Question

3. A function f(x) is represented by the Fourier series: f(x) = { (an sin "+ bn cos "E") on the range (-L, L). Express S-L f(x)?dx in terms of an and bn

Answer 1

Giren function is : f(x) = [(an sin(~23) + bn 603(***) 1. in interval -Lerci o Separating term for n=o, we get (by putting n=o in ff») = qo sin (ona) + be cos (0.92-) + ( an sin (1974) + bo, 60 (92) => in) = ao sino + , 6s (0) + E (09 th (2) [:: sinco) =0,4036) = 1] 1 + bn cas (nax)) (an sin ( * *2.) + by 603 (172) I lan sin (nan) + bn cos (194* )) = f(x) = ao sino + bo Cos (0) + ny = f(x) = 40(0) +bell) +. = f() = 0 + b + =) f() = bo + fan sin ( max.) + bn cos (172) ho equation 2 coefficients series of fla) is the fourier de fined by with f(x) dx ---- - - - (7 ' bo = 32 S +(7) dx bn = t fins 100 (934) dx ---- an = S +6w) sin (1974) dx ---- © - and -

ferom ③ , 9 and ①, we get = 2Lb. ---- I fln) cos( ngx) dx = Lbn ----☺ fla sin naa da Lan --- and [fla sin (max) dx = Lan ---- Now, multiply @ by f(x) on both sides, we get (flm))* = f(wp[bot (a, sin( pazaj + b, 60 (972)] > (419))? = bo tm) + Şamele) sin ( 1921) + Š bo f14) 60 (934) --- h=) h=) Now, integrating @ went xin limits -2 to L ((f(x))2 dx =( b. f(x) dx in + anf(x) sin (nan) du -2. - L =) + s bn fly) cas (nm) dx -Ź hal

- Brown's = bo jeme + an Stro einfazeja In7 h ) -L (nan **(a) as opin} lu By using ③, 6 and ® in , we get 2 dx = bo (albo) + ) an(lan) t) bn (Lbn) he co (2) 2 dx = albo t. 1 an L 2 Scaroy-da = by (ons) + anexans + (10) - jeres de sevë + a* * * - $ 19cm) - de - 2232 5 (ant thi) – jcerosa 2[*** [3] no ne) ) ² dx = 2262 + L 2) Ton / ho lau - This is the required answer