȱ Ȭ·ȱ ȱ ȱ ȱ àȱ ȱ [D ȱE] ¢ȱ ȱ ȱ ¤ [ȱL −[L− ¢ȱ ¤ȱ ȱ Δȱ[L ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱȱȱȱǰȱȱȱȱȱȱȱȱȱ [D ȱE]ǰȱȱǱ Q Δ[L = Δ[ +Δ[ + + Δ[Q = [ − [ + [ − [ + [ − [ + + [Q ∑ L = = E− D − − [Q− + [Q − [Q− = [Q − [ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ⎧ ⎫ ȱȱ [ȱ] ¢ 3 ȱ = ⎨ ⎬ ȱȱ ⎩ ⎭ ȱ ȱ ȱ ŝȱ ȱ ȱ ¤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ¢ȱ ⎢ȱ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ Δ[ȱ = Δ[ = Δ[ = Δ[ = Δ[ = Δ[ = Δ[ = ǯȱ ȱ ȱ ȱ àȱ ȱ ¢ȱ ȱ I ȱ[DE]→ ,5 ¢ȱ 3 =ȱ{[ [Q} ȱàȱȱǯȱȱȱ¤ȱȱȱȱȱȃȄȱ [D ȱE] ǰȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ¤ȱ ȱ øȱ [[L−ȱ [L] ȱ L=ȱQǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ [[L−ȱ [L]ǯȱ ȱȱȱȱȱȱ¡ȱȱ£ȱȱ¤¡ȱ¢ȱȱÇȱ ȱȱȱȱȱǰȱ 0ȱL ȱȱ¤¡ȱȱȱȱ ¢ PȱL ȱ ȱ Çȱ ȱ ȱ 0 L = I(α L ) PL =ȱ I(β L ) αȱL∈ [[L− [L] ¢ βȱL∈ [[L− [L]ǯȱ ȱȱȱȱȱȱȱȱȱȱàȱȱǱ Q Q L= L= 8I (3) = ∑ 0 Δ[L = ∑ I(αȱL )Δ[L ǯȱ ȱȱȱȱȱȱ ȱȱàȱȱǱ Q Q L= L= /I (3) =∑ P Δ[L = ∑ I(βȱL )Δ[L ȱ 0 LΔȱ[L PLΔȱ[L ȱ ȱ ȱ ·ȱ ȱ ȱ ¤ȱ ȱ ȱ ¤ǰȱ ¢ȱ ȱ Δȱ[L ¢ȱ ȱ 0ȱL ¢ PȱL ǯȱ Çǰȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ¢ȱ ȱ ǰȱ ȱ ȱ [D ȱE]ǰȱȱȱ ȱàȱȱ [D ȱE]ǰȱȱȱȱȱȱ¤ȱ ȱ ȱ ȱ ¤ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ [D ȱE] ȱ ȱ àȱ ǰȱ ¢ȱ ¢ȱ ȱ ȱ ȱ ¤¡ȱ ȱ ȱ ȱ àȱȱȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD 8I (3) = /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 0LΔ[L = 0 Δ[ + 0 Δ[ + 0 Δ[ + 0 Δ[ + + 0 Δ[ ∑ L =ȱ = I( [)Δ[+ I( [)Δ[ + I([)Δ[ + I([)Δ[ + I([)Δ[ + I([)Δ[ + I([)Δ[ + I([)Δ[ ǰȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ¢ȱ ȱ ǰȱ ȱ ȱ [D ȱE]ǰȱȱȱ ȱàȱȱ [D ȱE]ǰȱȱȱȱȱȱ¤ȱ ȱ ȱ ȱ ¤ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ [D ȱE] ȱ ȱ àȱ ǰȱ ¢ȱ ¢ȱ ȱ ȱ ȱ Çȱ ȱ ȱ ȱ àȱȱȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /I (3) = ( ) /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 PLΔ[L = P Δ[ + P Δ[ + P Δ[ + P Δ[ + + P Δ[ ∑ L =ȱ = I [ Δ[ + I( [)Δ[ + I( [)Δ[ + I( [)Δ[ + I( [)Δ[ + I( [)Δ[ + I( [)Δ[ + I( [)Δ[ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ǰȱ ȱȱȱȱȱȱȱȱȱȱȱ¢ȱȱȱ ȱȱȱȱȱȱȱǯȱ ȱǰȱȱȱà 3 =ȱ{[ [Q} [D ȱE]ǰȱȱ¤ȱȱȱȱȱ ȱȱ [ =ȱ D ¢ [ =ȱEǰȱȱǱ /I (3) ≤ȱ $≤ 8I (3)ǯȱȱ ȱ ȱȱȱȱȱȱȱȱàȱǰȱȱȱȱàȱȱ ȱ [D ȱE]ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǻàȱ Ǽȱ ŗǯ Řǯ I ( [)=ȱ [ [ȱ]ǰ Qȱ = Iȱ( [)=[ + [ȱ]ǰ Q = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱ I ȱ[DE]→ ,5 ȱ àȱ ȱ ¢ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ àȱ ȱ [D ȱE]ǰȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱȱȱȱȱȱàǯȱȱ /I (3) ≤ȱ8I ( 3)ǰȱȱȱ àȱȱ [D ȱE]ǯȱ ȱȱàȱȱȱȱ [D ȱE] ȱȱǰȱȱȱ ȱȱǰȱȱȱȱȱȱ /Iȱ (3) ¤ȱǰȱȱȱ ȱȱ 8Iȱ (3) ¤ȱ¢ǯȱ ·ȱ ȱ ȱ ȱ ȱ ¡àȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ǻȱ ȱ ȱ Ǽȱ ȱ ȱ £ȱ ȱ ǻȱ ȱ ȱ ȱ ȱ ȱ ȱ Ǽǯȱ ȱ ȱ ǰȱ ȱ ¡àȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ¡ȱ ǻȱ ȱ ȱ Ǽȱ ȱ ȱ ȱ ȱ£ȱ ǻȱȱȱȱȱȱ¤ȱȱȱǼǯȱȱ àȱ I ȱ[DE]→ ,5 ȱàȱȱ¢ȱȱǯȱȱ¤ȱȱȱ¤ȱȱ ȱ à \ =ȱ I( [) ȱ ȱ [D ȱE] ȱ ȱ øȱ øȱ ȱ ȱ /I (3) ≤ȱ $≤ 8I (3) ȱȱȱȱȱȱ [D ȱE]ǯȱ ȱǰȱȱ¤ȱȱȱ \ =ȱ I( [) [D ȱE] ȱȱøȱøȱȱȱ ¢ȱȱȱȱȱȱȱȱ [D ȱE] ¢ȱȱȱȱȱȱ ȱȱ ȱ [D ȱE]ǯȱ ȱȱȱàȱȱǰȱȱȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ £¤ȱȱǰȱȱȱ¤ȱȱàǰȱȱȱȱȱ¤ǯȱȱ ȱ I ȱ[DE]→ ,5 ȱàȱȱ¢ȱȱǯȱ 3 =ȱ{[ [[Q} ȱ ⎛ȱE− D⎞ ⎟⎟ L= Q ǻȱȱȱȱȱ àȱ [D ȱE] ȱȱ [L = D+ L⎜⎜ ⎝ Q ⎠ E− D Ǽǯȱ 8Iȱ (3) ȱȱȱȱàȱȱȱȱÛΔȱ[L = Q ¢ /Iȱ (3) ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ àȱǰȱǱ OLP 8 I 3 = OLP /I 3 ¢ȱȱȱȱȱȱȱ¤ȱ Q →ȱ∞ Q→ ∞ ȱȱàȱȱȱȱȱȱȱǽǰǾȱȱ¤ȱȱȱȱȱȱȱ ǽǰǾǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱ¤ȱȱȱ \ȱ =[ − ȱȱ [ȱ]ǯȱȱ àǱȱ ȱ ȱ ȱ ȱ ȱ ȱ £¤ȱ ȱ àȱ ȱ ȱ ȱ − ȱȱȱȱȱȱÛȱ Δȱ[L = = ȱ Q Q L=ȱQǯȱ ȱ ȱ àǰȱ ȱ ȱ [ȱ] ȱ ǰȱ ȱ ȱ ¤¡ȱ ȱȱ ȱ¤ȱȱȱ¡ȱȱȱȱǯȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ Ȭ· [L =ȱD+ LΔ[Lǰȱ ȱ ¤¡ȱȱȱȱȬ·ȱ¤Ǳ ⎛ ⎞ ⎛ ⎞ I⎜⎜ + L⋅ ⎟⎟ = ⎜⎜ + L⎟⎟ −= +L + L − = Q + LQ+ L ȱ Q Q Q⎠ ⎝ Q⎠ ⎝ Q Q ⎡ Q + LQ+ L ⎤ ⋅ ⎥ Q⎦ Q L= ⎣ ȱȱȱȱ¤Ǳ 8I ȱ3 =∑ ⎢ ȱȱ¤ǰȱȱȱȱǰȱǱ OLP /I 3 = OLP8I 3 Q →∞ Q→∞ Q = OLP Q→∞ Q + QL+ L ∑ Q L= ⎡ Q ⎤ Q + QL+ L ⎥ = OLP ⎢ Q→∞ ⎢ Q ⎥⎦ L= ⎣ ∑( ) ⎡ ⎛ Q(Q+ ) Q(Q+ )(Q+ )⎞⎟⎤ = OLP ⎢ ⎜ Q + + ⎟⎥ Q→∞ ⎢ Q ⎜ ⎠⎦⎥ ⎣ ⎝ ⎡ ⎞ ⎛ ⎞⎛ ⎞⎤ ⎛ = OLP ⎢+ ⎜+ ⎟ + ⎜+ ⎟⎜ + ⎟⎥ Q→∞ ⎣ Q⎠ ⎝ Q⎠⎝ Q⎠⎦ ⎝ = + + = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱ¤ȱȱȱ \ȱ =[ + ȱȱ [−ȱ]ǯȱȱ àǱȱ ȱȱȱȱȱȱ£¤ȱȱàȱȱȱȱ − − ȱȱȱȱȱȱÛǰȱ Δȱ[L = = ȱ Q Q L=ȱQǯȱ ȱ àǰȱ ȱ ȱ [−ȱ] ȱ ȱ ȱ ȱ Çȱ ȱ ȱ ǰȱ ȱ ȱ Ȭ·ȱ ȱ ȱ ¤ȱ ȱ ȱ ¡ȱ [L =ȱD+ LΔ[Lǯȱ ȱȱÇȱȱȱȱȱ¤Ǳ ⎞ ⎛ L⎞ L L L − QL+ Q ⎛ I⎜ −+ L⋅ ⎟ = ⎜ − + ⎟ + =ȱ − + + = Q Q⎠ ⎝ Q ⎠ Q Q ⎝ Q ⎡ L − QL+ Q ⎤ ⋅ ⎥ ȱȱȱȱ¤ /I ȱ3 =∑ ⎢ Q⎦ Q L= ⎣ ȱȱ¤ǰȱȱȱȱȱǱ Q OLP8I 3 = OLP /I 3 = OLP Qȱ→∞ Q→∞ Q→∞ L − QL+ Q ∑ L= Q ⎡ Q ⎤ = OLP⎢ • L − QL+ Q ⎥ Q→∞ ⎢ Q L= ⎣ ⎦⎥ ⎡ ⎛ Q(Q+ )(Q+ ) Q(Q+ ) ⎞⎤ = OLP⎢ ⎜ − + Q⎟⎥ ⎟ Q→∞ ⎢ Q ⎜ ⎠⎥⎦ ⎣ ⎝ ⎡ ⎛ ⎞⎛ ⎤ ⎞ ⎛ ⎞ = OLP⎢ ⎜+ ⎟⎜ + ⎟ − ⎜+ ⎟ + ⎥ Q→∞ ⎣ ⎝ Q⎠⎝ Q⎠ ⎝ Q⎠ ⎦ ∑( ) = − + = Ǳȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ǰȱ ȱ ¡ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ à 3 =ȱ{[ [Q} [D ȱE]ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ǰȱ ȱ ȱ øȱ ȱȱȱ·ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱǰȱȱȱ¤ȱǱȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱȱ ȱ ŗǯȱ ȱȱ¤ȱȱȱȱǰȱȱȱȱǯȱ Ǽ \ȱ = − [ + [− [− ] Ǽ \ȱ=[ − [+ Ǽ \ȱ = [ + [+ [− ] [] Řǯȱȱȱȱȱ ȱ¡ȱȱȱ¤ȱȱ I [ = [ + ǰȱȱ [ = −ȱ [ = ¢ȱȱȱȱȱǰȱ£ȱȱȱȱàȱȱ [ −ȱ] ȱ Ŝȱ ȱ ǯȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱȱȱ¤ȱȱȱǯȱ řǯȱ ȱȱ¤ȱ ȱ ȱ àȱȱ ȱȱ \ = − [ + [ + ǰȱȱ ȱȱȱȱȱȱȱȃ¡Ȅȱ¢ \ = ǯȱ Śǯȱ ȱȱ¤ȱȱȱàȱȱȱȱ \ = [− [ ȱ ¢ȱȱȱȱ ¡ǯȱ śǯȱȱȱȱȱȱȱÇȱȱȱȱ¤ȱȱȱàȱȱȱȱ ¤ȱȱȱà I [ = − [ +ȱ ¢ȱȱ ¡ǰȱ [ = ȱ [ = ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱȱ Ǳȱ ȱȱȱȱȱàȱ ȱȱȱȱȱȱȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ à Q Q \ =ȱ I( [) ȱȱ [D ȱE] ȱȱ $=OLPȱ∑ 0LΔ[L = OLP∑ PLΔ[L Q→ ∞ L= Q→ ∞ L= ȱ ȱ ȱ Çȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱȱȱȱȱȱȱàȱǯȱȱȱǰȱȱȱ ȱȱÇȱȱȱȱȱȱ¢ȱȱàȱǱȱȱ àǱȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ 3 ={D = [ȱ[[[Q = E}ȱàȱ [D ȱE]ǯȱ [D ȱE] ¢ Δȱ[L ȱ ȱ ȱ ȱ [[L−ȱ[L]ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ 3 ȱ ¢ȱ ȱ ȱ ȱ ¤¡ {Δȱ[L} ¢ȱ ȱ [L ∈ȱ[[L−[L] ȱ L=ȱQǯȱ E ȱȱȱȱǰȱȱȱȱǰȱ ∫ I([)G[= OLPȱ∑ I([L )Δ[L D Q 3 → L= ȱȱȱȱȱȱÇȱ¡ǰȱȱȱȱȱàȱȱȱȱ ȱȱ [D ȱE]ǯ ȱ I([L )Δ[ L ∑ L Q ŗǯ ȱ = ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ àȱȱȱ [D ȱE]ǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Řǯȱ ȱȱ 3 ȱ → ȱȱȱȱ [D ȱE] ȱ£ȱȱ ¤ȱ ǰȱ ȱ ǰȱ ȱ ȱ ȱ ȱ Qȱ→ ∞ ǯȱ E Q ⎛ ⎞ Q ⎛ ⎞ ∑ I⎜ [L⎟Δ[L ǯȱ ∫ I([)G[= OLPȱ ∑ I⎜⎝ [L ⎟⎠Δ[L = QOLP →∞ L= ⎝ ⎠ 3→L= D ȱ ȱ Çȱ ¡ǰȱ ȱ ȱ ǰȱ ȱ ȱ ǰȱ ȱ ȱ ȱ øȱ ȱȱȱȬǯȱ řǯȱ ȱ ȱ à \ =ȱ I( [) ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ [D ȱE] E ȱ ȱ ȱ ∫DIȱ( [)G[ ¡ȱ ¢ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ¤ȱȱȱàȱ [ =ȱ D ¢ [ =ȱEǯȱ E Śǯȱ ȱȱà ∫DIȱ( [)G[ ȱ • Iȱ( [) ȱȱ •ȱ ȱ [ =ȱ D ¢ [ =ȱE ȱ ȱ Çȱ ȱ àǰȱ [ =ȱD ȱÇȱȱ¢ [ =ȱE ȱÇȱǯ •ȱ ȱÇ G[ȱ ȱȱȱȱȱàǰȱȱȱ ȱȃȱ [ Ȅǯȱ E śǯȱ ȱȱ¡ȱȱȱȱȱ¡à ∫DIȱ( [)G[ ȱȱȃȱȄǱȱ E E D D ȱȱ ∫ Iȱ(X)GX ȱ¡ȱȱȱȱȱ ∫ Iȱ( [)G[ ȱȱ E ȱȱ ∫ Iȱ( \)G\ǯȱȱ D ȱ ȱȱàȱ ∫ ([ȱ ) −[+ G[ àǱȱ ȱ ȱ à 3 ȱ={= [ [L[Q = } ȱ ȱ ȱ ȱ ȱ ȱ ȱ− = = Δ[ǯȱ [L ∈ȱ[[L−[L] [[L−ȱ [L] ȱǰȱȱ Δ[L = Q Q L ȱȱ¡ȱȱȱȱ [L = [L = + L⋅ = Q Q 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ∫ ( [ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Q ) ( ) − [+ G[= OLP∑ I [L Δ[L Q→∞ L= ⎡⎛ ⎛ L⎞ ⎛ L⎞ ⎞ ⎤ = OLP ∑⎢⎜ ⎜ ⎟ − ⎜ ⎟ + ⎟ ⋅ ⎥ ⎜ Q Qȱ→ ∞ ⎝ Q ⎠ ⎟⎠ Q⎥⎦ L= ⎢ ⎣⎝ ⎝ ⎠ Q Q ⎡ ⎛ L L ⎞ ⎤ = OLP ∑ ⎢⎜⎜ − + ⎟⎟ ⋅ ⎥ Q→ ∞ Q ⎠ Q⎥⎦ ⎢⎝ Q L= ⎣ = OLP Q→ ∞ Q ∑ L − QL+ Q Q L= Q ⎞ ⎛ Q(Q+ )(Q+ ) Q(Q+ ) ⎜ + Q ⎟⎟ − ⎜ Q→ ∞ Q ⎝ ⎠ = OLP ⎛ ⎛ ⎞⎛ ⎞ ⎞ ⎛ ⎞ = OLP⎜⎜ ⎜+ ⎟⎜ + ⎟ − ⎜+ ⎟ + ⎟⎟ Q→ ∞ ⎝ Q⎠ ⎝ Q⎠⎝ ⎝ Q⎠ ⎠ = − + = ȱ ∫ ȱ−[ G[ ¤ȱȱȱ¤ȱȱȱǯȱȱ àǱ [ȱ]ǰȱ ∫ ȱ−[ G[ ȱȱȱȱȱ¤ȱȱȱȱȱȱ [ȱ]ǯȱ ȱ−[ = \ ȱ ȱ àȱ ȱ ȱ ¢ȱ ȱ \ = − [ ⇒ \ȱ + [ = ǰȱȱȱȱȱȱȱȱȱȱȱ ȱÇȱȱȱŗǯȱ ȱȱ ∫ − [ȱ G[= π () = π ǯȱȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ à ȱ \ =ȱ− [ǰȱȱ ¡ȱ ¢ȱȱȱȱà ¡ȱƽȱȬŗ ¢ ¡ȱƽȱřǯȱ àǱȱ ȱȱȱ¤Ǳ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ¤ȱȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱ ⋅ ⋅ = ¢ $ȱ = = ǯȱȱȱȱ ¤ȱ¤ǰȱ¢ȱ¤ȱǱ $ȱ = ƽȱŞǰśǯȱ ∫ ȱ−[ G[ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ − ¤ȱȱȱǰȱ¢ȱȱŘȱȮȱ¡ȱ ȱȱàȱȱȱȱȱȱ ȱȱȱ¡ȱȱȱȱǽȬŘǰȱřǾǱ Q Q Q ⎛ L⎞ ⎞ L⎞ ⎛ L⎞ ⎛ ⎛ − ⎜⎜ −⎜ − + ⎟ ⎟⎟ ƽ OLPȱ ⎜ ⎟ ⎜ − ⎟ ƽ OLPȱ Q→+∞ Q ⎠ ⎠ Q Q→+∞ L=⎝ Q ⎠ Q Q→+∞ L=⎝ Q Q ⎠ ⎝ ⎝ = L − Q(Q+ )⎞ ⎛ ƽ OLP ⎜ − = − = ǯȱ ⎟ Qȱ→+∞⎝ Q ⎠ ∫ ȱ−[ G[ ƽ OLPȱ ∑ ∑ ∑ ȱȱȱ ȱȱǽŘǰȱřǾǰȱȱȱàȱȱǰȱ ȱȱȱ¤ȱȱȱȱ ȱ ǰȱȱȱȱȱ ȱ−[ G[ Ǳ ∫ ∫ Q Q ⎛ L ⎞ ⎞ ⎛ L ⎞ ⎛ − Q(Q+ )⎞ ⎛ ⎜ −⎜ + ⎟ ⎟⎟ ƽ OLPȱ ⎜ − ⎟ ƽ OLP ⎜ ⎟=− ǯ ⎜ Q→+∞ Q⎠ ⎠ Q Q→+∞ L=⎝ Q ⎠ Qȱ→+∞⎝ Q ⎠ ⎝ L=⎝ ȱ−[ G[ ƽ OLPȱ ∑ ∑ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱ¤ȱȱȱȱ \ =ȱ− [ǰȱȱ ¡ȱ ¢ȱȱȱȱ ∫ à ¡ȱƽȱŘȱ¢ ¡ȱƽȱř ȱȱƽȱȬ ȱ−[ G[ ǯȱ ȱ ŗǯȱ Ǽ ǰȱȱȱàȱȱȱǱȱ ∫ (ȱ[ +)G[ − ∫ ([ +ȱ[− )G[ Ǽ ∫ ([ȱ −)G[ Ǽ − − Řǯȱ ȱȱȱǰȱ¤ȱȱȱ¤ȱȱȱ ǯȱ Ǽ ∫ Ǽ ∫ Ǽ ∫ −(− [ȱ)G[ řǯȱ ¡ȱȱÇȱȱȱȱȱȱȱȱǯȱ (+[ȱ)G[ (ȱ+ − [ )G[ − Q Ǽ OLP ∑ [LVHQ[LΔ[ [πȱ] Q→∞ L= Q H[L Ǽ OLPȱ ∑ Δ[ [] Q→∞ L=+ [L Q ⎡ ⎤ Ǽ OLPȱ ∑ ⎢[L∗ − [L∗ ⎥ Δ[ [] Q→∞ L=⎣ ⎦ ( ) Ǽ ∫ Ǽ ∫ −(−[ȱ)G[ Ǽ − ȱ−[ G[ − ∫ ȱ[ −G[ Q Ǽ OLPȱ ∑ [L∗ Δ[ Q→∞ L= Q Ǽ OLP ∑ FRV [LΔ[L 3 → L= [] [π] 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Śǯȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ¡ȱȱÇȱȱȱȱȱ Q Q ⎛ ⎛ Q+ L⎞ ⎞ ⎜⎜ OQ⎜ Ǽ OLPȱ ⎟ • ⎟⎟ Q→∞ L=⎝ ⎝ Q ⎠ Q⎠ ∑ Q+ L ⎛ ⎜ Q→∞ L=⎜ ⎝ Q Ǽ OLPȱ ∑ Q ⎛ ⎛ L− Q⎞ ⎞ ⎟⋅ ⎟ Q ⎠ Q⎟⎠ ∑ ⎜⎜⎝ VHQ⎜⎝ Q→∞ Ǽ OLPȱ L= ⎞ ⋅ ⎟ Q⎟ ⎠ E ∫ D[G[= Eȱ − D śǯȱ ȱ Ŝǯȱ ȱ ŝǯȱ ȱȱȱȱȱȱȱ E [ G[ = D ∫ȱ E − D ∫( Şǯȱ ȱȱȱȱǱ ∫ (−[ ) + ȱ[+ G[ǯ ) [ȱ +G[ǯȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱȱȱȱ ȱȱ ȱ¢ȱȱȱȱǯȱȱȱȱȱ ȱȱǱȱȱ D ∫D I ȱ[ G[ƽȱŖȱȱ ȱŗ Ǳ Q Q = ∑ I([L )Δ[L = QOLP ∑ I([L)⋅ = QOLP ∫ D Iȱ([)G[= QOLP →∞ →∞ →∞ D ∗ L= D ∫ D ȱŘ L= Iȱ( [)G[=− D ∫ E I([)G[ Ǳ Q Q ∑ I([L ) = QOLP ∑ I([L ) ∫ D I([)G[= QOLPȱ →∞ →∞ L= L= E ∗ Q = OLP− Q→∞ =− ∗ E− D Q Q D− E ∑ ( ) Q = − QOLP− ∑ I([L ) L L I [L∗ = →∞ ∗ = D ∫E I([)G[ E ∫ȱD FG[= F(E− D) ȱȱȱǯȱȱ ȱř Ǳ Q E ∫ FG[= OLP∑ F⋅ D Q→∞ L=ȱ E− D F(E− D) = OLP ⋅ Q = OLP F(E− D) = F(E− D) Q→ ∞ Q→ ∞ Q Q ȱŚ D E E E E ( I([) +J([))G[= ∫ I([)G[+ ∫ J([)G[ ∫ D D E E D E ∫ D( I([) −J( [))G[= ∫ D I([)G[− ∫ DJ([)G[ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ Ǳ D ȱ∫ E ( I( [) + J([))G[= D Q OLP Q→∞ Q ∑ ( I([L )+ J([L ))Δ[L = QOLP ∑ ( I([L)Δ[L + J([L)Δ[L) →∞ L= Q ⎛ L= ⎞ ⎜ I [ Δ[ + ⎟ J [ [ Δ L L L L ⎟ ⎜ Q→ ∞ L=⎝ L= ⎠ = OLP Q ∑ ( ) ∑ ( ) Q I([L)Δ[L + OLP ∑ J([L)Δ[L ∑ Q→ ∞ Q→ ∞ = OLP L= = L= E E ∫ D I([)G[+ ∫ DJ([)G[ Ǽȱȱȱȱȱ¤ȱȱǯȱȱ E E ∫ DFIȱ([)G[ =F∫ D I([)G[ ȱś Q E ( ) FI( [)G[= OLPȱ ∑ FI [L Δ[L ∫ D Q→∞ L= Q ∑ I([L)Δ[L = OLP F Q→∞ L= Q ∑ I([L)Δ[L = F∫ D I([)G[ Q→∞ E = F OLP L= ¤ȱ ŗǯȱ Iȱ( [) ¢ Jȱ( [) ȱȱȱȱȱ [D ȱE]ǰȱȱ E E E ∫ D I([)G[=ȱ ¢ ∫ DJ([)G[=− ǯȱ ∫ D(ȱI([) −J([))G[ Řǯȱ řǯȱ ∫ [ȱ G[ = [ȱ ∫ H G[=H ∫ (+[ȱ )G[ − H 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Ǽ [ȱ ∫ (H −)G[ Ǽ ∫ H Śǯȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ [ȱ+ G[ ȱ ∫ [ȱG[ = ∫ȱ W GWǯȱ ȱàȱȱȱȱǰȱȱǰȱȱȱǯȱ ȱ ȱ ǰȱ ȱ ȱ ȱ £ǰȱ ȱ £¤ȱ ȱ £ȱ ȱ ȱȱȱ·ȱȱȱȱȱǯȱ ȱŜ E ∫ D Iȱ( [)G[ = F ∫ D I( [)G[+ E ∫ F I([)G[ Iȱ( [)≥ ¢ D <ȱ F < E ȱȱȱȱȱȱȱ ¤ȱȱ \ =ȱ I( [) ȱȱ [D ȱE] ȱȱȱȱȱȱȱ¤ȱȱȱ \ =ȱ I( [) ȱȱ [D ȱF] ¢ [FȱE]ǯȱȱ ȱŝȱ Iȱ( [)≥ D ≤ȱ [ ≤ E E ∫ D I([)G[≥ ǯȱȱ àǱȱ Iȱ( [)≥ D ≤ȱ [ ≤ E E ∫D Iȱ([)G[ ȱ ȱ ¤ȱ ȱ ȱ ȱ ¢ȱ ȱ ¤ȱȱàȱȱȱøȱȱȱ E ∫ D I([)G[≥ ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱŞȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ I ( [) ≥ȱ J( [) D ≤ȱ [ ≤ E E ∫ D Iȱ( [)G[ ≥ E ∫ DJ([)G[ǯȱ I ( [) ≥ȱ J( [) ȱȱ¤ȱȱȱȱȱ¢ȱȱȱȱȱ¤ȱȱ ȱȱ \ =ȱ J( [) [D ȱE]ǯȱȱ ȱşȱ Pȱ(E−D) ≤ E P ≤ȱ I( [)≤ 0 D ≤ȱ [≤ E E ∫ D I([)G[≤ 0(E− D) P ≤ȱ I( [)≤ 0 E E D ≤ȱ [ ≤ E ȱ ȱ ȱ E ∫ D PG[ȱ≤ ∫ D I([)G[≤ ∫ D 0G[ ∫ DPG[ȱ=P(E− D) ǲȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ¤ȱȱȱȬȱ¢ȱȱȱ¢ ȱ ¤ȱ Pȱ(E−D) ≤ E ȱ ȱ ∫ D I([)G[≤ 0(E− D)ǯ E E ∫ D0G[ȱ= 0(E− D) ǲȱ ∫D 0G[ȱ ȱȱ¤ȱȱ Ȭȱ ¢ȱ ȱ ȱ ǯȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱ ŗǯȱ £ȱȱȱȱȱȱȱ ∫ȱ [G[ǯȱ Řǯȱ £ȱȱȱȱȱȱȱ ∫ OQ [G[ ǯȱ řǯȱ £ȱȱȱȱȱȱȱ ∫ Śǯȱ Iȱ([) ȱ àȱ ȱ ȱ+ [ G[ǯȱ ∫ Iȱ([)G[= ¢ ∫ Iȱ([)G[=ǯȱ ∫ Iȱ([)G[ǯȱ śǯȱ ȱȱ¡ȱȱȱȱȱǱȱ Ǽ ∫ Ǽ ∫ Ŝǯȱ Iȱ( [)G[+ ∫ Iȱ( [)G[− I( [)G[+ ∫ I( [)G[ ∫ I([)G[ Iȱ(W)GW =ǰ ∫ ∫ Iȱ(W)GW = − ¢ ∫ Iȱ(W)GW = ∫ Iȱ(W)GWǯȱ ŝǯȱ ȱȱȱȱ ≤ȱ ≥ ǯȱ π π Ǽ ∫ Ǽ ∫[ȱ G[ ȏȏȏȏȏȏȏ ∫ȱ Ǽ ∫ ȱ− [G[ ȏȏȏȏȏȏȏ π Ǽ ∫π ∫FRV [G[ȱ VHQ[G[ȱ ȏȏȏȏȏȏȏ VHQ[G[ȱ [G[ ∫ [ +ȱG[ π ȏȏȏȏȏȏȏ ∫πFRV [G[ȱ Ǽ ∫[ȱ G[ ȏȏȏȏȏȏȏ ∫[ȱ G[ Şǯȱ £ȱȱȱȱȱȱȱ ∫ [ȱG[ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱȱ¤ȱ Ǳȱ ȱȱȱȱȱ¤ǰȱȱȱȱȱȱ ȱȱȱȱǯȱ ȱ ȱ ȱ ǻȱ ȱ Ǽȱ ȱ ȱ ȱ ȱ ¤ǰȱȱȱȱǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ¤Ǳȱ ȱ ¤ȱǰȱȱȱȱȱȱȱȱȱȱǰȱ¢ȱ ȱ¤ȱǰȱȱȱȱȱȱȱȱ¤ǯȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ¡àȱ Ǳȱ ȱ ¤ȱ ȱ ¢ȱ ȱ ¤ȱ ǯȱ ȱ ¡àȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ £ǰȱ ¢ȱ ȱ ȱ £àȱ ȱ ȱȱȱ¢ȱȱȱȱ¤ǯȱ ¡ǰȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ àȱ¢ȱȱàȱȱȱǯȱ¤ǰȱȱȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ǯȱ ȱ ¢ȱ £ȱ ȱ ȱ àȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ·ȱ¤ȱ¤ǯȱ ȱ ȱ àȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ǰȱ ǻǼȱ W ∈ȱ,ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ à J ȱ, → ,5 ǰȱ ȱ ȱ ȱ ȱ ¡∈ȱ, ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ [D ȱ[]ǰȱ ȱ [ Jȱ([) =∫ I(W)GW D D ≤ȱ [ ≤ Eǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ àȱ ȱ ¤ȱ ȱ ǰȱ ȱ ȱ ȱ [ ∫D Iȱ(W)GW ¡ǯȱ¤ȱøǰȱȱȱȱ ȱȱà ȱȱ ȱ ȱ ǰȱ ȱ Iȱ(W)≥ ǯȱ Jȱ([) =∫ I(W)GW ȱ ȱ ȱ [ D ȱȱ¤ȱȱȱ¤ȱȱǰȱȱȱ ¡ǰȱȱȱ¡ȱȱȱȱȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ ȱȱȱà Jȱ( [) ¤ȱȱȱȱȱȱȱ ȱ àȱ ǰȱ ȱ ǰȱ ȱ àȱ ¤ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ¤ȱȱȱ¤ȱǯȱǶ·ȱȱȱȱ£ ȱȱ ȱ ȱ ¤ȱǰȱȱ ǰȱȱȱ ȱȱǰ J ′ȱ( [)ǰȱȱ ȱàǵȱ J ′ȱ( [) ȱȱȱȱàȱȱǱȱ ȱ ȱ Kȱ > ǰ J ( [+ȱK) − J( [) ȱ ȱ ȱ ¤ǰȱ ȱ ȱȱȱȱ¤ȱȱȱȱȱȱȱ¡ȱ [ +ȱKǯȱ ȱȱȱȱŖǰȱȱ¤ȱȱȱȱȱ¤ȱȱ ¤ȱȱȱ ǻ¡Ǽȱ¢ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱ J ( [+ȱK) − J([) ≈ KI( [)ǰȱ J ([+ȱK) − J([) ≈ I([)ǯȱ K ȱǰȱȱȱȱȱ J′( [) = OLPȱ K → ⎛ ȱ ⎜ ⎝ [ ∫ȱD ⎞ Kȱ→ ǰȱ J( [+ K) − J( [) = I( [) K ′ I(W)GW⎟ = I([)ǯȱ ⎠ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ øȱ ȱ ȱ ȱ ȱ ȱǰȱȱȱȱȱȱȱȱȱ¤ǯȱȱ ȱȱȱ¤ǰȱȱǱȱ ȱȱȱȱ [D ȱE]ǰȱȱàȱȱȱ Jȱ([) =∫ I(W)GWǰ D ≤ȱ [≤ E [ ȱȱ [D ȱE] ¢ȱȱ (D ȱE) ¢ J ′ȱ( [) = I( [)ǯȱ D ȱȱȱȱȱȱȱȱȃȄȱȱàȱȱ ȱ ȱ ȱ ȱ à Iȱ(W)ǯȱ ȱ ¤ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱȱ¡ȱȱȱȱȱȱȱ ȱȱ¡ȱȱȱàȱǻǼǰȱȱȱȱȱȱ àȱ W =ȱ [ǯȱ ȱ ȱ £ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱ ȱȱȱȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱȱȱȱà J( [)= ȱ [ ∫ ȱ+ WGW àȱ I (W)=ȱ + W ȱ ȱ ȱ ℜȱǰȱ ȱ àȱ ȱ ȱ J( [)= ∫ [ ȱ+ W GW ȱ ȱ ¢ȱ ȱ ȱ ℜȱ ¢ȱ ȱ ȱ ¤ J ′ȱ( [)= + [ ǯȱȱ ȱȱȱȱȱà Jȱ([) =∫ ȱ [ W GW àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ǰȱ ȱ à Jȱ( [) ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ¢ȱ Çȱ ′ J′ȱ([) = I′ [ [ = ⋅ [= ȱȱȱ¡ǰȱ [ȱǯȱ ȱȱǱ ( ) [ [ £ȱȱȱȱ£ȱȱȱȱǰȱȱȱȱ ȱ Iȱ([) ȱ ȱ àȱ ȱ ¢ ϕȱ( [) ȱ àȱ ǰȱ ¤ȱ øǰȱ Ǳ G ϕ ( [) I(W)GW = I(ϕ ([)) ⋅ ϕ ′([) G[∫ȱD ϕȱ( [)ǰ ψȱ( [) ȱ ȱ ȱ ¢ȱ ȱ ȱ ǰȱ ¢ȱ ȱ J([) = ϕ ( [) ∫ψ ([) ȱ I(W)GW = ∫ψ ([) =− Ǳ ȱ D I(W)GW+ ψ ( [) ∫ D ∫ D I(W)GW+ J ′ȱ( [) = − I(ψ ( [))ψ ′( [) + I(ϕ ( [))ϕ ′( [) = I(ϕ ( [))ϕ ′( [) − I(ψ ( [))ψ ′( [) J([)=∫ FRV[ VHQ[ ϕ ( [) ȱ+ W GW à J′( [)= − VHQ[ + FRV [− FRV[ + VHQ[ I(W)GW ϕ ( [) ∫ D I(W)GW 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ¤ȱ [ J([) =∫ I(W)GWǰȱ ȱ ȱ ȱ ȱ àȱ ¢ȱ ¤ȱ ȱ ȱ ȱ ȱ ŗǯȱ ȱ ȱǯȱ Jȱ()J()J()J()J()J() Řǯȱ ȱȱȱȱȱȱǯȱ Ǽ J([)=∫ [ ȱ+ WGW Ǽ [ Ǽ J([) =∫ OQȱWGW [ ȱ −W + W+ GW Ǽ Jȱ([) =∫ WFRV(W)GW [ [ Ǽ J([) =∫ Ǽ J([) = ∫ ∫ VHQ[ Ǽ J([) =∫ Ǽ J([) =∫ J( [) = ȱW GW W Ǽ J([) = (W+ ȱW)GW [X − ȱ GX [ X + [ ȱ[ + W GW ∫ VHQ[ ȱ(X )GX [ řǯȱ ȱ ȱ ȱ ȱ Çȱ Çȱ ȱ ȱ ¡ȱ ȱ ȱ ȱǯ • I([) = • I ( [) = ∫ (W − )GW [ ∫ − [ W GW + W 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Śǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ǰȱÇȱȱȱȱȱ¡àǱȱ [− ⎛ [+ ⎞ ǯ Jȱ( [) = ǯ Jȱ( [) = W + W + GW ⎜ W − W + W⎟GW ⎝ [ − ⎠ ∫ ∫ ∫ śǯȱ ȱȱȱȱàȱȱȱ ȱȱàȱ¡Çȱ Iȱ( [)ǯȱ [ ( ) [ I(W)GW = [Hȱ[ + ∫ H−W I(W)GW ȱȱ¡ǰȱ Ŝǯȱ ȱȱȱȱȱȱȱǱȱ [ W ȱ+ XGX⎞⎟GW Ǽ I( [) = ⎛⎜ ⎝ ⎠ [ ⎛ X ȱ+ XGX⎞ GW Ǽ I( [) = ⎟ ⎜ ⎝ W− ⎠ ∫ ∫ ∫ ∫ ∫ [ GWǰȱȱ I ′ [ ȱ − [ − W ŝǯȱ I ȱ[ = Şǯȱ Iȱ [ = ∫ J [ şǯȱ ŗŖǯȱ ŗŗǯȱ ŗŘǯȱ ∫ [ J ȱ[ = + W ∫ (+ VHQW ) GWǯȱ GW ¢ J ȱ[ = FRV[ ∫ W + GWǯȱȱ J′ [ ǯȱ [+ ∫ ŗŚǯȱ ) ȱ[ = ∫ [ ȱ [ W GWǯȱ ∫ (W [ ) − W+ GW − J ȱ[ = ∫ [+ HȱW GWǰȱȱ ǻŘǼȱȱ¢ K′ [ ǯȱ ) ′ ȱ[ VL ) [ = ŗřǯȱ ⎝ ȱ⎠ ǰȱȱ¡ǰȱȱȱȱ ¡ȱ ȱ¡£ȱȱ K [ = ⎛ π⎞ I ′⎜ ⎟ǯȱ [ GW ǰȱȱ J′ [ ǯȱ [ + W W + X GXǯȱȱȱ ) ′′ ȱ ǯ X I W GWǰȱȱȱ Iȱ W =∫ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ǰȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ Çȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ Çǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ǰȱ ȱ ȱ ȱȱȱȱȱȱǰȱȱȱ·ȱȱ¤ȱȱ ȱȱȱǯȱȱ ȱȱȱ¤ǰȱȱǱȱ ȱȱȱȱ [D ȱE] E ∫ D Iȱ([)G[= )(E) − )(D) ȱȱȱȱ ȱȱȱǰȱȱǰȱȱàȱȱ ) ′ = I àǱȱ Jȱ( [) = ∫ I(W)GW ȱȱȱȱ [D ȱE] ȱȱȱȱȱ [ D ȱȱȱȱ¤ȱȱȱȱ J ′ȱ( [) = I( [)ǰȱȱȱ ȱȱȱȱȱȱǯȱ ȱȱȱȱȱȱȱȱ [D ȱE] ȱȱ¢ȱȱȱȱ ȱǰȱȱ ) ( [) =ȱ J( [)+ F [∈ȱ(D E)ǯȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ) ( [) =ȱ J( [)+ F [ =ȱD ¢ȱ [ =ȱEǯȱ ) (E) − )(D) = [J(E) + F] − [J(D) + F] [D ȱE] ȱ · = J(E) + F− J(D) − F = J(E) − J(D) = = E ∫D E ∫D D I(W)GW− ∫ I(W)GW D E I(W)GW− ȱ= ∫ I(W)GW D ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱǰȱȱȱȱ·ȱȱȱȱȱȱ¢ȱȱȱ ȱȱ¤ȱȱȱ¤ȱǻȱȱ¤ȱȱȱ¢ȱȱȱ ¤ȱȱǼǯȱ ȱȱȱȱȱȱȱȱȱȱ ȱ ȱ ǰȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ǰȱȱȱȱȱ E ∫ȱD I([)G[ ȱȱȱȱȱȱȱ ȱȱ¡ȱȱǯȱ ȱȱȱȱȱȱȱȱ¢ȱȱȱ ǯȱ ȱ ∫ Iȱ([)G[ ȱ ȱ ȱ ȱ àǰȱ ȱ ȱ E ∫D Iȱ([)G[ ȱȱøȱȱȱȱȱȱÇȱȱȱȱȱǯȱȱ àȱ ȱȱȱȱàǱ ȱ ø ∫ E ∫ȱD I([)G[= ) [ D = ) E − ) D E [ȱG[ − àȱ ȱ ȱ ȱ ȱ ȱ ∫ [ȱ G[= − [ = − I ( [)=ȱ [ ) ( [)=ȱ [ − = π ȱ ø ∫ π VHF[ȱWDQ[G[ àȱ ȱ ȱ ȱ ȱ ȱ ȱ π π VHF[ WDQ[G[= ∫ π π VHF[π = VHF πȱ − VHF = − ȱȱ¤ȱȱȱ¤ Iȱ( [)= [ − [+ ȱŖȱȱřǯȱȱ à ∫( ) ⎛ [ ⎞ $ = [ − [+ G[ = ⎜⎜ − [ + [⎟⎟ = ⎝ ⎠ I( [)=ȱVHF [WDQ[ ) ( [)=ȱVHF [ǰȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ŗǯȱ øȱȱȱȱ ∫ (ȱ+\ − \ )G\ ŗǯ ȱ Řǯ ∫ řǯ ∫ (ȱ+[ [)G[ GW − W Śǯ śǯ OQ ∫ȱ OQ ∫ G[ȱ −[ π Ŝǯ ŝǯ I([)G[ ∫ π − ∫ GRQGH ȱȱȱÇȱȱȱȱȱȱȱ − Q+L Q ⋅H ŗǯ OLPȱ Q→∞ L= Q ∑ Q ⎞ ⎛⎜ ⎛ Q+L⎞ ⎜ + ⎟ ⎟ Řǯ OLPȱ ⎟ ⎜ Q→∞ L= Q⎝ ⎝ Q ⎠ ⎠ Q ∑ Q řǯ π ⎛ π L⎞ VHQ⎜ ⎟ ∑ ⎝ Q ⎠ L Q OLPȱ Q →∞ = Q Śǯ πȱ ⎛ −π Q+ π L⎞ VHF ⎜ ⎟ ∑ Q→∞ Q ⎝ ⎠ L= Q OLP Q śǯ ⎧ [ VL − π ≤ [≤ ȱ ⎩ VHQ[ VL < [≤ π I([) = ⎨ [ −ȱ[+ G[ Řǯȱ H[G[ OLP Q→ ∞ ⎛ Q ⎞ ⋅⎜ ⎟ ∑ Q ⎝ Q+ L⎠ L = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ àȱȱàȱ Ǳȱ ȱȱȱȱȱàȱȱȱ·ȱȱàǯȱ ȱȱȱȱàǯȱ ȱȱȱȱ¤ȱȱȱȱȱàȱȱ ȱ àǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱȱȱȱǯȱȱ ȱ ∫ ȱ[ + [G[ àǱȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ¡ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ∫ ȱ[ +[G[ǰȱȱ¤ǯȱ ȱȱȱ ȱ àȱȱ ȱ àȱȱȱȱȱ ȱ ȱ ȱȱȱȱȱàȱȱȱȱȱȱȱȱȱ \ =ȱ )(X) ¢ X =ȱ J( [) ȱ ȱ ȱ ȱ ȱ ȱ ȱ G [)(J([))] =ȱ )′(J([)) ⋅ J′([) G[ ȱȱàȱȱȱȱȱ ∫ȱ) ′(J([))J′([)G[= )(J([))+ &ǯȱ ȱ ȱ £ȱ ȱ ȃàȄ X =ȱ J( [)ǲȱ ȱ Ǳ ∫ȱ) ′(J([))J′([)G[= )(J([))+ & ȱ ∫ȱ) ′(X)GX= )(X)+ & ȱàȱȱȱȱȱȱǱȱȱ Ǳȱ àȱȱȱàȱȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ \ =ȱ I( J( [)) ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ∫ȱI (J([))⋅ J′([)G[= )(J([))+ & X =ȱ J( [) GX =ȱ J′( [)G[ ¢ ∫ I (X)GX = )(X)+ & 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ¡ȱ ȱ ·ȱ ȱ ȱ ȱ ǰȱ ȱ ȱȱȱǯȱ ȱǰȱȱȱȱȱȱàȱȱȱ ȱ ȱ ȱ ȱ ·ǰȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱȱȱȱȱȱȱǰȱȱȱàȱȱ Iȱ( J( [)) ¢ J ′ȱ( [) ȱȱǯȱȱȱ¤ȱ¡ȱȱ àǰȱȱȱȱȱȱȱ¤ȱǯȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ·¡ȱ ȱ ȱ ȱ £àȱ ȱ ȱ ǯȱȱ ȱ ∫ ȱ[ + [G[ àǱȱ X =ȱ+ [ ȱ ȱ Xȱ GX =ȱ [G[ ǰ ȱ ∫ȱ[ + [ G[= ∫ ([ ∫ XGX= +ȱ) ([)G[ àǱȱ X =ȱ+ [ GX =ȱ [G[ Ǳ X ∫ ([ +) ([)G[= ∫ X GX= + & (+ [ ) + & = ȱ ȱ ∫ȱFRV[G[ àǱȱ X =ȱ [ GX =ȱG[ ǰ ȱ X + &ƽ +[ + F ∫ FRV[G[ =∫ FRVXGX = VHQX+ & = VHQ([)+& 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ∫ [ VHQ([ ȱ +)G[ àǱȱ Xȱ =[ + GX =ȱ [G[⇒ ∫ ( ) [ VHQ [ +G[= GX = [G[ ( ) VHQXGX= − FRVX+ & =− FRV [ ȱ + + & ∫ ∫ ȱ[ − G[ ȱ àǱȱ ȱ[ − = X [ −ȱ= Xǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Ǳ G[=ȱ XGX ¢ȱȱȱȱ G[ =ȱ XGX ǰ ∫ [−G[ = ( ∫ ∫ ∫ [ ȱ[−G[ ȱ àǱȱ X = [− ⇒ Xȱ = [− ⇒ X + = [ XGX =ȱG[ X + ∫ [ [−G[= ∫ ⋅ X⋅ XGX = ∫(X + )XGX = ∫ (X + X)GX ǰ = = X X ( + + & ) ( [− + ) ȱ X [ − + & X⋅ XGX = X GX = ⋅ + & = ) [− + & 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ [ ∫ ȱ−[ G[ àǱȱ X =− [ GX = −ȱ[G[⇒ − GX = [G[ ǰ ∫ [ GX G[= − = − XGX −[ ∫ ∫ X ȱ =− X + & = − − [ + & ȱ ∫ +ȱ [ [G[ àǱȱ +[ȱ = X ⇒ [ = X− ¢ [G[ =ȱGX ⇒ [G[= ∫ ǰ +[ ([ ) GX ⋅ [G[= X(X− )GX ∫ ⎞ ⎛⎜ = X X − X+ GX = X − X + X ⎟⎟ GX ⎜ ⎝ ⎠ ∫ ( ) ∫ = X ⋅ − X ⋅ + X ⋅ + & ȱ = + [ − + [ + + [ ( ȱ ) ( ) ∫VHQ ([)FRV([)G[ ȱ àǱȱ X = VHQ( [) GX = FRV([) G[⇒ GX = FRV([) G[ ( ) + & 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ǰ ȱ ∫ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ VHQ ([) FRV([)G[= X X GX = ⋅ + & = VHQ ȱ([)+ & ∫ ∫[ FRVȱ([ )G[ àǱȱ X =ȱ [ GX =ȱ [G[⇒ ǰ GX = [G[ ∫ [ FRV([ ) G[= ∫ FRV(X) GX = VHQ(X) + & ȱ = VHQ ([ )+ & ȱ ∫ VHFȱ [ [ G[ àǱȱ X =ȱ [ GX=ȱ ǰ ∫ [ G[⇒ GX = [ G[ VHF [ [ G[=∫ VHF XGX= WDQX+ & ( ) =ȱWDQ [ + & ȱ ∫WDQȱ[G[ àǱ VHQ[ ∫ WDQȱ[G[=∫ FRV[G[ X = FRV [ ⇒ GX =ȱ− VHQ[G[ ǰ ∫ WDQ[G[=−∫ GX = − OQX + & =− OQ FRV[ + & = OQVHF[ + & X 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ∫ FRWȱ[G[ àǱ FRV[ ∫ FRWȱ[ G[ = ∫ VHQ[ G[ X =ȱ VHQ[ FRVȱ[G[ =GX ∫ FRW[ G[ = ȱ GX ∫X = OQX + & =OQȱVHQ[ + & = − OQFVF[ + & ∫ VHFȱ[G[ àǱ VHF[(VHF[+ WDQ[) G[ = VHF[+ WDQ[ ∫ VHF [ G[=∫ VHF[G[= ∫ ( ∫ VHFȱ [+ VHF[WDQ[ G[ VHF[+ WDQ[ ) X =ȱVHF [+ WDQ[⇒ GX = VHF[WDQ[+ VHF [ G[ ∫ ȱȱ VHF[ G[ = ∫ FVFȱ[G[ ȱ àǱ ∫ GX ∫ X = OQX + & =OQȱVHF[ + WDQ[ + & ∫ FVF [ G[ = FVF[ G[= ∫ FVF[(FVF[+ FRW[) G[ = FVF[+ FRW[ ∫ FVFȱ [G[+ FVF[FRW[ G[ FVF[+ FRW[ ( ) GX =ȱ −(FVF [FRW[+ FVF [) G[ X =ȱFVF [+ FRW[ GX = − FVF [FRW[− FVF [ G[ Ǳ ∫ FVF[ G[ =− ∫ GX = − OQX + & =− OQ FVF[ + FRW[ + & X 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱǱ ∫ VHF [G[=OQVHF[+ WDQ[ + & ∫ FVFȱ[G[ = − OQFVF[+ FRW[ + & ∫ WDQ [G[=OQVHF[ + & ∫ FRW[G[=− OQFVF[ + & ȱȱȱȱȱȱ Ǳȱȱȱȱȱȱȱ ȱȱà X =ȱ J( [) ȱȱȱȱȱȱ [D ȱE] ¢ Iȱ ȱȱȱȱȱ J ǰȱ E J(E) ∫ D Iȱ(J([)) J′( [)G[= ∫ J(D) I(X)GX ∫ ( ) ȱ [ȱ [ + G[ àǱȱ Xȱ =[ + GX = ȱ[G[⇒ GX = [G[ ¤ȱȱȱȱÇȱȱàǱ Xȱ() = + = X()= + = ∫ ( ) [ȱ [ + G[= ǰ ȱ ∫ ∫ X XGX= ⋅ = − = ȱ[ + G[ àǱȱ X = [+ ⇒ X = [+ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ XGX = G[ ¤ȱȱȱÇȱȱàǱ ȱXGX =G[⇒ ∫ [ +G[ = ∫ X X GX = ⋅ Xȱ( ) = X()= − = = =ȱ ¤ȱ ŗǯȱ ŗǯ øȱȱȱȱ ∫ ([ [ ) −ȱ G[ Řǯ ⎛ ⎞ ⎛ ⎞ řǯ ∫ ⎜ȱ+ ⎟ ⎜ ⎟GW ⎝ W ⎠ ⎝ W ⎠ śǯ ∫[ ȱ [+ G[ ŝǯ ∫ VHFȱ(−[) WDQ(− [) G[ şǯ ∫FRVȱ [G[ VHQ[ (ȱ+ [ ) G[ ∫ [ ŗřǯ VHFȱ( [) WDQ( [) +VHF( [)G[ ∫ ŗśǯ ∫ ȱ−\ G\ ŗŝǯ ∫ H[ VHQ(H[) G[ [ ȱ + G[ [ + [+ [ + Řŗǯ ∫ G[ ȱ [ + [− ŗşǯ ∫ ȱ[ + ∫ + [+ ) G[ ȱ[ − G[ [+ Ŝǯ ∫ ( [ +ȱ) − [ G[ Śǯ Şǯ FVFȱ [ G[ FRW [ ∫ ∫ ŗŖǯ WDQȱ [ VHF [ G[ ŗŗǯ ∫ ([ ŗŘǯ ∫ WDQȱ (θ)VHF (θ ) Gθ ŗŚǯ ∫ [ ȱ G[ −[ ŗŜǯ ∫[H[ G[ ȱ ŗŞǯ VHQ[ G[ +ȱFRV [ ŘŖǯ ∫ [ G[ [+ ȱ ŘŘǯ ∫ FVFȱ ([) H ŘŚǯ ∫ FVF ([) FRW([)G[ FRW[ G[ ȱ HW W Řřǯ ∫ GW 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Řǯȱ øȱȱȱȱ ∫ ŗǯ ∫ Řǯ řǯ Śǯ [ ȱ−[ G[ Ŝǯ [ G[ ȱ[+ ŝǯ πȱ ∫ [ ⎞ FRV ⎛ ⎜ ⎟ G[ ⎝ ⎠ ∫ Şǯ VHQθ Gθ FRVȱθ ∫ȱ ∫ȱ( \ + \ − G\ \ ∫ ∫ řǯ H ∫H [ ) [ + G[ ∫(FVF [− VHF[WDQȱ[)G[ ŗŚǯ ∫ ŗśǯ ⎧ [ VL ≤ [≤ ⎩− [ VL ≤ [≤ I [ = ⎨ [+ G[ ∫ȱ ∫ȱ Şǯ [ȱ − G[ FRV[ G[ + VHQ[ [ − [ + [ G[ [ [ + [ [ + [ + G[ [ [+ G[ ŗŝǯ ∫ȱ ŗŞǯ ∫ − ŝǯ ∫ ∫ȱ ŗŜǯ ([ȱ+) ∫ȱ [ȱ− G[ − Ŝǯ Wȱ W + GW Iȱ [ G[ FRQ ∫ − śǯ G[ ŗřǯ Śǯ ȱ− [ G[ȱ ȱ ŗǯ Řǯ VHQ −[ ∫ řǯȱ ∫ [ OQ[ G[ȱ şǯ ∫ [+ [ [−ȱG[ π śǯ [ ∫ −ȱ+[G[ ( ) [ + [ VHFȱ \ WDQ \G\ ∫ ∫( ŗşǯ G[ [ȱ [ G[ − ŘŖǯ [ ) [ + G[ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD şǯ ∫ (+ [) /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ [ G[ȱ ∫ Řŗǯ ŗŖǯ ŗŗǯ ∫ VHQȱ [ G[ FRV [ Řřǯ ∫ (FRVθ VHF θ )Gȱθ ∫ ([ȱ+ [− )G[ řŖǯ ∫ ∫ + [ G[ G[ [− [+ HVHQ[ G[ VHF [ ∫ (+ OQ [) [ H[ ∫ − H [ ∫[ + [+ G[ G[ [OQ [ ∫ G[ [ȱ + [G[ ∫ȱ ∫ [ + − ŘŚǯ Řśǯ ∫ G[ [ȱ + [+ − π ŗŘǯ [+ ŘŘǯ [ G[ȱ π ∫ ∫ VHQȱ[FRV[G[ [ G[ G[ ∫+ [ G[ ∫ VHQ −[ − [ G[ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱ Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ¢ȱȱ¤ȱȱǯ Iȱ( [) ȱȱàȱȱ¢ȱȱȱȱȱ [D ȱE]ǰȱȱ ȱ¤ȱȱȱȱȱ¡ǰȱȱ [ =ȱ D ¢ [ =ȱE ¢ȱȱ¤ȱȱȱàǰȱ E ȱȱȱȱ ∫ȱD I([)G[ǯȱ ȱȱàȱȱȱȱ¤ȱȱȱȱȱ ¢ ǯȱȱ I ( [) ≥ȱ J( [) ȱ [∈ȱ[D E]ǯȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ǯȱ ǰȱ ȱ ǰȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ǲ Iȱ([)< ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ( Iȱ( [) + N)> ¢ȱ Jȱ( [)> ( Jȱ( [) + N)> ǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱ¤ȱȱȱȱ¤ȱ I ( [)+ȱN ¢ J ( [)+ȱN [D ȱE] Ǳ E ∫ D = E E D D (J([) +N)G[− ∫ ( I ȱ[ + N)G[= ∫ ((J([) + N) − ( I([) + N)) E ∫ D E (J([) − I [ )G[= ∫ I [ − J [ G[ D àǱȱ ȱȱ¢ȱȱȱȱ [D ȱE]ǰȱȱ¤ȱȱȱàȱȱȱȱ¤ȱȱȱ¢ȱ ȱȱȱȱ [ =ȱD ¢ [ =ȱE ȱ ƽ E ∫ D I [ −J [ G[ ȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ȱ \ȱ = [ + \ = − [+ [= − ¢ [ȱ = ǯȱȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ [ȱ +≥ − [+ ⇔ [ + [+ > ȱ [ ∈ȱ,5 [ +≥ − [+ ȱȱǱ $ = = ∫ ([ + [+ )G[ = X$ [ + − − [ + G[ = [ + + [ − G [ ( ( ) ) ( ) ( ) ∫ ∫ −ȱ − − ȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ȱ I ( [)=ȱ− [ ¢ J( [)=ȱ− [ǯȱȱ àȱ ȱȱȱȱȱȱǰȱȱȱȱȱȱ ȱàȱ −[ = −[⇔ [ − [− = ⇔ ( [ȱ−)( [+ )= ⇔ [= [= −ǯȱ Çȱȱàȱȱȱȱ¤ȱ¢ȱȱȱȱȱ [ȱ = − ¢ [ȱ = ǯȱ ȱȱȱȱ − [ ≥ −[ −[ȱ ≥ −[ ⇔ [ − [− ≤ ⇔ [∈ []ǯȱȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD $= /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ =∫ ∫ (− [ + [)G[ −ȱ ⎛ ⎞ ⎜ − [ + [ + [⎟ = X$ ⎟ −⎜ ⎝ ⎠ ȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ I( [)=ȱ[ − [ − [ ¢ J( [)=ȱ − [ + [ǯȱȱ àȱ ȱȱȱȱȱȱǱ [ȱ −[ − [ = − [ + [ ⇔ [ − [ = ⇔ [([ − )= ⇔ [ = −[ = [= ǯȱ ȱȱȱ [ − [ − [≥ − [ + [ [ −[ − [≥ − [ + [ ⇔ [ − [≥ ⇔ [( [− )( [+ )≥ ȱ [ [ [ I ( [) ≥ȱ J( [) [ȱ∈[− ] ¢ I ( [) ≤ȱ J( [) [ȱ∈[]ǯȱ ȱȱƽ ∫ ([ȱ −[ − [− (− [ + [))G[+ ∫ (− [ + [− ([ − [ − [))G[ − ∫ − ([ȱ −[)G[+ ∫ (− [ + [)G[ ƽ ⎛ ȱ[ ⎞ ⎜ − [⎟G[+ ⎟ − ⎜ ⎝ ⎠ ƽ ȱX$ ƽ ∫ ⎛ − [ ∫ ⎜⎜⎝ ⎞ + [⎟G[ ⎟ ⎠ ȱ ȱȱ¤ȱ ȱ ȱ àȱȱȱȱ ¤ \ȱ =[+ ¢ȱȱ \ȱ=[− ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 àȱ ŗǼȱ ȱȱȱȱȱȱȱ \ȱ = [+ ⎫ ⎬ ⇒ ( [− ) = [+ ⇒ [ − [+ = [+ \ = [− ⎭ ⇒ [ − [+ = ⇒ ( [− )( [− )= ⇒ [ = R [ = ŘǼȱ ȱȱȱ¤ \ȱ =[+ ȱȱȱàǰȱȱȱ ȱȱȱȱȱǱ \ȱ = [+ \ = − [+ ǯȱ ¢ȱȱȱȱȱȱȱ ¤ȱ ȱȱ \ȱ =[+ ¢ \ȱ=[− Çȱ ȱ àȱ ¢ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ŗǯȱ [ȱ = − [ȱ = \ȱ = [+ \ \ = − [+ ǯ $ = ∫ −ȱ( /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ) [+ + [+ G[ = ȱ ȱ ȱ ȱ ¤ X$ [ȱ = [ȱ = ȱ ȱ ȱ \ȱ =[− ¢ȱ ȱ ¤ Řǯȱ \ȱ = [+ ǯ $ = ∫ȱ ( ) [+ − ( [− ) G[ = X$ǯȱ ȱȱ¤ȱȱȱàȱ $=ȱ + = X$ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àǯȱ ȱ£ȱȱȱȱ¢ȱȱàȱȱ¡ǰȱȱȱ¡ȱȱàȱȱ¢ǯȱ ȱ ǰȱ ȱ ȱ àȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ [ = I( \) [ = J( \) \ =F \ =ȱG ȱȱȱ¢ ȱȱȱȱ¤ȱǱ $ȱ= G ∫ F I( \) − J( \) G[ ȱȱȱȱȱȱȱȱȱȱȱàȱȱȱȱ ȱȱ¤ǯȱȱȱȱàȱ \ȱ = [+ ⇒ \ − = [⎫ ⎬ \ = [− ⇒ \+ = [ ⎭ ȱȱȱà \ −= \+ ⇔ \ȱ − \− = ⇔ ( \− )( \+ )= \ = R \ = ȱȱȱ \ȱ −≥ \+ Ǳ \ȱ −≥ \+ ⇔ \ − \− ≥ ⇔ \∈ ]− ∞−[ ∪ ]+∞[ ǰȱȱȱ [−] \ȱ −≤ \+ ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱ¤ȱȱǱ $ȱ= ∫ − (\+ − (\ − ))G\= ∫ − (− \ + \+ )G\ = ∫ ⎛ ⎞ ⎜− \ + \ + \⎟ = + = X$ ⎟ − ⎜ ⎝ ⎠ ¤ȱ ŗǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ \ ȱ = − [ ¢ ( \ȱ+) = [+ ǯȱȱ ǱȱŝŘ ǯǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ \ =ȱ VHQ[ ¢ \ =ȱFRV [ π π [ȱ = ¢ [ȱ = ǯȱȱ Ǳȱ X$ Řǯȱ řǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ \ȱ =( [− ) ¢ȱ ȱ \ȱ =( [+ )ǯȱȱ ǱȱřŜ ǯǯȱ Śǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ \ȱ =[ + [+ ¢ \ȱ =− [ − [− ǯȱȱ ǱȱŚ ǯǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ \ =ȱ [ǰ \ =ȱ [ ¢ȱ ȱ \ȱ = ǯȱȱ Ǳ Xȱ$ śǯȱ ȱȱ¤ȱȱȱàȱȱȱ \ =ȱH[ǰȱȱ \ =ȱ [ ¢ȱȱ ⎞ ⎛ ȱȱ [ȱ = ǰ [ȱ = ǯȱȱ Ǳ⎜ H −ȱ ⎟ X$ ⎠ ⎝ Ŝǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ \ =ȱ [ \ ¢ \ =ȱ[− [ ǯȱȱ Ǳ X$ ŝǯȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD π [ȱ = ǯȱȱ ŗŖǯȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ¤ȱȱȱà ȱȱȱ \ =ȱ VHQ[ǰ \ =ȱFRV [ǰ [ȱ = ǰ Şǯȱ şǯȱ ȱ ( ) Ǳ −ȱ X$ ȱȱȱ¤ȱȱȱȱ \ȱ =[− ¢ȱȱ¤ \ȱ = [+ ǱȱŗŞ ǯǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ \ȱ = [ [ + ȱȱ¡ȱ¢ȱȱ [ȱ = ǯȱȱ Ǳ −ȱ X$ ( ) ȱȱ¤ȱȱȱàȱȱȱȱ \ =ȱ [ − [ǰȱȱȱ¡ȱ¢ȱ Ǳ Xȱ$ ȱ [ȱ = ǰ [ȱ = ǯȱȱ ŗŗǯȱ ȱȱ¤ȱȱȱàȱȱȱȱ \ȱ = [ − [ − [+ ǰȱȱ ȱ¡ȱ¢ȱȱ [ȱ = − ¢ [ȱ = ǯȱȱ Ǳ Xȱ$ ŗŘǯȱ ŗřǯȱ ȱȱ¤ȱȱȱàȱȱȱȱ \ =ȱ [ǰ \ =ȱ − [ + [ǯȱȱ Ǳ Xȱ$ ŗŚǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ \ȱ = [− ¢ȱ ȱ \ȱ =[− ǯȱȱ ǱȱŗŞ ǯǯȱ ȱȱ¤ȱȱȱàȱȱȱȱȱ \ =ȱ [ − [ + [ ¢ \ =ȱ [ − [ǯȱȱ Ǳ Xȱ$ ŗśǯȱ ȱȱ¤ȱȱȱàȱȱȱȱ \ =ȱ [ ¢ȱȱ¤ \ =ȱ− [ǯȱȱ Ǳ Xȱ $ ŗŜǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ \ =ȱ [ ¢ȱ ȱ ¤ Ǳ \ ȱ =− [ǯȱȱ X$ ŗŝǯȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ŗŞǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ Iȱ( [)= [ − [ + [− ¢ Jȱ( [)=− [ + [− ǯȱȱ Ǳ Xȱ$ ŗşǯȱ ŘŖǯȱ Řŗǯȱ ¢ȱƽȱŖǯȱ ŘŘǯȱ ȱȱ¤ȱȱȱàȱȱǱ \ȱ =[+ ǰ \ȱ =( [− ) ǰ [ȱ = − ǰ [ȱ = ǯȱȱ Ǳ Xȱ$ ȱȱ¤ȱȱȱàȱȱǱ \ =ȱ [ ǰ \ȱ =[ − ǯȱȱ Ǳ Xȱ$ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ \ = − [ − [ +ȱ ¢ȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ \ = [ − [ + ǰ \ = − [ + [ −ȱ ¢ȱȱȱ ¡ ƽȱŗǰȱ ¡ȱƽȱśǯȱ Řřǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ \ = [ + [ + ǰ \ = [ + ȱ ¢ȱȱȱ ¡ ƽȱȮřǰȱ ¡ȱƽȱŘǯȱ ȱȱ¤ȱȱȱàȱȱȱȱ \ =FRVȱ[ ǰȱ ¢ȱƽȱŖȱȱ¢ȱȱ π \ = − ¢ \ = πȱ ǯȱ ȱ ŘŚǯȱ Řśǯȱ ȱȱ¤ȱȱȱàȱȱȱȱ \ = [− [ ȱ \ = [ ǯȱ ŘŜǯȱ ȱȱ¤ȱȱȱȱ \ = [ \ȱ [ = \ ǯȱ Řŝǯȱ ȱȱ¤ȱȱȱàȱȱȱȱ \ = [− [ ȱ \ = [ ǯȱ ŘŞǯȱ ȱ ȱ ¤ȱ ȱ \ = −( [− ) \ = [+ ȱ \ = [ −ǯȱ ȱ àȱ ȱ Řşǯȱ ȱȱ¤ȱȱȱàȱȱȱȱ \ = [ȱǰȱ¢ȱȱȱ \ = − [ \ = [ + ǯȱ řŖǯȱ ȱȱ¤ȱȱȱàȱȱ \ = [ȱǰ \ = [ + ǰ [ = −ǯȱ ȱ řŗǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ȱ \ = [ + ǰ \ = −ȱ[ ǰ [ =ȱ ¢ [ = ǯȱ řŘǯȱ ȱȱ¤ȱȱȱàȱȱȱȱ \ = [ ȱǰ \ = − [ȱ + [ ǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 øȱȱàȱȱàǱȱ ȱ·ȱȱȱ Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ¢ȱȱøȱȱàȱȱàǯ ȱ¤ȱȱ¤ȱȱȱȱȱȱȱȱȱȱǯȱ ȱ¢ȱȱȱ ¤ȱȱøȱȱàǯȱ ȱȱàȱȱȱȱȱȱȱȱȱǰȱȱàȱȱȱ ȱàȱȱàȱ¢ȱȱȱȱȱȱȱàȱǻȱȱȱǼǯȱ ȱ¤ȱȱȱ ȱȱȱȱǰȱȱȱȱ ȱ ¤ȱ ȱȱȱȱȱǯȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ àǯȱ ȱ ȱ àȱ ȱ àȱ ȱ ¢ȱ ȱ ¤ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ¤ȱ ȱ ȱȱȱȱàǰȱȱȱȱȱȱȱ Δ9 = πȱ5 Δ[ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ¡ȱȱȱȱàȱȱȱȱȱȱȱȱȱǰȱȱ Δȱ[ ¢ȱ 5ȱ( [L )ǰȱȱȱȱ Q Q ȱȱà ≈ π [5( [L )] Δ[ = πȱ∑ [5([L )] Δ[ ∑ L L= = ȱ ¡àȱ ȱ ȱ ȱ Δ → ȱ (Q→ ∞ ) ǯȱ ȱ ǰȱ ȱȱȱȱàȱȱȱ Q ȱȱà = OLP π ∑ [5( [L)]Δ =π Δ → L= E ∫D [5([L)] G[ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱǰȱȱ·ȱȱȱȱȱȱǱȱ àȱȱ ȱȱȱ 9 = π 5ω ȱȱ → Δ9 = π (5([L )) Δ[ ȱȱȱàȱȱ àȱȱà → 9 = πȱ E ∫ [5([)] G[ D ȱȱȱ ȱȱȱȱȱàȱ¤ǯȱȱ ·ȱȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ àȱ ȱ ȱ ·ȱ ȱ ȱ ǰȱ£ȱȱȱȱȱàǯȱ ȱȱàȱ£ȱ E ∫D Ǳ πȱ [5([)] G[ ȱȱàȱȱ Ǳ π G ∫F [5(\)] G\ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ·ȱȱȱȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ·ȱȱȱȱȱ¡ȱȱàȱǯȱȱȱȱȱȱ ȱȱ¤ȱȱȱȱȱ¡ǯȱȱȱ¢ȱȱȱȱȱȱ¢ȱ¡ȱ ȱȱȱ¢ ȱȱǰȱȱȱȱȱȱȱ ( ) ȱȱȱ ƽ πȱ 5 − U Z ȱ ȱ àȱ ȱ ȱ ·ǰȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ¡ 5ȱ( [) ¢ȱȱȱ U ȱ( [)ǰȱ ȱȱȱȱȱàȱȱȱȱȱàǱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱàȱȱȱȱ 9 =πȱ ∫D [(5([)) E ] − (U([)) G[ ¤ȱ ŗǼȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱȱȱ \ =ȱ [ ȱ ŗǰȱȱȱȱȱ¡ǯȱȱ Ǳ πȱ ŘǼȱ ȱȱ ȱȱàȱȱȱȱȱȱȱȱ àȱ ȱ \ =ȱ [ ǰ \ȱ = ¢ [ȱ = ȱȱȱȱ¢ǯȱȱ Ǳ πȱ řǼȱ ȱàȱ ȱȱ ȱ \ =ȱ [ǰ \ =ȱ [ ȱȱ ȱȱȱȱȱ¡ǯȱ ȱȱȱȱàȱǯȱȱ Ǳ πȱ ŚǼȱ ȱàȱȱȱȱ \ =ȱ [ǰ \ =ȱ [ ȱȱȱȱȱȱȱ \ȱ = ǯȱȱȱȱȱàȱǯȱȱ Ǳ π 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 śǼȱ ȱàȱ ȱȱȱ \ =ȱ [ǰ \ =ȱ [ ȱȱȱȱȱȱȱ¢ǯȱ ȱȱȱȱàȱǯȱȱ Ǳ πȱ ŜǼȱ ȱ ȱ ȱ ȱ àȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱȱȱ¤ I ( [)=ȱ VHQ[ ¢ȱȱȱ¡ ǻ ȱ≤[≤ π Ǽȱȱȱȱȱ¡ǯȱȱ Ǳ ȱπ ŝǼȱ ȱȱȱȱàȱȱȱȱȱȱàȱȱȱȱ ¤ȱ I ( [)=ȱ− [ ¢ Jȱ( [)= ȱȱȱȱ \ȱ = ǯȱȱ Ǳ πȱ ŞǼȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ȱ \ =ȱ [ ¢ \ =ȱ [ ȱȱȱȱ¡ȱȱ Ǳ şǼȱ πȱ ȱȱȱȱàȱȱȱȱȱàȱȱȱȱ¤ȱ \ȱ = [ + \ = [= ¢ [ȱ = ȱȱȱȱ¢ǯȱȱ Ǳ π 'HWHUPLQHHOYROXPHQGHOVyOLGRJHQHUDGRDOURWDUHQWRUQRGHOHMH [ ODUHJLyQSODQD DFRWDGDSRUODVFXUYDV \ = [ \ = [ = &DOFXODUHOYROXPHQGHOVyOLGRTXHVHREWLHQHDOKDFHUJLUDUODUHJLyQOLPLWDGDSRUODV FXUYDV \ = [ \ = [ HQWRUQRDODUHFWD \ = 'HWHUPLQHHOYROXPHQGHOVyOLGRJHQHUDGRDOURWDUHQWRUQRDOHMH [ ODUHJLyQDFRWDGD SRUODSDUiEROD \ = [ \ODUHFWD \ =[ 'HWHUPLQH HO YROXPHQ GHO VyOLGR TXH VH REWLHQH DO JLUDU OD UHJLyQ OLPLWDGD SRU ODV FXUYDV \ = [ \ = [ DOUHGHGRUGHOHMH [ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 àȱȱȱ Ǳ ȱȱ·ȱȱàȱȱȱȱȱȱ ȱȱȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ȱȱȱȱ¢ȱǯȱȱǱ • • • ∫ [ OQ[G[ ∫ [H G[ ∫ȱH FRV[G[ [ [ ȱȱ·ȱȱàǰȱȱȱȱȱȱȱ ȱȱȱǰȱ¢ȱ¤ȱȱǯȱȱàȱȱȱȱȱ ȱȱàȱȱȱȱȱȱǱȱ ȱ ¢ȱ ȱȱȱ ¡ǰȱǰȱȱȱȱ G [( IJ)( [)] =ȱ I( [) GJ( [) + J( [) GI([) G[ G[ G[ ȱȱǰ I ( [) GJ( [) G GI( [) =ȱ [( IJ)( [)] − J( [) ǯȱ G[ G[ G[ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ G [( IJ)( [)] G[ = I( [)J( [) ȱǱ G[ ∫ȱ ⇒ ∫ ∫ GJ([) G[( IJ)( [)] GI( [) G[= G[− J([) G[ G[ G[ G[ GJ( [) GI( [) I ( [) G[=ȱ I([)J( [) − J([) G[ G[ G[ I ([) ∫ ∫ ∫ ȱøȱȱȱȱȱȱàȱȱȱàȱȱ ǯȱȱȱ X =ȱ I( [) ¢ Y =ȱ J( [)ǰȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱǱ ∫ XGY =ȱXY− ∫ YGYǲ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ·ȱ ȱ ȱ ǯȱȱàȱȱȱȱȱȱ¢ȱȱǻȱȱȱȱȱ Ǽȱȱȱȱȱȱȱȱ¡ǯȱȱȱȱȱ ȱ ¢ȱ ȱ ȱ Çȱ ȱ ȱ ȱ ǰȱ ȱ ¤ȱ ȱ ȱ ȱ àȱǱȱȱ ȱȱàȱȱǯȱ ŗǼȱ ȱ ȱ ȱ ȱ ȱ àȱ ¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ àȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ŘǼȱ ȱȱȱȱàȱȱȱ¢ȱȱȱȱàȱ¤ȱ ȱ ȱ¢ȱȱȱȱȱȱȱǯȱȱ ȱ ∫ȱ [H[ G[ àȱ H[ G[=ȱGY H[ =Y X = [ GX =G[ ∫ȱ [H G[= [H − ∫ H G[= [H − H + & [ ȱ [ [ [ [ ∫ȱ[ H G[ [ àȱ X =ȱ [ ¢ GX =ȱ [G[ ¢ H[ G[=ȱGY H[ =ȱY ∫ [ H G[=[ H − ∫ [H G[ ȱ ȱ ȱ ∫ȱ[H G[ ȱ ȱ ȱ ȱ ȱ ǲȱ ȱ ǲ ȱ [ [ [ [ ȱ ȱ ȱ ǰȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ǻȱȱǼǯ ∫[ H G[= [ H − ∫ [H G[= [ H − ⎜⎝ [H − ∫ H G[⎟⎠ [ [ [ ∫ [ ⎛ [ = [ H[ − [H[ + H[G[= [H[ − [H[ + H[ + & [ ⎞ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱ ∫ȱ OQ [G[ àȱ ȱȱȱ OQȱ[G[ XGYȱ ȱȱ ȱ¢ ǰȱȱ àȱ¤ȱȱ X =OQȱ[ \ G[= GYǯȱȱȱȱ ǰȱȱȱ ȱȱǰȱÇȱȱȱȱ ȱȱȱȱȱà ǯȱ GX =ȱ G[ ¢ [ =ȱY [ ǲ ȱ ∫ OQ[G[ = àȱ ∫ XGY= XY− ∫ YGX = [ OQȱ[− [⋅ G[= [OQ[− G[= [OQ[− [+ & ∫ [ ∫ ∫ȱH FRV [G[ [ X=ȱFRV [ ¢ GX =ȱ − VHQ[G[ ¢ H[ G[=ȱGY H[ =ȱY ∫ H FRVȱG[=H FRV[+ ∫ H VHQ[G[ [ ǰ [ [ ¢ H[ G[=GY X =ȱ VHQ[ GX=ȱFRV [G[ ¢ H[ =ȱY ȱȱǱ ∫ H FRV [G[=ȱH FRV[+ H VHQ[− ∫ H FRV[G[ [ [ [ [ ȱ ȱ ȱ àȱ ȱ ȱ ¢ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱȱȱȱǯȱȱǰȱȱȱ¡àȱȱȱ ∫ȱH[ FRV [G[ ȱ ȱ ȱ ȱ ȃȱ àȄȱ ¢ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱȱ· ∫ȱH FRV [G[ ȱǱ [ ∫ ȱ H[ FRV[G[ = H[ FRV[+ H[VHQ[ ∫ H[ FRVȱ[G[= H[ FRV[+ H[VHQ[ + &ǯ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ àȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ǰȱ ȱ Řǰȱ ȱ ȱ ȱ ȱ ȱ ȱ Ǳ ∫ȱ E D E D I([)J′([)G[= I([)J( [) − ∫ȱ WDQ ȱ − ∫ E D I′( [)J( [)G[ [G[ àȱ X=ȱWDQ − [ GX=ȱ + [ ∫ Ǳ G[ = GY ¢ G[ [ =ȱ Y ¢ WDQ − [G[= [WDQ− [ − [ G[ + [ [ = [ȱWDQ− [ − G[ + [ ∫ ∫ ( ) = WDQ−() − − = ȱ OQ+ [ ( ) π − OQ() ( [)G[ ∫ȱVHQOQ àȱ \=ȱOQ [ ǰ G\ =ȱ G[ [ ∫VHQ(OQȱ[)G[= ∫ VHQ( \)⋅ H G\ \ ȱ X=H\ GX= H G\ \ VHQ\G\ =GY − FRVȱ\ = Y ǰ ∫ VHQ(OQ[)G[=∫ (VHQ\)H G\ = −H FRV \+ H FRV\G\ ∫ \ \ \ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱȱȱàȱȱǱ H\ =X FRVȱ\G\ =GY H\G\ = GX VHQ\= Y ǰ ∫ (VHQ\)H G\=−H FRVȱ\+ H VHQ\− ∫ H VHQ\G\ \ \ \ \ ǰ ∫ H\ VHQ\G\ = ȱȱǱ ȱ − H\ FRVȱ\+ H\VHQ\ + & ∫VHQ(OQȱ[)G[= ȱȱ − [FRV(OQ[) + [VHQ(OQ[) + & ∫ȱ VHF [G[ àȱ ȱ ∫ȱ VHF [G[= ∫ VHF[⋅ VHF [G[ ȱ àȱ ȱ ǯȱ X =ȱVHF [ ¢ GY=ȱVHF [G[ǰȱ GX =ȱVHF [WDQ[G[ ¢ Y=ȱWDQ [ Ǳ ∫ VHF [G[ = VHF[WDQ[− ∫ VHF[WDQ[WDQ[G[ ∫ = VHF[WDQ[− ∫ = VHF[WDQ[− ∫ =ȱVHF [WDQ[− VHF[WDQ [G[ VHF[(VHF [− )G[ VHF [G[+ ∫ = VHF[WDQ[+ OQVHF[+ WDQ[ − VHF[G[ ∫ VHF [G[ ∫ȱ VHF [G[= VHF[WDQ[+ OQVHF[+ WDQ[ ȱȱǱ Ǳ ∫ VHF [G[ = VHF[WDQ[+ OQVHF[+ WDQ[ + & 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ¤ȱ ȱȱ ȱ ȱ ǻǱȱ ȱȱ ȱ ·ȱ ¤ȱ ǯȱ ȱȱȱàȱȱǼǯȱ ŗǯ Řǯ řǯ ∫ȱ[ FRV [G[ ∫ȱ VHQ [G[ ∫ȱW H GW ∫ȱ W OQWGW ∫ȱ[ OQ[G[ ŗŘǯ ∫DUFVHQ[G[ ŗřǯ ∫ȱ[ VHQ[G[ − ȱ W Śǯ śǯ ∫ π [FVF [G[ ∫H G[ ∫ȱ[WDQ [G[ ∫ȱ ([+ )DUFWDQ([)G[ Ŝǯ ŝǯ Şǯ şǯ ŗśǯ ŗŜǯ [ ȱ ŗŝǯ − ∫ȱ [ DUFVHQ[G[ ∫ȱ( ) + [ G[ ∫ȱWOQ(W+ )GW ∫ȱ[ FRV [ G[ ∫ȱ[VHQ[ FRV [G[ VHQ ( [) G[ ∫ − ȱ − [ ∫ȱ ∫ȱ [(OQ[) G[ ∫ [ [− G[ ŗŞǯ ŗşǯ ŘŖǯ DUFWDQ [ H ŗŖǯ ŗŚǯ πȱ ŗŗǯ [H[G[ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱȱ·ȱ Ǳ ȱ ȱȱȱȱ ·ǯȱ ȱ £¤ȱ ȱ ȱ ·ȱ ȱ ȱ ȱȱȱǱ •ȱ ∫ȱVHQ [FRV P Q ∫ [G[ VHFP [WDQQ [G[ǯ ȱȱȱȱȱȱ¢ȱȱǯȱ ȱ £¤ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱ ∫ȱVHQ [FRV P Q [G[ǰȱȱȱȱ¢ȱȱȱøȱ ȱǯȱ Ǽȱ ȱȱȱȱȱȱǰȱȱȱȱȱ¢ȱȱȱ¤ȱ ȱǰȱȱȱǯȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ¤ȱȱǰȱȱȱǯȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ǰȱ ȱ ȱȱǱ − FRV[ VHQ [=ȱ FRV [ + FRV [= ȱȱȱȱȱȱȱ ȱȱȱǯȱȱ ȱ ∫ȱVHQ [G[ à , = ∫ȱVHQ [G[= ∫ VHQ [⋅ VHQ[G[= ∫ ( − FRV [)VHQ[G[ X=ȱFRV [ GX =ȱ− VHQ[G[ ǰ , =− X ( = −X+ − X )GX ∫ + & =− FRV[ + FRV [ + & 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱ ∫ȱ FRV [VHQ[G[ à , = ∫ ∫ FRV [VHQ [FRV[G[ = (FRV [) VHQ [FRV[G[= (− VHQ [) VHQ [FRV[G[ ∫ȱ ∫ FRV [VHQ[G[ = X =ȱ VHQ[ GX =ȱ FRV [G[ ∫ ( − X ) X GX= ∫ (− X , = = = ∫ (X − X + X)GX= + ȱ VHQ [ VHQ [ VHQ [ ȱ ∫ − π FRV [ πȱ VHQ[ + X − ) + X XGX X X + + & + & G[ à , = ∫ π πȱ FRV [ VHQ[ G[= ∫ π FRV [ π VHQ[ FRV[G[= ∫ π − VHQ[ FRV[G[ π VHQ[ X = VHQ[ ⇒ GX = FRV [G[ ⎛ π ⎞ ⎛ π ⎞ ⎛π ⎞ ⎛π ⎞ X⎜ ⎟ = VHQ⎜ ⎟ = X⎜ ⎟ = VHQ⎜ ⎟ = ⎝⎠ ⎝ ⎠ , = ∫ − X X ⎝ ⎠ GX = ∫ ⎝ ⎠ ⎛ − ⎞ ⎛ ⎞ ⎜ X − X ⎟GX = ⎜ X − X ⎟ ⎜ ⎟ ⎜ ⎟⎠ ⎝ ⎠ ⎝ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎟ − ⎜ ⎟ + ⎜ ⎟ ⎟ − ⎜ = ⎜⎜ ⎟ ⎜⎝ ⎟⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ = − 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱ à ,= ∫ȱ FRV [G[ + FRV[ G[ [ VHQ[ = + + & ∫ FRV [G[= ∫ ȱ ȱ ∫ȱVHQ [G[ à ∫ ∫ (VHQ [) G[= ∫ = (− FRV[+ FRV [)G[ ∫ VHQȱ[G[ = ,= ⎛ − FRV[⎞ ⎜ ⎟ G[ ⎝ ⎠ [ VHQ[ ∫ ∫ FRV [G[ + [ VHQ[ + FRV[ G[ = − + [ VHQ[ [ VHQ[ = − + + ⋅ + & [ VHQ[ VHQ[ = − + + & = ȱ − ∫ȱVHQ [FRV [G[ à ⎛ − FRV[⎞⎛ + FRV[⎞ ⎜ ⎟⎜ ⎟G[ ⎝ ⎠⎝ ⎠ [ + FRV[ ( − FRV [)G[= − = G[ [ VHQ[ = − [− ⋅ + & [ VHQ[ = − + & ,= ∫ ∫ VHQ [FRV [G[= ∫ ∫ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 •ȱȱ ȱȱȱȱȱȱȱ¢ȱȱȱȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ǰȱ ȱ ȱ VHFȱ [ ¢ȱ ȱȱȱȱȱǯȱȱàǰȱȱȱǯȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ǰȱ ȱ ȱ VHFȱ[ WDQ[ ¢ȱȱȱȱȱȱǯȱ Ǽȱ ȱ ȱ ¢ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ǰȱ ȱȱ WDQȱ [ ȱȱ VHFȱ [ǰȱȱ¢ȱȱȱȱȱȱ ǯȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ ȱ ∫ȱ VHF P [G[ǰȱ ȱ ȱ ȱ¢ȱ ǰȱ ȱ ȱǯȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱȱȱȱȱ¢ȱǯȱȱ ȱ ∫ȱ WDQ [VHF [G[ à ∫ȱ WDQ [VHF [G[= = ∫ ∫ WDQ [VHF [VHF [G[ WDQ [(+ WDQ [)VHF [G[ X=ȱWDQ [ ǰȱǰ GX=ȱVHF [G[ ǰ ∫X ( + X )GX= ∫ (X + X )GX = X ȱ X + & WDQ [ WDQ [ = + + & ȱ ∫ȱ WDQ + [VHF [G[ àǱ ∫ ∫ WDQ [⋅ VHF [⋅ VHF[WDQ[G[ = (WDQ [) VHF [VHF[WDQ[G[ ∫ȱ = (VHF [− ) VHF [VHF[WDQ[G[ ∫ WDQ [VHF [G[= X=ȱVHF [ GX= VHF [WDQ[G[ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ∫ WDQ [VHF [G[ = ∫ (X − ) X GX = (X − X + X − )X GX ∫ = (X − X + X − X )GX ∫ ȱ X X − + & VHF [ VHF [ VHF [ VHF [ = − + − + & = ȱ X X ∫ − + πȱ WDQ [G[ à ∫ π WDQ [G[= ∫ = = π (WDQ [)(WDQ [)G[ ∫ ∫ π (VHF [− )WDQ [G[ π WDQ [G[+ ∫ πȱ = − ∫ (VHF π WDQ [VHF [G[ ) ∫ [ − G[+ π π (− WDQ[+ [) + ∫ π WDQ [VHF [G[ WDQ [VHF [G[ X=ȱWDQ [ G[=ȱVHF [G[ ȱȱǰ ȱ ∫ π ⎛ ⎛π ⎞ π ⎞ WDQ [G[= ⎜⎜ − WDQ⎜ ⎟ + ⎟⎟ + ∫ XGX ⎝ ⎠ ⎠ ⎝ π π Xȱ π =−+ + = −+ + = − + ∫ȱ WDQ [G[ à ∫ ∫ = ∫ WDQ [G[= WDQ[WDQ [G[ ( ) ∫ WDQ[VHF [− G[ = WDQ[VHF [G[− ∫ WDQ[G[ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 X =ȱ WDQ [ ∫ WDQ[ VHF [G[ = ∫ XGX =ȱ X WDQ [ + & = + & ǰ ∫ ȱ WDQȱ [G[ = ∫ȱ WDQ [ VHF[ WDQ [ − OQ(VHF[)+ & G[ àȱ ȱ ∫ ȱ WDQȱ [ VHF[ G[= ∫ ȱ ȱ ¡ȱ (WDQ [)⋅ VHF[G[= VHF[ VHF[ ∫ ȱ ȱ ȱ ⎛ WDQ [ ⎞ ⎜ ⎟G[= VHF [ WDQ [ ⋅ ⎜ ⎟ ⎝ VHF [ ⎠ ∫ Ǳ ⎛ VHF [− ⎞ ⎜ ⎟G[ VHF [ WDQ [ ⋅ ⎜ ⎟ ⎝ VHF [ ⎠ X =ȱVHF [ GX=ȱVHF [WDQ[G[ ǰ VHF [− VHF[WDQ[G[ = VHF [ ∫ X − GX X ∫ = ∫ ⎛⎜ X − X− ⎞⎟GX= ȱX + X− + & ⎠ ⎝ ȱȱȱȱȱȱȱȱ¢ȱȱȱȱ ȱȱȱȱȱ¤ȱȱȱȱȱ ȱȱȱȱ£ȱȱ +ȱFRW [= FVF [ǯȱȱ ȱ ∫ȱ FRW [FVF [G[ àǱ ∫ ∫ FRW [FVF [FVF [G[ [G[ = ∫ȱ FRW [(+ FRW [)FVF FRW [FVF [G[ = X=ȱFRW [ GX= − FVF [G[ ∫ȱ VHF Q [WDQP [G[ǰȱ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ∫ FRW [ FVF [G[= −∫ X (+ X )GX= −∫ (X + X )GX =− X − X + & FRW [ FRW [ =− − + & •ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ǯȱ ȱȱȱǱȱ ∫ȱVHQQ[ FRV P[G[ Ǽ VHQQ[ FRV P[G[ ∫ȱ Ǽ FRV P[ FRVP[G[ ∫ȱ Ǽ ȱȱȱȱȱǱ VHQ$FRVȱ%= [VHQ( $− %) + VHQ( $+ %)] VHQ$VHQ%= [FRV( $− %) − FRV( $+ %)] FRV$FRV%= [FRV( $− %) + FRV( $+ %)] ȱ ∫ȱVHQ [FRV[G[ à ∫ VHQ [FRV[G[= ∫ (VHQ([− [) + VHQ([+ [))G[ = VHQ([) + VHQ([))G[= ( (VHQ[+ VHQ[)G[ ∫ ∫ = FRV[ FRV[ ⋅ − ⋅ + & = FRV[ FRV[ − + & 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ¤ȱ ȱȱȱȱȱȱȱȱ ŗǯ ෾ȱFRV [VHQ [G[ Řǯ ∫ ⎛ [⎞ FRV ⎜ ⎟G[ ⎝ ⎠ řǯ ∫ȱ ȱ ∫ȱ (WDQ [− VHF [)G[ VHF[ ∫ȱ FRV[G[ ŗŗǯ ∫ VHF [ WDQ[G[ πȱ FRV[ G[ + VHQ[ π ∫ VHQ[FRV[G[ ŗŚǯ ∫ (VHQ [+ )G[ π FRV[G[ ŗśǯ VHQ [+ ∫ȱ −πȱ ⎛ [⎞ WDQȱ⎜ ⎟G[ ⎝ ⎠ ŗŖǯ ∫ π ∫ śǯ ∫ȱ WDQ [VHF [G[ Ŝǯ ∫ȱ VHF [ WDQ[G[ ŝǯ ∫ȱ WDQ [VHF [G[ Şǯ ∫ȱ FVF [FRW[G[ şǯ ŗŘǯ ŗřǯ VHQ [G[ Śǯ πȱ − ŗŜǯ πȱ ∫ G[ ȱ− VHQ[ ∫ȱVHQ [ FRV[G[ VHQ ( [) ŗŞǯ ∫ȱ [ G[ G[ ŗşǯ ∫ȱ VHQ [ − WDQ [ ŘŖǯ ∫ȱ VHF [ G[ FRV [+ VHQ[ Řŗǯ ∫ȱ VHQ[ G[ ŘŘǯ ∫ FRV [FRV[FRV[G[ ŗŝǯ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 àȱ·ȱ Ǳȱ ŗǼȱ ȱȱȱȱȱȱȱȱȱȱȱȱȱȱ ȱǱ • [ =D VHQθ • [ =D WDQθ • [ȱ =D VHFθ ŘǼȱȱȱȱȱȱȱȱ·ȱȱàȱ ·ǯȱ ȱȱȱ¤ȱȱȱÇȱȱȱȱǰȱȱȱȱȱ ȱ ෾ D ȱ [ G[ǲȱȱ D ȱ> ǯȱȱȱȱȱȱ ȱȱȱȱȱ·ǰȱȱȱȱ·ȱ ȱȱȱ·ȱȱȱȱȱȱǯȱ ȱ ȱ ȱ ȱ ȱ ·ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ θȱ ȱ ȱ à [ȱ =D VHQθ ǰȱ ȱ ȱ − VHQȱθ = FRVθ Ǳ D −ȱ[ = D(− VHQθ) = D FRVθ = DFRVθ ȱ ȱ ȱ à [ =ȱ DVHQθ ǲȱ ȱ ȱ ȱ ȱ ȱ à VHQȱθ ȱàȱǲȱȱǰȱȱȱ¢ǲȱ¢ȱȱȱȱ ⎡ πȱ π ⎤ ȱȱȱ θȱ ȱ ⎢− ⎥ ǯȱ ⎣ ⎦ π π FRV ϑȱ > ǰȱ −ȱ ≤ θ ≤ ȱ ∫ȱ − [ [ D −ȱ [ = DFRVθ ǯȱȱ G[ àǱȱ [= VHQθ ⇒ G[=ȱ FRVθGθ ¢ −[ = − VHQθ = FRVθ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱȱǰ ∫ − [ ∫ ∫ FRVθ ⋅ FRVθ Gθ (VHQθ ) G[= [ = = = FRV θ VHQ θ ∫ Gθ = ∫ FRWθ Gθ (FVF θ − )G θ = FRWθ − θ + & FRVθ VHQθ = − θ + & = − [ ȱ [ − [ − VHQ−⎛ [⎞ + & ⎜ ⎟ [ ⎝ ⎠ ⎛ [⎞ − VHQ−⎜ ⎟ + & ⎝ ⎠ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ·ȱ ȱ ȱ ȱȱȱȱ¢ȱȱȱ¡ȱȱȱ D −[ ¡àȱ D −[ D + [ [ −D ȱ D ȱ + [ [ − D FRQ D > ǯȱ àȱ π π [ =DVHQθ − ≤ θ ≤ π π [ =DWDQ θ − < θ < [ =DVHFθ ≤ θ < ∫ȱ − VHQθ = FRVθ +WDQθ = VHFθ π π π ≤θ < VHF θ −ȱ= WDQθ [ + G[ àȱ [ȱ = WDQθǰȱǰ G[ = VHF θGθ ∫ ∫ WDQ θ + ⋅ VHF θ Gθ = ∫ VHF θ ⋅ VHF θ Gθ π π = VHF θ Gθ VHFθ > VL < θ < ∫ [ + G[= 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ ǻȱ Ǽȱ [ ǲ [ ȱ + G[ = [ + + OQ [ + + [ + & ( ∫ ȱ ∫ ) G[ȱ [ + àȱ [ȱ = WDQθ ǰȱ ǰ G[ =VHF θGθ ∫ G[ G[= [ + ∫ VHFθ Gθ G[= ∫ VHFθ Gθ VHFθ WDQθ + VHFθȱ Gθ = OQVHFθ + WDQθ + & = OQ [ + + [ + & = ȱ ∫( ∫ G[ȱ [ + ) àȱ [ȱ=WDQ θǰȱ ǰ G[= VHF θGθ ∫ ( VHF θȱ Gθ ) ⎡ WDQθ + ⎤ ⎥⎦ ⎢⎣ = ȱ ∫ = ∫ VHFθ Gθ (VHFθ ) FRVθ Gθ = VHQθ + & = ∫ȱ [ ( [ ) + [ [ + = ∫ + & G[ àȱ [ȱ =WDQθ ǰȱǰ G[=ȱVHF θGθ ǯȱ ȱȱȱȱȱÇȱȱàǱ Gθ VHFθ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ⋅ WDQθ ⇒ θ = ⋅ ∫ [ π = WDQθ ⇒ = WDQθ ⇒ θ = π ∫ ∫ G[ = ( [ + ) π ȱ = = ⋅ VHFθ ∫ VHQ θ Gθ = FRV θ π ∫ WDQθ ⋅ VHFθ Gθ π WDQθ Gθ VHFθ − FRVθ FRVθ ⋅ VHQθ Gθ πȱ X=ȱFRVθ ⇒ GX= − VHQθ Gθ ǯȱ FRV = \ FRV = ȱ ∫ ∫ [ȱ G[= − ([ + ) − X X ∫ GX= (X − − )GX ⎛ ⎞ = ⎜ X− − X⎟ = ⎠ ⎝ ȱ ∫ [ G[ȱ [ − à [ȱ= VHFθ ǰȱǰ G[= VHFθWDQθ Gθ ∫ G[ [ [ − = ∫ VHFθ WDQθ VHFθ VHFθ − = ȱ ∫ȱ ∫ [ − [− [ Gθ = FRVθȱ Gθ = ∫ VHFθ WDQθ VHFθ WDQθ VHQθ + & = Gθ [ − + & [ G[ àȱ ( ) ȱ −[ − [+ = − [ + [− = − ( [ + ) + = − ( [+ ) 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ǰȱȱȱȱȱȱȱȱȱǱ ∫ȱ [ − [− [ G[ = ∫ [ − ( [+ ) G[ [+ȱ = VHQθ ǰȱǰ G[ȱ=FRVθ Gθ ¢ȱȱȱǱ ∫ = [ − [− [ ∫ G[ = ∫ VHQθ − − VHQθ VHQθ − ⋅ FRVθ Gθ = FRVθ = − − [ − [ȱ − VHQ− ⋅ FRVθ Gș = ∫ ∫ (VHQθ − ) ⋅ FRVθ Gθ Gș = − VHQθ VHQθ − Gθ = −FRVθ − θ + & [+ + & ¤ȱ ȱȱ ȱȱ ŗǼ ∫ȱ[ ŘǼ ∫ȱ řǼ ∫ ŚǼ − [G[ [ − D [ G[ G[ȱ [ [ − ∫ȱ( [ [ + ) śǼ ∫ȱ( ŜǼ ∫ G[ G[ ) + [ G[ȱ [ + [+ ŝǼ ∫ȱ HW − HW GW ŞǼ ∫ȱ H şǼ ∫ ŗŚǼ − GW ŗśǼ [ + ∫ȱ ŗŘǼ ∫ȱ ∫ȱ − [ [ [ − G[ [− [ − H[ H[ ∫ȱ G[ [ [ + G[ȱ G[ [G[ ŗŗǼ [ OQ [− [ȱ G[ ∫ ∫ȱ OQ [ W ŗŖǼ ŗřǼ G[ ŗŜǼ ∫ȱ [ + [+ G[ [ + [ + ŗŝǼ ∫ȱ G[ + [ + [ ŗŞǼ ∫( − + [ ) G[ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 àȱȱȱȱ Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ·ȱ ȱȱ ȱ àȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǲȱ ȱ ȱ ȱ ȱ ȱ ȱ I ( [) =ȱ S( [) ǰȱ Sȱ\T ȱȱȱ Tȱ( [)≠ ǯȱ T( [) ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ Tȱ [ ȱ¢ȱȱȱȱȱ Sȱ [ ǯȱȱȱȱȱ ȱȱȱȱǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ¤ȱ ȱ ȱ Ǳȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱàȱȱ¤ȱȱàȱȱǯȱȱ ȱ ∫ȱ [ + [+ G[ [ + àǱȱ ȱ ȱ ȱ ȱ àǰȱ ȱ ȱ Ǳ [ȱ +[+ ǰȱ ȱ [ + [ȱ +[+ [ = + ǯ [ + [ + ∫ ⎛ ⎞ ⎜ + [ ⎟G[ = [+ ⎜ ⎟ [ + ⎠ ⎝ =[ + OQ[ȱ + + & [ + [+ G[= [ + ȱ ∫ȱ ∫ [ ∫ [ + G[ [ − [ + [+ G[ [ − àǱȱ ȱȱȱȱȱȱȱàȱȱǱ [− [ −[ + [+ = [ − + [ − [ − 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ∫ [ − [ + [+ G[ = [ − = ∫ [ [− ⎞ ⎛ ⎟G[ ⎜ [ − + [ − ⎠ ⎝ − [+ ∫ȱ [− G[ [ − ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ S [ Sȱ [ ȱȱȱ G[ǰȱ ȱȱàȱȱǯȱ T [ T [ ∫ȱ Sȱ [ ȱ ȱ T [ ȱȱȱǯȱȱȱȱȱȱȱȱ ȱ£ Tȱ [ ȱȱȱ ȱȱȱ¢Ȧȱ¤ǯȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ à ȱ ȱ ȱ ȱ ȱ Çȱ ȱ ȱ ȱ Tȱ [ ǲȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ȱȱ ȱ ȱ ¡ȱȱ ȱ ȱȱ ȱȱ ¢Ȧȱ ¤ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ £ȱ Tȱ [ ȱ ȱ £ȱ ȱ ȱ ȱ ȱ ȱ ¢Ȧȱ¤ǰȱȱ·ȱȱȱȱȱȱȱȱ ȱ£ȱȱȱǯȱ ȱ ȱ Tȱ [ ȱ ȱ ȱ ȱ ȱ ȱ ǲȱ ȱ T [ = (Dȱ[+ E)(D[+ E) …⋅ (DN [+ EN )ǲȱȱȱȱ¢ȱȱȱȱǯȱȱ ȱǰȱȱȱȱȱȱȱȱȱ¡ȱ $ $ $[ S ȱ[ $ȱ$…$N ǰȱȱǱ ǯȱ = + + … T [ D[+ E D[+ E DN [+ EN ȱȱȱȱȱȱȱȱȱȱǱȱȱ ȱ ø ∫ȱ [− G[ [ − [− à [ȱ− ȱ ȱ àȱ ǯȱ ȱ £ȱ ȱ ȱ ȱ [ − [− Ǳ [ −[− = ([− )( [+ ) 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱȱȱȱȱǰȱ¡ȱȱøȱȱȱ¢ȱȱ [ȱ− $ % = + ȱ ǯȱ ([− )([+ ) [− [+ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ $ȱ % $([+ ) + %( [− ) + = ([− )([+ ) [− [+ $ % $( [+ ) + %( [− ) [ȱ− = + = ([− )([+ ) ([− )([+ ) [− [+ ǰ [ −= $( [+ ) + %( [− ) ⇒[ȱ− = $[+ $+ %[− % ⇒ [− = ( $+ %)( [) + $− % ⎧ $+ %= ⇒⎨ ⎩ $− %= − ȱȱȱȱȱ $ȱ= %= Çǰ ∫ [− G[ = [ − [− ∫ G[+ [− ∫ G[ [+ =ȱOQ [ − + OQ [+ + & Ǳȱ ȱȱ£ȱȱ·ȱȱȱȱȱȱȱȱ ¢ȱǯȱȱà [ȱ−= $( [+ ) + %( [− ) ȱȱǲȱȱȱȱȱ ȱȱ¡ǯȱ ȱȱȱȱȱȱ¡ȱȱȱǱȱ [ȱ=− (−) − = $(ȱ) + %(− )ǰȱ − = −%ȱ⇒ %= ǯȱ [ȱ = () −ȱ= $() + %()ǰȱ =$ȱ⇒ $= ǯȱȱ ȱ ∫ȱ [ + [− G[ [ − [ + [ à [ +ȱ[− ȱ ȱ àȱ ǰȱ ¢ȱ ȱ £àȱ ȱ ȱ [ − [ + [ ȱǱ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 [ȱ −[ + [ = [([ − [+ ) = [( [− )( [− ) ǰȱ¡ȱøȱȱǰȱȱ¢ȱȱȱǱ [ +[− $ % & = + + [( [− )( [− ) [ [− [− = $(ȱ [− )( [− ) + %[( [− ) + &[( [− ) [( [− )( [− ) ȱȱȱ [ +[− = $( [− )( [− ) + %[( [− ) + &[( [− ) 6L [ = ⇒ = −& ⇒ & = − 6L [ =⇒ −= $⇒ $= − 6L [ = ⇒ = %⇒ % = ∫ G[ + [ ∫ [ + [− G[= − [ − [ + [ ∫ G[ +− [− ∫ G[ [− ȱ =− OQ[ + OQ[− − OQ[− + & ȱ Tȱ [ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ȱ ȱ ȱ ȱ ȱ ǰ (D [+ȱE)ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ $ £àȱ Tȱ [ ǯȱ ǰȱ ȱ ȱ ȱ ·ȱ ø ȱ D[+ȱE $ $ $Q ǯȱȱ ¤ + + …ȱ+ D[+ E (D[+ E) (D[+ E)Q ȱ ∫ȱ [ + [− G[ [( [+ ) à $( [+ ) + %[( [+ ) + &[ [ +[− $ % & = + + = [ [+ ( [+ ) [( [+ ) [( [+ ) 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ǲ [ +[− = $( [+ ) + %[( [+ )+ &[ ( ) = $ [ + [+ + %[ + %[+ &[ [ +ȱ[− = ( $+ %)[ + ($+ %+ &)[ + $ ⎧ $ȱ+ % = ⎪ ⎨$+ %+ & = ⎪$= − ⎩ ǰ \ & = − ȱȱȱȱȱ $ = − % = ∫ [ + [− [([+ ) G[= − ∫ ∫ G[ + [ ∫ G[ − [+ G[ ( [+ ) ȱ = − OQ [ + OQ[+ + + & ([+ ) ȱ ø ∫ȱ [ ⎛ ⎞ ⎜ ⎟ G[ ⎝ [ − [+ ⎠ à ∫ [ ⎛ ⎞ ⎜ ⎟ G[ = ⎝ [ − [+ ⎠ ∫ [ G[ ([− )([− ) [ $ % & ' = + + + ([−) ([− ) [− ([− ) [− ([− ) $(ȱ [− )( [− ) + %( [− ) + &( [− )( [− ) + '( [− ) ([− )([− ) = ǰ [ȱ = $([− )( [− ) + %( [− ) + &( [− )( [− ) + '( [− ) [ȱ= %ȱ = [ȱ= 'ȱ= [ȱ = − −& + $= [ȱ = &ȱ−$= − 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ⎧ ⎪⎪&ȱ− $= − ȱȱȱ ⎨ ⎪− & + $= ⎪⎩ ȱ $ȱ= ¢ &ȱ =− Çǰ ∫ ∫ = ȱ ∫ [ G[= ([− ) ([− ) ∫ȱ G[ +− ([− ) ∫ G[ + [− G[ ([− ) − ȱ − OQ[− − + & ( [ − ) ( [− ) [ − [ + [+ G[ [ − [ − [+ àȱ [ȱ −[ + [+ ȱȱàȱǰȱȱȱȱȱȱȱ [ − [ − [+ [ȱ −[ + [+ [ = [+ + ȱàȱȱ ǯȱ [ − [ − [+ [ − [ − [+ Ǽ [ȱ ȱ ȱ àȱ ǰȱ ȱ £àȱ ȱ [ −[ − [+ [ȱ − [ − [+ = ( [− ) ([+ )ǯȱ Ǽ Ǽȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ȱ [ȱ $ % & = + + ([−) ([+ ) [+ [− ([− ) [ȱ = $( [− ) + %([+ )([− ) + &( [+ ) [ȱ = ⇒ & = [ȱ = − ⇒ $= − [ȱ = ⇒ % = ȱȱÇȱǱ ∫ [ − [ + [+ G[ = [ − [ − [+ ∫ [+ G[− = [ ∫ G[ + [+ ∫ ∫ G[ + [− + [− OQ [+ + OQ [− − G[ ([− ) + & [− 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱ Tȱ( [) ȱȱ¤ȱǰȱȱȱȱȱȱǯȱ Tȱ( [) ȱȱ D[ ȱ +E[+ Fǰȱȱ Eȱ −DF< ȱȱ¡à Sȱ( [) ȱȱ¡ȱȱȱȱȱǰȱ¤ȱȱ·ȱȱ T( [) $[ +% ǰȱȱȱ¢ȱȱȱǯȱȱ ȱǱ ȱ D[ + E[+ F ȱ ∫ȱ [ − [− G[ [([ + ) à [ −[− $ %[+ & = + [([ + ) [ [ + = ǰ $(ȱ [ + ) + [(%[+ &) [([ + ) [ − [− = $[ + $+ %[ + &[ =ȱ( $+ %)[ + &[+ $ Çǲ $ȱ+ %= ǰ &ȱ =− ǰ $ȱ=− ǰ $ȱ= − ǰ %ȱ = ǰ & = − ∫ ∫ ∫ G[ ∫ [ ∫ [ − [− G[= − [([ + ) G[ + [ + =− [+ − G[ [ + [ ∫ G[− [ + ȱ G[ [ + ⎛ [⎞ = − OQ [ + OQ[ + − WDQ−⎜ ⎟ + & ⎝ ⎠ ȱ ∫ȱ [ [ G[ − àǱ $ %[+ & [ [ = + = [ − ( [− )([ + [+ ) [− [ + [+ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 = ( ) ([− )([ + [+ ) $ȱ [ + [+ + ( %[+ &)( [− ) ǰ [ =( $+ %)[ȱ + ( $− %+ &)[+ $− & ǰ ⎧ $ȱ+ % = ⎪ ⎨ $− %+ & = ⎪ $− & = ⎩ ȱȱȱȱȱ $ȱ= ǰ %ȱ = − ǰ & = ∫ [ [ ∫ ∫ G[ȱ [− = +− G[ ∫ [− ∫ [ + [+ G[ [+ = − G[+ ∫ [− ∫ [ + [+ ∫ G[ = − − [+ G[ [ + [+ G[ + [− ȱ [ + [+ = [ + [+ G[ [ + [+ + ⎞ ⎛ =⎜ [ȱ+ ⎟ + ⎠ ⎝ ǰ ∫ [ ȱ [ G[ = − ∫ ∫ G[ − [− ∫ [+ G[+ [ + [+ G[ ⎞ ⎛ ⎜ [+ ⎟ + ⎠ ⎝ ⎛ [+ ⎞ = OQ[ − − OQ[ȱ + [+ + ⎟⎟ + & WDQ−⎜⎜ ⎝ ⎠ ∫ȱ [ − [+ G[ [ − [+ àǱ [ȱ −[+ ȱ ȱ àȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ [ − [+ àȱȱǱ [− [ȱ −[+ ǯ = + [ − [+ [ − [+ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ∫ ǰ [ − [+ G[ = [ − [+ ∫ ∫ ∫G[ ∫ ∫ G[ ∫ ∫ G[ ∫ [− G[ [ − [+ G[+ = = = + [− G[ [ − [+ + [− G[− [ − [+ G[ [ − [+ + [− G[− [ − [+ G[ ȱ ∫ ∫ ([− ) + ⎛ [− ⎞ WDQ−⎜ = [+ OQ[ − [+ − ⎟ + & ⎝ ⎠ ȱ ∫ȱ G[ [ + [ + à $[ + % &[+ ' = = + [ +[ + [ + [ + [ + [ + ( )( ) =ȱ( $[+ %)([ + ) + (&[+ ')([ + ) = ( $+ &)[ + (%+ ')[ + ( $+ &)[+ %+ ' ȱÇȱȱȱȱȱȱȱǱ ⎧ȱ= $+ & ⎪= %+ ' ⎪ ⎨ ⎪= $+ & ⎪⎩= %+ ' ȱȱȱȱǰȱȱ $ȱ= ǰ %ȱ=− ǰ &ȱ = ǰ '= ∫ G[= − [ + [ + ∫ G[ [ + + ∫ G[ ⎛ [ ⎞ ȱ =− WDQ−⎜ ⎟ + WDQ−( [)+ & [ + ⎝ ⎠ ȱ Tȱ( [) ȱ ȱ ȱ ¤ȱ ȱ ǯȱ ȱ Tȱ( [) ȱ ȱ ( ) U D[ ȱ +E[+ F ǰȱ Eȱ −DF< ǰȱ ȱ ȱ àȱ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 Sȱ [ $[ +% ǰȱȱȱȱȱàȱȱø ȱ ǰȱ T [ D[ + E[+ F $[+% $[+ % $U[+ %U ȱȱȱ ǯȱȱ + + …ȱ+ D[ + E[+ F (D[ + E[+ F) (D[ + E[+ F)U ȱȱ ȱ àǱ [ + ([ ) + ∫ȱ ([ + ) G[ [ + ( ) $[+ % &[+ ' =( $[ + %) [ȱ + + &[+ ' + [ + ([ + ) = ǰ [ ȱ + = $[ + %[ + ($+ F)[ + %+ ' Ǳ $ȱ= ǰ %ȱ= ǰ &ȱ =− ǰ ' = ∫ [ + ([ + ) G[= ∫ = [G[ + [ + ∫ ∫ ([ + ) = ∫ ǰ ∫ ([ [ + ) + G[ = ∫ = G[ [ ([ ) + ∫ G[+ ([ G[ ) + G[ȱ=VHF θGθ ¢ȱȱȱȱ [WDQ θ + ] = ∫ VHF θ Gθ = ([ + ) [G[ȱ − [ + [ȱ = WDQθ G[ ∫ − [+ ∫ = FRV θGθ = VHQ θ + θ VHQθȱFRVθ + VHFθ Gθ VHFθ ∫ (FRVθ + )Gθ + & θ + & = ( [ + ) [ + ⎛ [⎞ WDQ−⎜ ⎟ + & ⎝ ⎠ [ ⎛ [⎞ OQ [ + + WDQ−⎜ ⎟ + & + + [ + [ + ⎝ ⎠ ( ) ( ) 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ¤ȱ ·ȱȱȱȱȱǯȱ ∫ȱ ŗǼ ∫ [ + ȱ G[ [ − [ ŘǼ ∫ȱ [ + [ − [+ G[ [ + [− ŗŖǼ řǼ ∫ G[ȱ ([+ ) ([− ) ŗŗǼ ∫ȱ VHQ [ ŚǼ ∫ȱ [ G[ ([+ ) ŗŘǼ ∫ȱ VHQ[FRV [ G[ + FRV [ śǼ ∫ȱ [ + G[ [ + [ ŗřǼ ∫ȱ H[ G[ (H[ − )(H[ + ) ŜǼ ∫ȱ [ − [+ G[ ([− )([ + ) ŗŚǼ ∫ȱ H[ G[ (H [ + )(H[ − ) ŝǼ ∫ȱ [ − [ + [+ G[ [ + [ + ŗśǼ ∫ȱ FRV[ G[ VHQ [+ VHQ[− ŞǼ ∫ȱ [ G[ [ − ŗŜǼ şǼ [− G[ ([ + [+ ) ∫ȱ [ + [ G[ [ + [ + (VHQ[− )FRV[ G[ ∫ − VHQ[+ DUFVHQ[ G[ [ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 àȱȱȱȱȱȱȱ Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ·ȱ ȱ à ⎛ [⎞ Wȱ = WDQ⎜ ⎟ ǯȱ ⎝ ⎠ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱàȱȱȱȱȱȱȱ ȱȱǱȱȱ ȱ ∫ G[ȱ [− − [− àȱ X=ȱ [− ⇒ X = [− ⇒ XGX= G[ ǰȱȱȱȱȱȱȱȱȱǱ ∫ G[ = [− − [− ∫ XGX X − X ∫ ∫ ⎞ X ⎛ = ⎜ X+ + ⎟ GX= ∫ ⎝ X− ⎠ + X+ OQX− + & = ȱ X GX = X(X− ) X X− = [− + [+ OQ [− + & ȱ ∫ [ [+ ȱ + [ G[ àȱ [ȱ+ = X ⇒ [+ = X ⇒ [ = X − ǯȱȱ G[ =ȱXGXǯ ǰ ∫ [ [+ ȱ + [ G[ ƽ (X − )X GX= ∫ X+ X − ∫ȱ X + X− GX X − X ȱȱȱàȱȱȱȱǱ ⎛ ⎞ X− ⎜ ȱX − + X− ⎟GX ƽ ⎛⎜ X −⎞⎟GX+ GX ⎜ ⎟ ⎝ ⎠ X+ X− ⎠ X+ X− ⎝ ∫ ∫ ∫ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ( ) Xȱ +X− = (X− ) X + X+ ȱȱȱ¡ Xȱ− ȱ X + X− ȱȱǱ Xȱ− ȱ $ %X+ & = + ȱȱƽ −ȱ ǰȱȱƽ ¢ȱȱȱƽȱŞǯȱ X + X− X− X + X+ ȱȱǱ ∫ ⎛⎜ ȱX −⎞⎟GX+ ⎝ ⎠ ∫ GX= ⎛⎜ X − ⎞⎟GX− GX+ ⎝ ⎠ X− X + X+ ∫ X− GX ƽ X + X− ∫ ⎡ ⎢ X+ ⎛⎜ X − ⎞⎟GX− GX+ ⎢ GX+ ⎝ ⎠ X− ⎢ X + X+ ⎣⎢ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ X+ ⎤ ⎥ GX ⎥= X + X+ ⎥ ⎦⎥ ∫ X+ ⎛⎜ X − ⎞⎟ GX− GX+ GX+ GX= ⎝ ⎠ X− X + X+ ⎛ ⎞ ⎜ X+ ⎟ + ⎠ ⎝ ⎛ ⎞ ⎜ X+ ⎟ ⎜ ⎟ + & = X − X− OQX− + OQX + X+ + WDQ− ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ X − X− OQX− + OQ(X+ ) + + ⎛ X+ ⎞ WDQ− ⎜⎜ ⎟⎟ + & = ⎝ ⎠ ( ) [ + − ȱ [+ − OQ [+ − + OQ [+ + + + −⎛⎜ [+ + ⎞⎟ WDQ ⎜ ⎟ + & ⎝ ⎠ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱȱȱàȱȱȱȱȱȱȱȱ¡ȱ¢ȱ¡ǰȱ ȱǱ VHQ[ + FRV[+ VHQ[ VHQ [+ FRV[ • J [ = VHQ[+ FRV[+ • I [ = 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǰȱ ȱ ǰȱ ȱ àȱȱȱȱȱȱȱȱȱȱȱàȱǯȱ ȱàȱ [ ⎛ [⎞ W = WDQ⎜ ⎟ ⇒ WDQȱ− W = ⇒ WDQ− W = [⇒ GW = G[ ⎝ ⎠ + W ⎛ [⎞ ⎛ [⎞ ⎛ [⎞ ⎛ [⎞ W =ȱ WDQ⎜ ⎟ ⇒ W = WDQ⎜ ⎟ ⇒ W = VHF⎜ ⎟ − ⇒ W + = VHF⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ [⎞ ⎛ [⎞ ⎛ [⎞ ⇒ − VHQ⎜ ⎟ = ⇒ VHQ⎜ ⎟ = − ⇒ FRV⎜ ⎟ = ⎝ ⎠ + W ⎝ ⎠ + W ⎝ ⎠ + W W ⎛ [⎞ ⇒ VHQ⎜ ⎟ = ⎝ ⎠ + W W ⎛ȱ[⎞ ȱȱǰ VHQ ⎜ ⎟ = ⎝ ⎠ + W ¢ ⎛ȱ[ ⎞ FRV⎜ ⎟ = ⎝ ⎠ + W W ⎛ [⎞ ⎛ [⎞ ⎛ [⎞ VHQ[ =ȱVHQ⎜ ⋅ ⎟ = VHQ⎜ ⎟ FRV⎜ ⎟ = ⎝ ⎠ ⎝ ⎠ + W ⎝ ⎠ Ǳ ⎛ [⎞ ⎛ [⎞ ⎛ [⎞ − W FRV[= FRV⎜ ⋅ ⎟ = FRV⎜ ⎟ − VHQ⎜ ⎟ = ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ + W ȱǱ ȱ W − W ⎛ [⎞ VHQ[ [ = \ FRV W =WDQ⎜ ⎟ ⇒ GW G[ ǰ =ȱ = + W + W ⎝ ⎠ + W ȱ ∫ȱ G[ + VHQ[+ FRV[ àǱȱ ⎛ [⎞ ȱȱ£ȱȱà Wȱ = WDQ⎜ ⎟ ǰȱȱǱ ⎝ ⎠ ∫ȱ G[ = + VHQ[+ FRV[ + W GW = W+ + W ∫ ∫ ∫ + W W − W + + + W + W GWƽ GW [ = OQ+ W + & = OQ+ WDQ + & + W 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ¤Ǳȱ ·ȱȱȱȱ ŗŗǼ ∫ȱ + FRV [ ŗŘǼ ∫ȱ G[ FRV [+ VHQ[FRV[+ VHQ[ ŗřǼ ∫ȱ G[ ( + FRV[)(+ FRV[) ŗŚǼ ∫ȱ + FRV[ ∫ȱ [− G[ ŗśǼ ∫ȱ G[ VHQ[+ FRV [ ∫ȱ+ ŗŜǼ ∫ȱ ( ŗŝǼ ∫ G[ (ȱ− VHQ[)(− VHQ[) ŗŞǼ ∫ȱ − VHQ[+ FRV[ G[ + VHQ[− FRV[ ŗşǼ ∫ȱ H [ G[ H[ + H[ + ŘŖǼ ∫ VHF [ G[ + VHQ[ ŗǼ ∫ȱ ŘǼ ∫ řǼ ∫ ŚǼ ∫ȱ śǼ [ G[ [− G[ȱ ([+ ) [+ + [ȱ+ + ([+ ) − [+ [− G[ [+ [ G[ G[ ŜǼ ŝǼ ŞǼ şǼ ∫ȱ ∫ȱ ∫ȱ ŗŖǼ ∫ [ G[ [ − [ [+ [ G[ G[ G[ − VHQ[ π πȱ G[ + VHQ[− FRV[ VHQ[ G[ − FRV[) 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD ȱȱ (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ȱ ȱȱ1ȱȱ Xȱ Ǳȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ·ȱ ȱ à ∫ + H G[ [ ∫+ [ G[ [ ∫OQ [ G[ ∫ VHQ[ G[ FRV[ ∫ ∫VHF [WDQ[ G[ ∫[ − [ ∫ [− [ ∫ G[ G[ [ − [ ∫[ + [+ [− G[ ∫ ([+ ) G[ ∫[ OQ [G[ G[ [ + ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ VHQ [ [ − [ G[ π [ H− [G[ ∫ ∫ [ G[ G[ + FRV [ [ − [ + G[ [⋅ [ − G[ WDQ(W) ⋅ VHF(W)GW [ + [ G[ [ − [ − [ VHF W WDQW + WDQW [ − [ GW G[ [ + G[ [ + [ + [ VHQ[ G[ − VHQ[ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ∫( [+ )( [ + ) ∫WDQ [G[ G[ ∫ ∫ ∫ ∫ ( ∫ ∫ ∫ ∫ ∫( ) ∫ ∫ ( ) ∫ ( ) ∫ ( ) ([+ )H[ G[ π WDQ[⋅ VHF [ G[ [ ∫ − [ G[ ∫ [− [ ∫[ − [+ G[ G[ [+ ∫ − [ G[ ∫[ OQ [ G[ ∫ VHQ [G[ ∫[ + [ + [ G[ ∫( ෾ ) −[ [ G[ [ + [+ ∫[ − [ + [− G[ G[ − VHQ[ + H − H− [ [ G[ ( [+ VHQ[) G[ [ + [ − [G[ [ + [+ G[ [ + ( [+ ) [ − [+ ) [ + [+ [ + [− − G[ [ + [+ ∫ [ − [+ G[ [ − [ − [ G[ G[ +[ + [FRV [ G[ HVHQ[ VHQ [ G[ FRV [ G[ G[ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ∫ [ − [G[ ∫( [ + [+ )( [+ ) ෾ ∫ [ + [G[ DUFWDQ[ [ G[ π ∫ [ FRV [G[ − [ ∫ ∫ [ (+ OQ [) ∫ H[ − H [ ∫ − [ ∫ ∫ WDQ [ G[ FRV [ ෾ [ + G[ [ − [ − [ ∫ − [ G[ + [ [ H[ G[ H[ ∫ ( ∫ ∫ ∫ ∫ ) [− [ G[ [ − [ + − [ G[ H[ G[ H[ − +FRV[ G[ VHQ[ VHQ[G[ π [FRV[G[ OQ WDQ [ G[ VHQ[FRV[ ∫ [+ + [+ G[ ∫ π G[ π [ ෾ π + [ G[ G[ ෾H[H G[ VHQ −[ [ −[ G[ + [ G[ ∫ ∫ [ − G[ [ G[ ∫ ∫ ∫ [G[ +H[ G[ − H[ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 ∫ [ − [ G[ ∫ ∫ [ [ + [+ G[ G[ ( [+ )( [− ) ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ G[ VHQ[+FRV[ [WDQ−[ G[ FRW [FVF [ G[ [ + [ − [ [ + G[ G[ G[ [ + HW − H W GW [ [ + G[ [ + G[ [ ∫ ∫ ∫ ∫( ∫ ∫ ∫ ∫ ∫ ∫ ∫ [ + G[ [ − [ [ G[ [ + [− G[ [ − [+ −[ ) − [ G[ −[ + − [ [ G[ + [ [ G[ G[ FRV [ H[ G[ H[ −H[ + G[ H −H[ [ G[ [ + + [− ∫( G[ [H[ ) + [ G[ 8QLYHUVLGDGGH&RVWD5LFD /L]HWK6DQFKR0RUD (VFXHODGH0DWHPiWLFD3UR\HFWR0$7(0 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱŘŖŖśȱ ȱ ŗǯȱȱȱ¤ȱȱȱàȱȱȱȱȱȱ ǻŜǼȱ Ǽȱ ȱ ȱȱȱǱ ]−ȱ∞−[ ǰ ]−ȱ[ ¢ ]ȱ+∞[ Ǽ I −ȱ= \ I ȱ¡ȱ Ǽ OLP I [ =ȱ ¢ [→−∞ OLP I [ = +∞ [ȱ→+∞ Ǽ OLP I [ = +∞ ¢ [ȱ→−− OLP I [ = +∞ [ȱ→−+ Ǽ OLP I [ = ¢ OLP I [ = − [ȱ→ − [ȱ→ + Ǽ Iȱ = \ I = ŘǯȱȱȱȱÇȱǻȱȱ¡ȱǼǯȱ [ − [+ VHQ[ Ǽ OLP [→ȱ [ Ǽ OLPȱ ⎛⎜ [ + [− − [ − [⎞⎟ ⎝ ⎠ [ → −∞ Ǽ ⎛ ⎞⎟ [ȱ→ −[ − [⎟ ⎝ ⎠ OLP ⎜⎜ − [ − [+ − [ + [− [ − [+ WDQπȱ[ Ǽ OLP [→− [+ Ǽ OLP [ȱ→ řǯȱȱȱȱȱȱȱǻȱȱȱȱȱǼȱ [+ Ǽ Iȱ [ = [ − [+ ( ) ǻřȱǼȱ ǻŚȱǼȱ ǻřȱǼȱ ǻřȱǼȱ ǻŚȱǼȱ ǻřǼȱ Ǽ I [ =ȱ [ + VHF [ ǻŚȱǼȱ Ǽ I [ =ȱ VHQ[+ FRV [ ǻśȱǼȱ Ǽ I [ =ȱ [HFRW[ ǻŚȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Śǯȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱȱȱà Kȱ ȱȱȱȱ ȱ ǻŜȱǼ ⎧ D[ + VL [< ⎪ [ − ⎪ K [ = ⎨ E VL [= ⎪D[ − VL [> ȱ ⎪ ⎩ śȱȱǼȱȱȱȱȱȱȱȱ ȱ Ǽȱ ȱ ǻřǼȱ [ + S[+ T=ȱ ȱ ȱ Ç£ȱ ȱ ǻŘȱǼȱ S> \ T >ȱ Ŝǯȱ ȱ ȱ ȱ ȱ I [ = [ + D[+ E \ J [ = F[− [ȱ ȱ ȱ ȱøȱȱȱǻŗǰŖǼǯȱ DȱE \ F ǻśǼȱ ŝǯȱȱȱàȱ ǰȱ ¢ȱȱȱȱȱȱȱȱøȱǰȱȱ ȱȱȱȱǱ I [+ K =ȱ I [ I K ǰȱȱȱȱȱ ¡ȱ ¢ I =ȱ ǯȱ Ǽȱȱȱȱ I ȱ ¡ȱȱȱ I [ = I [ I ȱ ǻŚȱǼȱ Ǽȱ·ȱȱȱȱȱà ȱȱȱȱǯȱ ǻŗȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱŘŖŖśȱ ȱ ȱ ŗǼȱ ȱȱà Iȱ ¢ȱ¤ȱȱǰȱȱȱȱǰȱȱȱ ¡àȱȱȱȱȱȱȱȱȱȱȱȱȱȱǯȱ ǻŗŘȱǼȱ Ǽ Ǽ Ǽ Ǽ OLP I [ = ȏȏȏȏȏȏȱ [ȱ→− OLP I [ = ȏȏȏȏȏȏȱ [ȱ→− OLP I [ = ȏȏȏȏȏȏȱ [ȱ→−∞ OLP I [ = ȏȏȏȏȏȏȱ [ȱ→∞ Ǽȱ ȱ ȱȱȱȱ ¡ȱ ȱȱȏȏȏȏȏȏȏȱ Ǽȱ ȱ ȱȱȱȱ ¡ȱ ȱȱȏȏȏȏȏȏȏȱ Ǽȱȱȱȱȱȱȱȱȱȱȱ ǼȱȱÇȱǰȱȱȱ¤ȱȱȱǰȱȱȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ ȱȱȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽ Iȱ − Ǽȱ ȱÇȱ£ǰȱȱȱ¤ȱȱȱǰȱȱȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽȱȱȱȱ¡ȱȱȱȱȱȱȱȱȱȱǱȱȏȏȏȏȏȏȱ Ǽȱ ȱȱ Iȱ ȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ŘǯȱȱȱȱÇȱǻȱȱ¡ȱǼǯȱ [ − D+ [+ D Ǽ OLP [→ȱD Ǽ ⎛ ⎞ ȱ OLP ⎜⎝ W − W + W ⎟ ⎠ ǻŚȱǼȱ Ǽ ⎡ ⎤ − [→⎣ [− [ + [ − ⎥ ⎦ ǻřȱǼȱ Ǽ OLPȱ Ǽ [ − D [→ −∞ OLPȱ⎢ − + [ [→ − −[ OLP VHQ π ȱ[ [→ − [ ǻřȱǼȱ ǻŚȱǼȱ ǻśȱǼȱ řǯȱȱȱȱȱȱȱǻȱȱȱȱȱǼȱ Ǽ Iȱ [ = VHQ[ [ − ǻřȱǼȱ [ + VHF[ Ǽ I [ =ȱH ǻŚȱǼȱ Ǽ I ȱ[ = [ + [ + [ ǻśȱǼȱ [−ȱ \ J [ = F[+ E ȱ ȱ ȱ Śǯȱ ȱ ȱ ȱ ȱ I [ = DH øȱȱȱǻŗǰŗǼǯȱ DȱE \ F ǻśȱǼȱ [ śǯȱȱȱȱȱȱȱȱ I ([)=ȱH ¢ J([)=ȱ WDQ [ ȱȱ ǻśȱǼȱ Ŝǯȱ ȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à Kȱ ȱȱȱȱ ȱ ǻŞȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ⎧ VL [ = ⎪ Kȱ [ =⎨ ⎪D− [VHQ[ VL [ ≠ ⎩ ŝǯȱȱ ȱ¢ȱȱȱȱ¢ȱȱȱȱȱȱȱȱøȱ ǯȱ¤ȱȱǱȱ Ǽ J [ = [I [ + ȱ Ǽ J D+ E =ȱ J D J E ȱȱȱ¢ ǯȱ Ǽ ȱ J′ [ = J [ ȱ OLP I [ ȱ= [→ ǻśȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱŘŖŖŜȱ ȱȱ ŗǼȱȱȱȱȱ·ȱȱȱàȱȱȱȱȱȱ ¢ȱȱȱ¡àȱǯȱǻȱśȱǼȱ Ǽȱȱȱȱȱ Ǽȱ OLPȱ [→D I [ = O I ȱD =O ǯȱ Ǽȱ Iȱ D ȱ¡ǰȱ OLP [→ȱD Ǽȱ OLP [→ȱD I [ ȱ¡ǯȱ I [ ȱ¡ǰȱ Iȱ D ȱ¡ǯȱ ȱȱȱȱ ȱ ǻǼȱȱ ȱ¢ȱ ȱ ǻǼȱȱȱ ȱ¢ȱ ȱ ǻǼȱȱȱ ȱ¢ȱ ȱ ǻǼȱȱȱ ǰ ȱ¢ȱ ȱ Ǽȱȱ Ǽ OLP I [ = [ȱ→ Ǽ Iȱ ¢ Jȱ ȱȱȱȱȱ ⎧ VL [ = ⎩ I [ VL [≠ Ǽ Jȱ [ =⎨ ȱȱȱȱȱȱȱȱȱ ǻǼ Iȱ = ǻǼ Jȱ = J [ = [ȱ→ ǻǼ OLP J [ = [→ ǻǼ OLP 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǽȱȱȱ¤ȱȱ ȱȱȱȱ¤ȱǰȱȱȱȱȱ ǻǼ OLP IJ [ ȱ¡ȱ [→ȱ ǻǼ OLP I [ + J [ = OLP I [ + OLP J [ [→ȱ [→ [→ J [ = ǻǼ OLP [ȱ→ I [ ǻǼ OLP J [ [→ȱ I [ Ǽȱ OLP [ȱ→ ȱ¡ȱ I [ = ∞ ¢ OLP J [ = ∞ ȱȱȱȱȱ [ȱ→ ǻǼ OLP I [ − J [ = [ȱ→ I [ = ǻǼ OLP [ȱ→ J [ ǻǼ OLP I [ + J [ = ∞ [ȱ→ ǻǼ OLP [ I [ = [→ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 Ǽȱȱȱȱȱ Ǽȱȱ Iȱ ȱȱ D ǰȱ Iȱ ȱȱ D ȱ Ǽȱ I ′ȱ U ¡ǰȱ OLP [→ȱU I [ = I U ȱȱȱȱȱȱȱȱȱ $ ȱȱ ǯ % ȱȱȱ ǯ & ȱȱ ȱ¢ȱȱ ǯ ' ȱȱȱȱǯȱ ŘǼȱȱȱà Iȱ ¢ȱ¤ȱȱǰȱȱȱȱǰȱȱȱ ¡àȱȱȱȱȱȱȱȱȱȱ ǯȱ ǻŗŘȱǼȱ Ǽ OLP I [ = ȏȏȏȏȏȏȱ Ǽ OLP I [ = ȏȏȏȏȏȏ [ȱ→ [ȱ→− 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Ǽ Ǽ ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 OLP I [ = ȏȏȏȏȏȏȱ [ȱ→−∞ OLP I [ = ȏȏȏȏȏȏȱ [ȱ→∞ Ǽȱ ȱ ȱȱȱȱ ¡ȱ ȱȱȏȏȏȏȏȏȏȱ Ǽȱ ȱ ȱȱȱȱ ¡ȱ ȱȱȏȏȏȏȏȏȏȱ Ǽȱȱȱȱȱȱȱȱȱȱȱ ǼȱȱÇȱǰȱȱȱ¤ȱȱȱǰȱȱȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ ⎛ ⎞ Ǽ Iȱ ⎜ − ⎟ ȱȱȏȏȏȏȏȏȏȏȏȏȏȏȱ ⎝ ⎠ Ǽȱ ȱÇȱ£ǰȱȱȱ¤ȱȱȱǰȱȱȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽȱȱȱȱ¡ȱȱȱȱȱȱȱȱȱȱǱȱȏȏȏȏȏȏȱ Ǽȱ ȱȱ Iȱ −ȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ řǼȱȱȱȱÇȱǻȱ¡Ǽǯȱ Ǽ Ǽ − + [ OLPȱ ǻŚȱǼȱ [+ VHQ[ [ [→ȱ ǻŚȱǼȱ [→ − −[ OLP ȱȱȱ ŚǼȱȱȱȱȱàȱȱȱȱ¤ȱǰȱȱ K′ȱ + [J I [ ȱ K [ =ȱ ǻśȱǼ + J [ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 śǼȱȱȱȱȱȱàȱȱȱǰȱ¡ȱ·ȱ £àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǲȱ ȱ ȱ ȱ ȱ ȱȱȱȱǯ [− Iȱ [ = [+ [− ǻśȱǼȱ ŜǼȱ ȱ ȱ ȱ à Iȱ ¤ȱ ȱ ȱ ȱ ȱ (ȱ) ¢ VHQπ[ ȱ I [ = ǰȱ Iȱ ȱŖȱ ¢ȱȱŗȱ ȱȱ Iȱ ȱȱȱȱ [ [− [ȱ]ǯȱ ǻŜȱǼȱ ŝǼȱǰȱȱ¡ǰȱȱÇȱ£ȱȱȱ¤ȱȱȱàǯ ⎧ [ + [− VL [≥ ⎪⎪ Kȱ [ =⎨ [ − ⎪ ⎪⎩ [+ [ + [+ VL [< ǻŜȱǼȱ ŞǼȱȱȱȱǰȱȱȱȱȱ H[ȱVHQ[ Ǽ I [ = [ + ǻŚȱǼȱ Ǽ I [ =ȱ[ + VHF [ + FRV WDQ[ ǻŜȱǼȱ şǼȱ ȱ ȱ ȱ ¡ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ à \ =ȱ [ ȱȱȱȱ (ȱ)ǯȱ ǻŜȱǼȱ ŗŖǼȱȱȱȱȱȱȱ ȱ ¢ȱ ȱ ȱȱȱȱà Kȱ ȱȱȱŗǯ ⎧D[ + E VL [≤ ⎪ Kȱ [ =⎨ ⎪ [ VL [ > ⎩ ǻśȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱ ȱ ȱȱ ȱ ȱȱȱȬ ŘŖŖŝȱ ·ȱŗŗȱȱȱȱŘŖŖŝǯȱ ȱȱ¡ ȱ ȱǱȱřȱȱ •ȱ ȱǰȱȱàȱ¢ȱȱǰȱȱȱǯ •ȱ ȱ øȱ Çȱ £ȱ ȱ ǯȱ ȱ ȱȱ ¡ȱ ȱ ȱ ȱȱȱ ¤£ȱ ȱȱȱȱǯ •ȱ ȱ ȱ ȱ ¢ȱ ȱ ¢ȱ ȱ ǯȱ ȱ ȱȱ ȱ ¤ȱȱ ǰȱ·ǰ ȱȱ¤ǯ •ȱ ȱ ȱ ȱ ¡ȱ ȱ ǰȱ ȱ ȱ ǰȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ¢ȱ ȱ ȱ ȱ ȱȱȱǯ •ȱ ȱ ȱ ȱ øȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱ¤ǯ •ȱ ȱ¡ȱȱȱ£ȱǻŞǼȱÇȱ¢ȱȱȱȱŝŜȱǯȱȱǰȱȱ ȱǰȱȱ·ȱǯ •ȱ ȱȱȱ¢ȱȱȱȱ¢ȱȱȱ·¡ǯȱ ŗǼȱȱȱ¤ȱȱȱàȱȱȱȱȱȱ ǻŜȱǼȱ ]−ȱ∞−[ ǰ ]−ȱ[ ¢ ]ȱ+∞[ Ǽȱ ȱ ȱȱǱ Ǽ Iȱ = \ I = Ǽ OLP I [ = ¢ [ȱ→−∞ Ǽ OLP I [ = −∞ ¢ [ȱ→− OLP I [ = +∞ [ȱ→+∞ OLP I [ = +∞ [ȱ→+ Ǽ OLP I [ = ¢ OLP I [ = [ȱ→− [ȱ→ Ǽ Iȱ − ȱ¡ǰ Iȱ = ¢ I = 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ŘǼȱȱȱà Iȱ ¢ȱ¤ȱȱǰȱȱȱȱǰȱȱȱ ¡àȱȱȱȱȱȱȱȱȱȱȱǯȱ ǻŜȱǼȱ Ǽȱ ȱȱȱȱ ¡ȱ ȱȱȏȏȏȏȏȏȏȱ Ǽȱȱ ȱȱȱȱ ¡ȱ ȱȱȏȏȏȏȏȏȏȱ Ǽȱȱȱȱȱȱȱȱȱȱȱ Ǽ ȱÇȱǰȱȱȱ¤ȱȱȱǰȱȱȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ ǼȱȱÇȱ£ǰȱȱȱ¤ȱȱȱǰȱȱȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Ǽȱ ȱȱȱ¡ȱȱȱȱȱȱȱȱȱȱǱȱȏȏȏȏȏȏȱ řǼȱȱȱȱÇȱǻȱ¡Ǽǯȱ [− [− [ȱ→ [ − [ − [+VHQȱ[ Ǽ OLP [ [→ Ǽ Ǽ Ǽ OLP ( ) OLP [+ [ + [ [→ȱ−∞ ⎡ HVHQ[ − ⎤ OLPȱ⎢ ⎥ [→π ⎣ [− π ⎦ ǻśȱǼȱ ǻśȱǼȱ ǻśȱǼȱ ǻśȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ŚǼȱȱȱȱǰȱȱȱȱȱ H[ȱ +[ Ǽ \ = FRV [− Ǽ Iȱ ǻśȱǼȱ [ =(WDQ[) VHF[+ ǻśȱǼȱ ⎧ VL[< E ⎪⎪ [ śǼȱ I ȱ[ =⎨ ⎪− [ VL[≥ E ⎪⎩ Ǽȱȱȱȱȱȱȱȱ ȱȱȱ ǯȱ ǻśȱǼȱ ǼȱǶȱ ȱ ȱȱȱȱȱȱȱȱȱȱǵȱ ǻśȱǼȱ −[ȱ ŜǼȱȱȱȱà H = [ ȱàȱȱȱ ]ȱ[ ǻŚȱǼȱ ŝǼȱȱ OLP (− [)= [ȱ→− ǻśȱǼȱ ŞǼȱȱȱȱǱȱ Ǽȱ ȱ ȱ ȱ ȱ ȱ \=ȱ + H[ − [ ȱ ȱ ȱ ȱ ȱ ȱȱȱ [ȱ−\ = ǻśȱǼȱ Ǽȱȱȱȱȱ¡ȱȱȱȱȱ¤ȱ I [ =ȱ [+ VHQ[ ȱȱ ȱ£ǯȱ ǻśȱǼȱ Ǽȱȱȱȱȱȱȱȱȱȱȱ (ȱ) ¢ȱȱȱ ȱȱȱȱȱ I ( [)=ȱ − [ǯȱ ǻśȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱŘŖŖśȱ ȱȱ ȱȱ¤ȱȱȱà ȱȱȱàȱ¢ȱǱŞȱǯȱ ǯȱ ȱȱ¤¡ȱȱȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ ǯ OLPȱ I( [) ƽȱȱȏȏȏȏȏȏȏȏȏȏȏȱ [→−+ ǯ ǯ ǯ ǯȱ ǯȱ ǯȱ OLPȱ I( [)ƽȱ ȏȏȏȏȏȏȏȏȏȏȏȏȱ [→− OLPȱ I( [) ƽȱ ȏȏȏȏȏȏȏȏȏȏȏȏȱ OLPȱ I( [)ƽȱ ȏȏȏȏȏȏȏȏȏȏȏȏȱ [→−∞ [→+∞ ȱȱàȱȱȱà ǻ¡Ǽÿ ÿŖ Ǳȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ ȱȱàȱȱȱà I ′(ÿ [)ÿ ÿŖ Ǳȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ ȱȱàȱȱȱà I ′′ÿ( [)ÿ ÿŖȱǱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȱ Řǯȱ ȱȱȱȱȱȱ¡ȱŘȱ ƸȱŘ¢ȱŘȱ ƽȱŗȱȱȱȱȱȱȱȱ ȱȱȱȱȱŗǯȱ Ŝȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ řǯȱ ȱȱȱàǯȱ ǯ ǻȱȱȱǼȱ I( [) = /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 śȱ ([+ ) WDQ−ȱ([) [VHQ ( [) + [ Śǯȱ Ƕȱ·ȱ£àȱȱȱ¤ȱȱȱ¤ȱ¤ȱȱȱȱ ȱ ȱŗŖȱ ȱȱ ȱȱǵȱ Ŝȱǯȱ øȱ ¢ȱ ȱ ¢ȱƽȱ¡ȱřȱ ȮȱŘ¡ȱŘȱ Ƹȱŗȱ ǰȱ ¡ ƽȱŘȱȱȱ¢ȱ ¡ȱƽȱŖǰŘǯȱ śǯȱ řȱǯȱ Ŝǯȱ ȱȱȱà ȱȱȱȱȱȱǽȱȬŗȱǰȱŘȱǾǰȱȱ ǻ Ȭŗ Ǽȱƽ ŗȱ ¢ȱ ǻȱŘ Ǽƽ śǯȱȱȱȱ¡ȱȱȱȱȱ¤ȱȱȱȱȱȱȱ śȱǯȱ ȱȱȱȱȱȱ¢ȱƽȱŘ¡ǯȱ ⎛ [ ⎞ OLPȱ⎜ ⎟ [→+∞ [+ ⎝ ⎠ ŝǯȱ [ Ŝȱǯȱ Şǯȱ ȱȱȱȱȱ¤ȱȱȱȱȱŜȱȱ ȱ¢ȱȱȱŚȱȱǯ ȱȱ¤ȱȱȱȱȱȱȱřŞŚȱȱŘȱǰȱ Ƕ¤ȱȱȱȱȱȱȱ¤ȱ¤¡ǵȱ Ŝȱǯȱ Ǳ I ( [)= şǯȱ ǯȱ ǯȱ ǯȱ ǯȱ [ + [− ȱ ǰ − [ + [ I ′ȱ( [)= − [ [ ¢ I ′ȱ′( [)= − [ ȱȱà ǻ¡ǼǱȱ ŗśȱǯȱ ǰȱȱ¡ǰȱȱÇȱȱȱ¤ǯȱ ȱȱȱȱÇȱ¢ȱȱ¡ȱǻàȱǰȱ ǰȱȱ¤¡ȱ¢ȱÇȱǼǯȱ ȱȱȱ¢ȱȱȱȱǯȱȱ¢ȱ ȱȱ¡àȱÇǯȱ ȱȱȱȱàȱȱ¢ȱȱ¤ȱȱȱȱȱ ¤¡ȱǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱŘŖŖśȱ ȱ ȱ £ȱȱàȱȱ¢ȱȱ¤ȱȱȱàȱȱ ǻȱŗŖȱǼ OLP I ( [) = −∞ OLP I ( [) = +∞ [→ − − [→ − + OLP [ → − I ( [ )= OLP I ( [) = −∞ OLP I ( [) = −∞ [→ + [→ −∞ OLP I ( [ )= OLP I ( [)− [ = [ → ∞ [→ −∞ I = I − = − 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱà Iȱ ¢ȱ¤ȱȱǰȱȱȱȱǰȱȱȱ ¡àȱȱȱȱȱȱȱȱȱȱȱȱȱȱǯȱ ǻȱŗŖȱǼȱ Ǽȱ ȱȱȱ¡àȱȱȏȏȏȏȏȏȏȏȏȏȏȏǯȱ Ǽ Iȱ £ȱȱÇȱȱȱ¡ȱȱȱȏȏȏȏȏȏȏȏȏǯȱ Ǽȱ ȱȱ Iȱ′ − ȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏǯȱ Ǽȱ ȱȱ Iȱ′′ − ȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏǯȱ Ǽȱ ȱȱàȱȱȱà Iȱ′ [ < ȱȏȏȏȏȏȏȏȏȏȏȏǯȱ Ǽȱ ȱȱàȱȱȱà Iȱ′′ [ < ȱȏȏȏȏȏȏȏȏȏǯȱ Ǽ OLPȱ [ → ∞ I ( [)ƽȏȏȏȏȏȏȏȏǯȱ Ǽȱ ȱàȱȱȱÇȱȱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏǯȱ Ǽȱ ȱàȱȱȱÇȱ£ȱȱȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏȏǯȱ Ǽȱ ȱøȱȱÇȱȱȱȱȱȱà \ =ȱ I [ ȱȱ ȱȏȏȏǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱàȱȱȱȱȱȱȱ [− \ = ǻřǰŗǼǯȱ [+ȱ \ ȱȱȱ ǻȱŜȱǼ ȱ I ′ȱ I [ =ȱ [+ [+ [VHQ[ ǻȱśȱǼ ȱ ȱȱȱŗŖȱȱřȱ ȱàȱȱ¤ȱ¡ȱȱȱȱȱ ·ȱȱȱȱ ǯȱȱ ȱȱȱȱȱȱȱȱśŖȱ ǰȱȱ¤ȱ¤ȱȱȱ£àȱȱŚȦǯȱȱǶ¤ȱ¤ȱ¢ȱȱ ȱȱȱȱȱȱǵȱ ǻȱŝȱǼ −ȱ ǯȱ ȱ ȱȱȱȱȱȱ¡ȱ WDQ ǻȱśȱǼ ȱ ȱȱǰȱȱ¢ȱȱȱȱȱȱȱȱȱ¤ȱȱȱȱ ŜŖķȱ¢ȱȱȱȱȱ¢ȱȱȱȱřȱǯȱȱȱȱȱǰȱ·ȱȱǰȱ ȱȱȱȱȱȱŗŞȱȦǯȱȱȱȱȱȱȱȱǰȱ ·ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ Řŝȱ Ȧǯȱ ȱ ȱ ȱ ȱȱȱȱȱȱȱ¤ȱà¡ȱȱǯȱ Ǳȱ • VHQ$ VHQ% VHQ& = = D E F • D = E + F − EF FRV$ • F =ȱD + E ǻȱŝȱǼ ȱ ȱ = + Fȱ ȱ ø F ∈ ] ȱ[ǯȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǻȱ Ŝȱ Ǽ −ȱ ȱ I [ = [FRV − [ − − [ I ′ȱ [ = FRV [ ǻȱŚȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱŘŖŖŜȱ ȱ ŗǼȱȱȱȱàǱȱǻȱȱȱǼ [ − I([) =ȱ [WDQ( [) + WDQ− [ ǻȱśȱǼȱȱ ŘǼȱȱȱȱȱȱ [ȱ+ [\ + \ = ȱȱȱȱȱȱȱ ȱȱȱ¡ǯȱ ǻȱśȱǼȱȱ řǼȱȱ£àȱ·ȱ¢ȱȱȱȱŗŖȱǰȱȱȱȱȱȱȱ ȱȱ¡ȱǯȱȱȱȱȱȱȱȱ£àȱ¤ȱȱȱȱàȱ πȱ [ − [ ǯȱ Ǳ 9 = ȱȱȱȱȱȱȱȱ£àȱ¢ȱȱȱȱȱśȱȱ ȱ ȱȱ¤¡ȱȱȱȱ ȱ ǯȱȱȱȱȱ¤¡ȱȱȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ £àȱ ¢ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ǻȱŝȱǼȱȱ ŚǼȱȱàȱ ȱ ȱȱȱȱ ȱ ȱȱśŖŖȱȦȱ ¢ȱȱ ȱȱȱŚȱ ŖŖŖȱǯȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ ȱǯȱȱȱȱ£ȱȱȱȱȱȱàǰȱǶȱ·ȱ£ȱȱȱ £ȱȱȱȱȱàȱȱȱȱȱȱǰȱȱȱȱ£ȱ ȱŘȱŖŖŖȱǵȱ ǻȱŝȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 śǼȱ ȱ ȱ à Iȱ ¢ȱ ¤ȱ ȱ ǰȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ¡àȱ ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ ǻȱśȱǼȱ Ǽȱ ȱȱȱ¡àȱȱȏȏȏȏȏȏȏȏȏȏȏȏǯȱ Ǽȱ ȱȱàȱȱȱà