!"## $%&'# ( ) !"#$%#& ''( ')** • !"#$#% &'()*!+ #,(-!./0 123.4!/$3# .!).!/!#434$'#5 z = x + iy Re(z) = x Im(z) = y z = Re(z) + iIm(z) i2 = −1 6!'(!4.$& $#4!.)3434$'#7 p 8!#%49 ': z = x + iy 3/ 3 ;!&4'. $# R2 $/ |z| = x2 + y2 • !"#!< 3&4$'#/0 =<<$4$'#>?,-4.3&4$'# @ A#4,$4$;!7 (,*4$)*$&34$'# @ A#4,$4$;!7 2'()*!+ &'#B,%34!0 $: z = x + iy 49!# z = x − iy C.')'/$4$'#0 D'. !;!.E z ∈ CF z · z ∈ R+ 7 $;$/$'# @ (,*4$)*$&34$'# -E 49! &'#B,%34! ': 49! <!#'($#34'.7 G+3()*!/0 z1 = 2 + i F z2 = 1 − 3i7 z1 $;$<$#% z • • G+!.&$/!0 = 2+i (2 + i) (1 + 3i) (1 − 3i) (1 + 3i) √ |z| ≤ |Re(z)| + |Im(z)| ≤ 2 |z| C.'':0 |z| = q 2 1 − 3i = = − 1 + 7i 10 = −1 10 +i 10 q q q 2 2 2 2 2 (Re(z)) + (Im(z)) = |Re(z)| + |Im(z)| = (|Re(z)| + |Im(z)|) − 2 |Re(z)| · |Im(z)| ≤ (|Re(z)| + |Im(z)|) = |Re(z)| + |Im(z)|7 q q 2 2 2 |Re(z)| + |Im(z)| = (|Re(z)| + |Im(z)|) = |Re(z)| + 2 |Re(z)| · |Im(z)| + |Im(z)| 2 A4 .!(3$#/ 4' /9'H 49340 2 |Re(z)| · |Im(z)| ≤ |Re(z)|2 + |Im(z)|2 =#< 49$/ $/ '-;$',/ /$#&! (|Re(z)| − |Im(z)|)2 ≥ 0 3*H3E/7 • 7 C'*3. .!).!/!#434$'#0 I !"#$ !"%&'#()'*+&, z = x + iy %-+ -./' 0" #")#"/"+&"1 -/2 3 4"%&'#5/ ."+6&78 9'#:-..$ 1";+"1 r = r p x2 + y 2 8 +'&*%" 9'# "!"#$ z ∈ C r ≥ 0 y <-:)."2 z = −3 + 3i8 F7"+ θ = − %-.%?.-&*+6 &7" )'.-# #")"#"/"+&-&*'+ r = π 3π , = 4 4 ! G'.-# #")"/"+&-&*'+ *+ &7*/ %-/" */2 )#'!"1 *+ &7" ."%&?#",= • y 3 4"%&'#5/ -+6." (#".-&*!" &' &7" #"-. -<*/,8 9'#:-..$ 1";+"1 θ = arctan 8 '# tanθ = = x x >'&*%" &7-& tanθ 7-/ &7" /-:" !-.?"/ 9'# θ5/ *+ &7" @/& -+1 &7" A#1 B?-#&"#/8 -+1 &7" /-:" 9'# &7" C+1 -+1 &7" D&7 B?-#&"#/= E" 1";+" θ ∈ [−π, π]= θ x = r · cos θ y = r · sin θ 3 √ = −1 188 tanθ = −3 ⇒ z = r · eiθ = r · (cos θ + isin θ) &7*/ 7-))"+/ ( ?."#5/ 9'#:?.- -/ F-/ H":-#I/2 |z| = r = p x2 + y 2 z · z = x2 + y 2 = r2 = |z|2 • G#')'/*&*'+2 eiθ = eiη ⇐⇒ θ = η + 2πk 8 k ∈ Z= G#''92 eiθ = cosθ + i · sin θ 8 eiη = cos η + i · sin η= e • • iθ =e iη ⇐⇒ cos θ = cos η sin θ = sin η ⇐⇒ θ = η + 2πk 8 k ∈ Z= J";+" &7" "#$%&'() '9 - %':)."