16.8: Fourier Series
- Page ID
- 107067
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For a periodic function of period \(2L\):
\[\begin{array}{c} f(x)=\frac{a_0}{2}+\sum\limits_{n=1}^{\infty}a_n\cos\left ( \frac{n \pi x}{L} \right )+\sum\limits_{n=1}^{\infty}b_n\sin\left ( \frac{n \pi x}{L} \right ) \\ a_0=\frac{1}{L}\int\limits_{-L}^{L}f(x)dx \\ a_n=\frac{1}{L}\int\limits_{-L}^{L}f(x)\cos{\left(\frac{n \pi x}{L} \right)}dx \\ b_n=\frac{1}{L}\int\limits_{-L}^{L}f(x)\sin{\left(\frac{n \pi x}{L} \right)}dx \end{array} \nonumber\]