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Fourier series

2022-07-28 03:48:00 【Super size monkey K】

$f(x)=a_{0}cos0x+a_{1}cosx+a_{2}cos2x+..... +b_{0}sin0x+b_{1}sinx+..... =a_{0}+\sum_{n=1 }^{\infty}(a_{n}cosnx+b_{n}sinnx)$

{1,cosx,sinx,con2x,sin2x,....,cosnx,sinnx,....}

Orthogonality

$\int_{-\pi }^{\pi}cosnxdx = 0 (n=1,2,3....)$

$\int_{-\pi }^{\pi}sinnxdx = 0 (n=1,2,3....)$

$\int_{-\pi }^{\pi}sinkx*cosnxdx = 0 (n,k=1,2,3....)$

$\int_{-\pi }^{\pi}coskx*cosnxdx = 0 (n,k=1,2,3....)$

$\int_{-\pi }^{\pi}sinkx*sinnxdx = 0 (n,k=1,2,3....)$

Integral sum difference formula

$sin\alpha cos\beta =\frac{1}{2}[sin(\alpha +\beta )+sin(\alpha -\beta )]$

$cos\alpha cos\beta =\frac{1}{2}[cos(\alpha +\beta )+cos(\alpha -\beta )]$

$cos\alpha sin\beta =\frac{1}{2}[sin(\alpha +\beta )-sin(\alpha -\beta )]$

$sin\alpha sin\beta =-\frac{1}{2}[cos(\alpha +\beta )-cos(\alpha -\beta )]$

Series three ：an,bn=？

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