< +?:0"# z &' 0"2 arg(z) = J";+" &7" *#$%&'() '9 - %':)."< +?:0"# z &' 0"2 θ | tan θ = Arg(z) = )#*+%*)." -#6?:"+& (ארגומנט ראשי,= y x θ ∈ (−π, π] | tan θ = y x 8 -./' %-.."1 &7" >'&*%" &7-& *9 θ0 ∈ Arg(z) &7"+ 9'# /?#" θ0 + 2πk ∈ arg(z) 9'# /':" k ∈ Z= K+ '&7"# F'#1/2 Arg(z) + 2πZ = {Arg(z) + 2πk | k ∈ Z} = arg(z)= Arg(0) */ +'& 1";+"1 '# 1";+"1 &' 0" L"#' /':"&*:"/= E" /"& 9'# "!"#$ z ∈ R− Arg(z) = π ⇒ Arg(z) */ +'& %'+&*+?'?/" '+ R− = • G'F"# '9 %':)."< +?:0"#/2 H*/*+6 - %':)."< +?:0"# z &' - )'F"# */ 1'+" *+ &7" 9'..'F*+6 F-$2 C !"# $%& #'!()*# z = 1 + i+ ,# ,!-. .% /!*/0*!.# z 203 1 !√ 2# /3!-4# .% )%*!& /%%&56-!.#78 z 203 = 2 · ei 203 π 4 √ √ 3π 3π = 2101 2e50iπ+i 4 = 2101 2ei 4 = 2101 · !√ ! 3π = 2101 (1 − i) 2 · cos 3π 4 + isin 4 9- .3# 4#-#&!* /!7#8 • ! n n z n = (x + iy) = reiθ = rn einθ 1 :%%.7 %$ /%()*#' -0(;#&78 9- .3# 4#-#&!* /!7#+ .3# #<0#.6%- z n = a ,6** !*,!=7 3!># - 0-6<0# 7%*0.6%-71 √ ,# ,!-. .% !-7,#& .3# <07#.6%- z =?+ !/.0!**= ,# ,%0*5 !-7,#& .3# <0#7.6%- z = wn = rn einθ + w = r · eiθ 1 ?%.6/# .3!. 8 n z = |z| · ei·Arg(z) + .3#&#$%&# |z| = rn ⇒ r = p n |z| @ Arg(z) = nθ ⇒ θ = 76-/# z = |z| · ei·(Arg(z)+2πk) + .3# 4#-#&!* 7%*0.6%- ,%0*5 ;# θ = Arg(z) n Arg(z) + 2πk n 1 + ,3#&# k ∈ Z, n ∈ N1 %>#&!** ,# 4#. - 0-6<0# 7%*0.6%-7 .% .3# *!7. #<0!.6%-1 A'!()*#8 √ 3 −2 + 2i =? 3π √ |z| = 8+ Arg(z) = arctan(−1) = 4 31 √ √ π 1 3π 3 = 8 6 ei 4 1 −2 + 2i = 8 · ei 4 9- 4#-#&!*8 zk = 8 ei( + ) B!*/0*!.6-4 .3# &#7. %$ .3# 7%*0.6%-7 ,# 4#.8 z0 = 8 ei + z1 = 8 ei( + ) + z2 = 8 p √ % /%-/*05#+ ,# 3!># 73%,- .3!. z = |z| · ei 1 √ Arg(z) 67 -%. /%-.6-0%07 %- R− .3#&#$%&# z 67 !*7% -%. /%-.6-0%07 .3#&#1 1 6 π 4 2πk 3 1 6 n Arg(z) n n n C π 4 1 6 π 4 2π 3 1 6 π ei( 4 + 4π 3 )