Fourier Series

Tags: #math #fourier series

Equation

$$y(x)=c_{0}+\sum^{M}_{m=1}c_{m}\cos mx+\sum^{M^{'}}_{m=1}s_{m}\sin mx \\ c_{0}=\frac{1}{2\pi}\int^{\pi}_{-\pi}y(x) \mathrm{d} x \\ c_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \cos mx \mathrm{d} x \\ s_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \sin mx \mathrm{d} x$$

Latex Code

                                 y(x)=c_{0}+\sum^{M}_{m=1}c_{m}\cos mx+\sum^{M^{'}}_{m=1}s_{m}\sin mx \\
c_{0}=\frac{1}{2\pi}\int^{\pi}_{-\pi}y(x) \mathrm{d} x \\
c_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \cos mx \mathrm{d} x \\
s_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \sin mx \mathrm{d} x
                            

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Introduction

If y(x) is a function defined in the range then the above fourier series expansion formula holds.


  • Fourier series for ranges t belongs to [0,T]
  • Fourier series for ranges x belongs to [0,L]

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