Homework Help: Fourier Series to Fourier Integral

1. Jun 16, 2012

AlfredVioleta

1. Question

$$Consider\ any\ periodic\ function\ f(x)\ of\ period\ 2L\ that\ can\ be\ represented\ by\ a\ Fourier\ series:\ {f(x)= a_0 + \sum_{n=1}^\infty\ a_ncos\ w_nx + b_nsin\ w_nx}\ ,\ w_n= {n\pi x\over \ L }$$

$$How\ do\ I\ get\ this\ form\ :\ f(x)= \int_{0}^{\infty}\ [A(w)cos\ wx + B(w)sin\ wx ]\ dw\ ,\ A(w)= {1\over \pi}\int_{-\infty}^{\infty}\ f(v)cos\ wv\ dv\ , B(w) = {1\over \pi}\int_{-\infty}^{\infty}\ f(v)sin\ wv\ dv?$$

2. The attempt at a solution

$$Denoting\ the\ variable\ of\ integration\ by\ v(why\ so?)\ i.e.\ f(x)= {1\over 2L}\int_{-L}^{L}\ {f(v)}\ dv + {1\over L}\sum_{n=1}^\infty\ [(cos\ w_nx) \int_{-L}^{L}\ {f(v)cos\ w_nv}\ dv + (sin\ w_nx) \int_{-L}^{L}\ {f(v)sin\ w_nv}\ dv]$$

$$To\ convert\ to\ a\ Fourier\ integral,\ set\ \Delta w= w_{n+1} - w_n = {\pi\over \ L}\ ,it\ follows\ that\ f(x)= {1\over 2L}\int_{-L}^{L}\ {f(v)}\ dv + {1\over \pi}\sum_{n=1}^\infty\ [(cos\ w_nx) \Delta w\int_{-L}^{L}\ {f(v)cos\ w_nv}\ dv + (sin\ w_nx) \Delta w\int_{-L}^{L}\ {f(v)sin\ w_nv}\ dv]$$

Last edited: Jun 16, 2012