Iȱ′ [ > ȱȏȏȏȏȏȏȏȏȏȏȏǯȱ Ǽȱȱȱàȱȱȱà Iȱ′′ [ < ȱȏȏȏȏȏȏȏȏȏǯȱ Ǽ OLP [→ȱ π I ( [)ƽȏȏȏȏȏȏȏȏǯȱ Ǽȱ ȱøȱȱÇȱȱȱȱȱȱà \ =ȱ I [ ȱȱ ȱȏȏȏǯ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ŜǼȱȱȱǰȱȱ¢ȱȱȱȱȱȱȱȱȱ¤ȱȱȱȱŜŖķȱ¢ȱ ȱȱȱȱ¢ȱȱȱȱřȱǯȱȱȱȱȱǰȱ·ȱȱǰȱȱȱ ȱȱȱȱŗŞȱȦǯȱȱȱȱȱȱȱȱǰȱ·ȱȱǰȱȱ ȱȱȱȱȱŘŝȱȦǯȱȱȱȱȱȱȱȱȱȱ ȱ¤ȱà¡ȱȱǯȱ Ǳȱ • VHQ$ VHQ% VHQ& = = D E F • D = E + F − EF FRV$ • F =ȱD + E ǻȱŝȱǼȱȱ ŝǼȱ ȱ ȱ à Iȱ [ = [ + [− [ − ǰȱ ȱ ȱ ȱ ȱ ¤¡ǯȱ ȱ ȱ ȱȱȱȱ£ǰȱȱǰȱȱȱȱȱ ȱȱà ȱ ȱȱȱàǱ − Iȱ [ = ¢ Iȱ [ = [+ [+ ǯȱ ǯȱ ǯȱ ǯȱ ǯȱ ȱȱȱ¤¡ȱȱȱàȱ¢ȱȱȱȱàȱ ȱȱǯȱ ǻȱřȱǼȱ ǰȱȱ¡ǰȱȱÇȱȱȱ¤ǯȱ ǻȱśȱǼȱ ȱȱȱȱÇȱ¢ȱȱ¡ȱǻàȱǰȱ àȱ ǰȱ ¤ȱ ȱ ȱ ȱ ¤¡ȱ ¢ȱ Çȱ Ǽǯȱ ǻȱŚȱǼȱ ȱ ȱ ȱ ¢ȱ ȱ ȱ ȱ ǯȱ ȱ ¢ȱ ȱȱ¡àȱÇǯȱ ǻȱŚȱǼȱ ȱ ȱ ȱ ȱ àȱ ȱ ¢ȱ ȱ ¤ȱ ȱ ȱ ȱ ȱ ¤¡ȱǯȱ ǻȱřȱǼȱȱ ŞǼȱȱȱàȱǱȱ Iȱ′ [ = ȱȱ¡ȱȱȱ (D ȱE) Iȱ ȱȱ (D ȱE)ǯȱ ǻȱśȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD 81,9(56,'$''(&267$5,&$ (6&8(/$'(0$7(0È7,&$ 352<(&720$7(0 /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 0LpUFROHVGHGLFLHPEUHGHO 6HJXQGR([DPHQ3DUFLDO 7LHPSRSUREDEOHKRUDV ,16758&&,21(6 • /HDFXLGDGRVDPHQWHFDGDLQVWUXFFLyQ\FDGDSUHJXQWDDQWHVGHFRQWHVWDU • 8WLOLFH ~QLFDPHQWH EROtJUDIR D]XO R QHJUR 6L HO H[DPHQ FRQWLHQH SDUWHV HVFULWDV FRQ OiSL] RKDFHXVRGHFRU UHFWRUOtTXLGRXVWHG S LHUGHHOGHUHFKRDUHFODPDU • 7UDEDMHFRQHOPD\RURUGHQ\DVHRSRVLEOH6LDOJXQD SUHJXQWDHVWi GHVRUGHQDGD pVWDQRVHFDOLILFDUi • /RVtWHPHVGHEHUiQVHUU HVXHOWRVHQHOFXDGHUQRGHH[DPHQORVPLVPRVVRQGH GHVDUUROOR SRU OR WDQWR GHEH DSDUHFHU WRGR HO SURFHGLPLHQWR TXH MXVWLILTXH FRUUHFWDPHQWHODVROXFLyQ\ODUHVSXHVWDGHFDGDXQRGHHOORV • 5HFXHUGHTXHQRGHEHXWLOL]DUFDOFXODGRUDSURJUDPDEOHQLJUDILFDGRUD • (VWHH[DPHQFRQVWDGHRFKR tWHPHV\XQWRWDOGHSXQWRV5HYLVHDQWHVGH LQLFLDUTXHHVWpFRPSOHWR • 7UDEDMHFRQFDOPD\OHGHVHDPRVHOPD\RUGHORVp[LWRV 'HULYHODVIXQFLRQHVVLJXLHQWHV 1RHVQHFHVDULRVLPSOLILFDU [− D J [ = VHQ−([)+ OQ [+ E \ = [+ [+ [VHQ[ 9DORU SXQWRV 9DORUSXQWRV ȱȱȱȱȱȱȱȱȱȱà [ + \ + =ȱ [\ ȱ ȱȱȱ£ǯ 9DORUSXQWRV ȱȱȱȱȱàȱȱȱȱ£àȱȱŘȱȱřȱȦǯȱȱȱȱȱ ȱȱŗŞȱȱ¢ȱȱȱȱȱȱȱȱȱȱȱŘŚȱǯȱǶȱ·ȱ ȱ¤ȱȱȱȱȱȱȱȱȱȱȱȱȱȱŜȱ ǵ 9DORU SXQWRV ȱȱȱȱ¡ȱȱȱ WDQȱ°ǯ 9DORU SXQWRV ȱȱȱȱàȱ ȱ ȱȱ [DȱE] ¢ȱ I ′ [ =ȱ ȱȱ ¡ȱȱ ǻǰȱǼǯȱȱ I [ =ȱ [− D+ I D ȱȱ¡ȱ [DȱE] ǯ 9DORUSXQWRV 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 &RQVLGHUHODIXQFLyQ I FX\DJUiILFDVHDGMXQWDFRQEDVHHQHOODFRPSOHWHODH[SUHVLyQ TXHVHOHGDGHPRGRTXHVHFRQYLHUWDHQ YHUGDGHUD 9DORUSXQWRV S 8QYDORUPi[LPRORFDO FRUUHVSRQGHDBBBBBBBBBBBB T (OFRQMXQWRVROXFLyQGHODLQHFXDFLyQ I′ [ > FRUUHVSRQGHDBBBBBBBBBBB U (OFRQMXQWRVROXFLyQGHODLQHFXDFLyQ I′′ [ < FRUUHVSRQGHDBBBBBBBBB V OLP [→ I ( [) BBBBBBBB W (OQ~PHURGHDVtQWRWDVYHUWLFDOHVGHODFXUYDGHHFXDFLyQ \ = I [ HVLJXDODBBB 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱȱŚȱ¢ȱŜȱȱȱȱȱȱȱŝǰśȱǯȱȱȱȱȱȱ ȱȱǲȱȱȱ¤ȱȱȱȱ¡ȱȱȱȱ¢ȱ ȱ ȱ ǰȱ ȱ ȱ ȱ ȱ ȱ ȱ ǯȱ Ƕȱ ·ȱ ȱ ȱ ȱ ȱ ȱ ȱȱȱȱȱȱȱȱȱȱ£ȱȱÇǵ 9DORU SXQWRV &RQVLGHUH OD IXQFLyQ I [ = [ − [ + GHILQLGD HQ VX GRPLQLR Pi[LPR 3DUD [− UHVSRQGHU HVWD SUHJXQWD SXHGH XWLOL]DU VLQ FDOFXODUODV ODV GRV SULPHUDV GHULYDGDV GH OD IXQFLyQ I TXHDSDUHFHQDFRQWLQXDFLyQ [ − [ + I [ = \ I [ = ( [−) ( [−) M N O P Q 'HWHUPLQHHOGRPLQLRPi[LPRGHODIXQFLyQ 9DORUSXQWRV 'HWHUPLQHVLH[LVWHQODVDVtQWRWDVGHODJUiILFD 9DORUSXQWRV 'HWHUPLQH ORV LQWHUYDORVGH PRQRWRQtD \SXQWRVH[WUHPRV GyQGHFUHFHGyQGH GHFUHFHFXiOHVVRQORVSXQWRVPi[LPRV\PtQLPRVUHODWLYRV 9DORUSXQWRV $QDOLFH OD FRQFDYLGDG \ HVFULED ORV LQWHUYDORV FRUUHVSRQGLHQWHV 6L KD\ SXQWRV GHLQIOH[LyQLQGtTXHORV 9DORUSXQWRV &RQ EDVH HQ OD LQIRUPDFLyQ REWHQLGD FRQVWUX\D OD JUiILFD GH I HQ VX Pi[LPR GRPLQLR 9DORUSXQWRV 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱŘŖŖśȱ ȱȱ ȱ ǻ¡Ǽȱ ȱȱȱǱȱ I [ = − H[ ȱ ȱ¤ȱȱ ȱ ȱȱȱȱ ( Hȱ) \ () ǻśȱǼȱ Q Řǯȱ ȱȱȱÇǱ ⎛ πȱ ⎛ ⎜⎜ ⋅FRV⎜ ∑ Q Q→∞ ⎝ = ⎝ OLP L π L+ Q ⎞ ⎞ ⎟⎟ Q ⎠ ⎟⎠ ǻśȱǼȱ řǯȱ ȱȱȱȱȱȱ¤ȱȱȱàȱȱȱȱ \ȱ=− [ + [− ¢ȱȱȱȱ¡ǯȱ Śǯȱ ) ȱ[ = ∫ [ = I W GW ǰȱȱȱ Iȱ W ǻŜȱǼȱ ∫ W + X GXǯȱȱ ) ′′ ȱ ǯȱ X ǻŚȱǼȱ śǯȱ ȱȱȱȱ ǯ ∫ ǯ ∫ȱ ǯ ∫( WDQ− [ȱ + [ G[ G[ [OQ [ ǻŚȱǼȱ ǻřȱǼȱ [ȱ+ [− )G[ ǻŚȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD Ŝǯȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ ȱȱȱȱ ǯȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ \ = [ + [+ \ = [+ ȱ [ = − \ [ = ǯȱ ǻş Ǽȱ ȱȱ ȱ ȱàȱȱȱǰȱȱ ȱȱ ¡ǰȱȱȱàȱȱȱȱ \ = [ ȱ ¢ \ =ȱ ǻřȱǼȱ ǯȱ ȱȱȱ£ȱȱŚŖȱȱȱȱȱȱȱ ȱȱȱȱŗŖȱ ȱȱȱȱŗśȱǶ·ȱȱ ǻŚȱǼȱ ȱȱȱȱȱȱȱŗśȱȱŗŞǵȱ ǯȱ ȱȱ ȱ ȱȱȱ −ȱ ǯȱȱȱȱȱȱȱȱ ȱ [ ȱȱȱȱ ǰȱȱ ȱ ] ȱ ∫ − I [ G[ǯȱ ǻřȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱŘŖŖśȱ ȱ ȱ ȱàȱ ȱȱ ȱȱȱȱȱ ȱȱȱǻŗǰřǼȱ \ȱ=[+ ǯȱ ȱ ȱ ȱ ȱ ȱ ǻ¡ǰȱ ¢Ǽȱ ȱ ȱ ȱ ȱ ȱ Gȱ \ G[ àȱȱȱǯȱ ȱ = ǰȱ ȱ ȱ ǻȱŜȱǼ ȱȱȱÇǱ Q OLP Q→∞ ∑ L= ⎛ π VHQ⎛⎜ πȱQ+π L⎞⎟ ⎞ ⎛ π Q+ π L⎞ ⎟ ⎜ Q ⎠ ⎝ ⋅ FRV⎜ ⎟ ⎜⎜ Q ⋅H Q ⎠ ⎟⎟ ⎝ ⎝ ⎠ ǻśȱǼ ∫ȱ ( [ + )G[ ȱ ȱȱàȱȱǰȱ J [ ȱ I [ = + W ȱ I ⎛⎜⎝ π⎞⎟ȱ⎠ ǻŗŖǼ FRV [ ∫ GW ¢ J [ = ∫ (+ VHQW )GW ȱ ǻŜȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD śǼȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱȱǯȱȱǻȱŗŜȱǰȱŚȱȱȱǼȱ ∫ ǯ ȱȱ VHQ − [ − [ G[ȱ ǯ ∫ȱ [ − G[ − ȱ ǯ ǯ ȱ ∫ ( [ + ) [ + G[ − ∫ [+ G[ [ + [+ ȱȱȱȱ Ǽȱ ȱȱ¤ȱȱȱàȱȱȱȱ¤ \ = [ − ȱ \ = [−ȱ ¢ȱȱ ǻśȱǼȱ Ǽȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱȱȱȱ ǻŝȱǼ \ = [ \ \ = [ȱ ȱȱȱȱȱ \ =ȱǯȱ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱŘŖŖŜȱ ȱȱ ŗǼȱ J( [) =∫ [ I Wȱ GWǰȱȱȱȱȱàȱ¢ȱ¤ȱȱǯȱ ȱȱȱȱ¤ȱǰȱǱJȱ ǰJȱ ǰJȱ ǰJȱ ǰJȱ ¢Jȱ ǻȱŜȱǼȱ ŘǼȱȱȱȱȱǯȱȱǻȱŗŜȱǰȱŚȱȱȱǼȱ HVHQ[ G[ȱ G[ Ǽ Ǽ VHF [ [− ∫ ∫ȱ ∫ H Ǽ H G[ȱ [ OQ[ Ǽ ∫ − − [ + [ȱ + řǼȱǰȱȱ¡ǰȱȱȱȱ ¡ȱ ȱ¡£ȱȱ ∫ G[ [+ [ ȱ GWǯȱ W ǻŜȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱ Q ŚǼȱȱȱȱÇǱ OLP Q→ ∞ πȱ ⎛ −π Q+ π L⎞ VHF ⎜ ∑ Q ⎝ L = Q ⎟ ⎠ ǻŜȱǼȱ E ∫ȱD śǼȱȱȱȱ£ȱȱàȱȱȱȱ [G[= E − D ǻŗŖȱǼȱ ⎧ VL[< ⎪⎪− [ + [ = ȱ ⎨ ŜǼȱȱ K ȱ[ = [ ¢ +ȱ ȱǱ ⎪ [ VL [≥ ⎪⎩ ȱ Iȱ ȱȱȱ Iȱ (−∞ȱ∞ ) ǯȱ ǻŜȱǼȱ ŝǼȱȱȱȱȱ Ǽȱ ȱ ȱ ¤ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ \ȱ =[− ¢ȱ ȱ \ȱ =[− ǯȱ ǻśȱǼȱ Ǽȱ ȱ ȱ ȱ ȱ àȱ ȱ àȱ ȱ ȱ ȱ ȱ ȱ àȱ ȱ ȱ ȱ ¤ I ( [)=ȱ VHQ[ ¢ȱ ȱ ȱ ¡ȱ ǻ ȱ≤[≤ π Ǽȱ ȱ ȱ ȱ ȱ ¡ǯȱ ǻśȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 81,9(56,'$''(&267$5,&$ (6&8(/$'(0$7(0È7,&$ 352<(&720$7(0 0LpUFROHVGHDEULOGHO 7HUFHU([DPHQ3DUFLDO 7LHPSR3UREDEOHKRUDV ,16758&&,21(6 • /HDFXLGDGRVDPHQWHFDGDLQVWUXFFLyQ\FDGDSUHJXQWDDQWHVGHFRQWHVWDU • 8WLOLFH ~QLFDPHQWH EROtJUDIR D]XO R QHJUR 6L HO H[DPHQ FRQWLHQH SDUWHV HVFULWDV FRQ OiSL] XVWHGSLHUGHHOGHUHFKRDUHFODPDU • 7UDEDMHFRQHOPD\RURUGHQ\DVHRSRVLEOH6LDOJXQD SUHJXQWDHVWi GHVRUGHQDGD pVWDQRVHFDOLILFDUi • (VWHH[DPHQHVGHGHVDU UROORSRUORTXHGHEHU iVHUUHVXHOWRHQHOFXDGHUQRGH H[DPHQ \ GHEH DSDUHFHU WRGR HO SURFHGLPLHQWR TXH MXVWLILTXH FRUUHFWDPHQWH OD VROXFLyQ\ODUHVSXHVWDGHFDGDXQRGHHOORV • 5HFXHUGH TXH ~QLFDPHQWH SXHGH XWLOL]DU FDOFXODGRUD TXH VRODPHQWH WHQJD ODV RSHUDFLRQHVEiVLFDV • (VWHH[DPHQFRQVWDGHVLHWH tWHPHV\XQWRWDOGHSXQWRV5HYLVHDQWHVGH LQLFLDUTXHHVWpFRPSOHWR • 7UDEDMHFRQFDOPD\OHGHVHDPRVHOPD\RUGHORVp[LWRV $FRQWLQXDFLyQVHPXHVWUDODJUiILFDGHI&RQEDVHHQHOODFDOFXOH ∫J [ G[ 5HFXHUGHTXHGHEHDSDUHFHUWRGRHOSURFHVR (QFXHQWUH I [ VLVHVDEHTXH I [ = − H[ ODJUiILFDGH I SDVDSRUORVSXQWRV ( H ) \ () SXQWRV 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 SXQWRV 8VDQGRODGHILQLFLyQFDOFXOHHOiUHDEDMRODFXUYD \ = [ − [+ HQHOLQWHUYDOR [−] SXQWRV 'HWHUPLQHHOVLJXLHQWHOtPLWH Q ∑ OLP Q→∞ L= ⎛ π VHQ⎛⎜ πQ+π L⎞⎟ ⎞ ⎛ π Q+ π L⎞ ⎟ ⎜ Q ⎠ ⎝ ⋅ ⋅ H FRV ⎜ ⎟ Q ⎠ ⎟⎟ ⎜⎜ Q ⎝ ⎝ ⎠ SXQWRV [ 6L ) [ = ∫ I W GW HQGRQGH I [ = ∫ [ + X X GXHQFXHQWUH ) SXQWRV 5HVXHOYDORVSUREOHPDVVLJXLHQWHV F 'HWHUPLQHHOiUHDGHODUHJLyQOLPLWDGDSRUODVFXUYDV \ = [ + [− \ = − [+ [ = − \ [ = SXQWRV G &DOFXODUHOYROXPHQGHOVyOLGRTXHVHREWLHQHDOKDFHUJLUDUODUHJLyQOLPLWDGDSRU ODVFXUYDV \ = [ \ \ = [ HQWRUQRDODUHFWD \ = SXQWRV &DOFXOH D ∫ I [ G[ FRQ ⎧⎪ [ VL ≤ [ ≤ I [ = ⎨ ⎪ ⎩− [ VL ≤ [ ≤ SXQWRV E ∫ [ − G[ SXQWRV − F ∫ ( [ + ) − [ + G[ SXQWRV 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD G[ G ∫ H ∫ I ∫ VHF \ WDQ\ G\ ( ) [ + [ FRV[ G[ +VHQ[ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 SXQWRV SXQWRV SXQWRV 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱ ȱȱŘŖŖśȱ ȱ ȱȱȱȱȱ ŗǯ Řǯ ∫ Ŝǯ ∫ȱ ŝǯ ∫ [ȱ −[G[ ǻśȱǼȱȱ ∫ȱ [ OQ [ G[ ǻśȱǼȱȱ πȱ ∫ řǯ WDQ[⋅ VHF [ G[ Śǯ śǯ Şǯ ∫ ( [+ VHQ[) ∫ ȱ G[ G[ ǻŝȱǼȱȱ G[ + FRV [ ǻśȱǼȱȱ + H − H− [ȱ ∫ ȱ ǻŞȱǼȱȱ [ πȱ ǻśȱǼȱȱ [ − [ G[ ǻŜ Ǽȱȱ G[ −[ − [+ ǻśȱǼȱȱ şǯ ∫ [ ȱ G[ [ − ǻşȱǼ 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ ȱȱŘŖŖśȱ ȱ ȱȱȱȱȱ ŗǯ Řǯ ∫ȱ [ ∫ȱ ∫ȱ ∫ȱ ∫ȱ ∫ȱ HW [ + G[ − H W ǻśȱǼȱȱ GW ǻśȱǼȱȱ řǯ [ + [ − [G[ ǻśȱǼȱȱ − Śǯ śǯ Ŝǯ VHF W WDQW + WDQW [ − [ Şǯ şǯ G[ [ + [ G[ [ − [ − [ ∫ȱ ∫ ŝǯ GW [ [ + [+ G[ G[ ∫ ȱ+ [ [ ǻśȱǼȱȱ ǻŞȱǼȱȱ ǻŝȱǼȱȱ ǻśȱǼȱȱ − VHQȱ[ ǻŜȱǼȱȱ G[ ǻşȱǼ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱŘŖŖśȱ &DOFXOHODVLQWHJUDOHVVLJXLHQWHV ∫ [ − [G[ SXQWRV ∫ ([+ )DUFWDQ([)G[ SXQWRV ∫ (WDQ [− VHF [)G[ SXQWRV ∫ SXQWRV − H[ H[ G[ (VHQ[− )FRV[ G[ ∫ VHQ [ SXQWRV ∫ [ − [ + [+ G[ [ + [ + SXQWRV ∫ − VHQ[+ FRV[ G[ + VHQ[− FRV[ SXQWRV ∫ − VHQ[+ [ − [ + G[ SXQWRV 8QLYHUVLGDGGH&RVWD5LFD (VFXHODGH0DWHPiWLFD ȱȱ /L]HWK6DQFKR0RUD 3UR\HFWR0$7(0 ȱȱŘŖŖŜȱ &DOFXOHODVLQWHJUDOHVVLJXLHQWHV ∫ [ − [G[ SXQWRV ∫ ([+ )DUFWDQ([)G[ SXQWRV ∫ (WDQ [− VHF [)G[ SXQWRV ∫ SXQWRV − H[ H[ G[ (VHQ[− )FRV[ G[ ∫ VHQ [ SXQWRV ∫ [ − [ + [+ G[ [ + [ + SXQWRV ∫ − VHQ[+ FRV[ G[ + VHQ[− FRV[ SXQWRV ∫ − VHQ[+ [ − [ + G[ SXQWRV ȱ¤ȱ ǰȱ ȱ ¢ȱ ǰȱ ǯȱ ¤ȱ ¢ȱ Çȱ Çǯȱ ·¡Ǳȱ Ǳȱ Ȭ ǯȱȱàȱŗşşŖǯȱ ǰȱ ǯȱ ȱ ¤ȱ ȱ ȱǯǰȱŗşŞŝǯȱ Çȱ Çǯȱ Ǳȱ Ǳȱ ǰȱ ǯȱ ¤ȱ ȱ ȱ ǯȱ ȱ ·¡Ǳȱ Ǳȱ ȱ ·ǰȱŗşşŞǯȱ ȱ ǰȱ ǯȱ ¤ȱȱȱǰȱȱǯȱ Ǳȱ ǱȱǰȱȱàǰȱŗşşŞǯ

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