In the previous post on Fourier Series, we looked at functions periodic with period 2pi. Now, we’ll take a look at Fourier series for functions having an arbitary period, lets say, some period 2L.

The general formulas we would need for finding the Fourier series are as follows…

The Series:

$\150dpi \large f(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty }(a_{n}cos\frac{n\pi x}{l}+b_{n}sin\frac{n\pi x}{l})$

The Constants:

$\150dpi \large a_{0}=\frac{1}{L}\int_{-L}^{L}f(x)dx$

$\150dpi \large a_{n}=\frac{1}{L}\int_{-L}^{L}f(x)cos\frac{n\pi x}{L}dx$

$\150dpi \large b_{n}=\frac{1}{L}\int_{-L}^{L}f(x)sin\frac{n\pi x}{L}dx$

These apply when the period given in the question are [-L, L]. We would modify the limits of integration in the above depending on the given interval.

Further, for functions periodic with a period 2pi, we only need to put L=pi in the above formulas. You can find those formula here.

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