advertisement

2 PROBLEM 1 (a) Derive the first two of the (general) orthogonality equations pT/2 m27r n2rr cos() cos(x) 4 -T/2 - pT/2 m2ir n2rr cos(—--x) sin(-T-x) J—T/2 T/2 1 sin( -T/2 m27r n2r x) sin(— T T )$ ifmn ifm=n T/2 . lo 1T/2 ifm ifm = n n You may use the following identities: cos a cos b [cos(a — b) + cos(a + b)], sinacosb [sin(a — b) + sin(a + b)]. (b) Assuming f(x) is a piecewise-smooth and continuous T-periodic function, we have the Fourier series (in the real form) f(x) =ao+ Use (a) to show that for n 2 / n2u n2r cos(——x) + b s1n(Tx) 1 T/2 1 n2ir f(x)cos(x)c b = f/2 c> 2 T/2 2 TIT/ n2rr f(x) siii(--)4 ir (1v’ —N d -r 5,h (v)2 k ( — (I11+h)27r _75v1cp (112 Jc.I) L - LLL (a)] -r 5, * (-)c) x C J , I 1’ 5 -Th )ç1 — >c jc H1 C C-’ — - 3 t (I Li 0 (. s_f, \j - 4- _L t\ I I 1 + N — lH 111 4 PROBLEM 3 (a) Find the complex form of the Fourier series of the given 2n-periodic function f(x) = if —‘it < x < it. (Hint: you might not need any calculation in part (a).) (b) Find the complex form of the Fourier series of the given 2ir-periodic function e° if —n <x <‘it. You may use the fact the (complex) Fourier coefficients are X 2 C_ T/2 1 Cj (c) Use (b) to find the complex form-&f—t1ie--Fo.u:ier series of the given 2’ir-periodic function f(x)=cosh2xif—n<x<n. () .4 1 f. (12 (r 2 c 2: (1 —° c’ 2-7 r.____ L — J ci ‘ /7 77c -7Tci ) 227 -f (Cl iy 1 F (—I)’ () n e ) ii — h