Fourier Series to Fourier Integral

[AlfredVioleta]

1. Question

[tex] 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 } [/tex]



[tex] 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? [/tex]



2. The attempt at a solution

[tex] 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][/tex]


[tex] 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] [/tex]

 

[lippyka]

Taking\ the\ limit\ as\ n\rightarrow \infty \ ,\ f(x)= {1\over 2L}\int_{-L}^{L}\ {f(v)}\ dv + {1\over \pi}\int_{-\infty}^{\infty}\ [cos\ wx\int_{-L}^{L}\ {f(v)cos\ wv}\ dv + sin\ wx\int_{-L}^{L}\ {f(v)sin\ wv}\ dv]\ dw Defining\ A(w)= {1\over \pi}\int_{-L}^{L}\ {f(v)cos\ wv}\ dv\ and\ B(w)= {1\over \pi}\int_{-L}^{L}\ {f(v)sin\ wv}\ dv\ ,\ the\ expression\ for\ the\ Fourier\ integral\ form\ of\ the\ function\ is\ given\ as\ :\ f(x)= {1\over 2L}\int_{-L}^{L}\ {f(v)}\ dv + {1\over \pi}\int_{-\infty}^{\infty}\ [A(w)cos\ wx + B(w)sin\ wx ]\ dw