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Find the Fourier series for a function f(x)=f(x+2),f(x)=(x-1)(x-3)\, on [1,3]\,.

Make the change of variables z=x-2\,.

Now, look for the Fourier series of the function f(z+2)=f(z+4),f(z+2)=(z+1)(z-1)\, on -1\leq z\leq 1\,

f(z)={\frac  {a_{0}}{2}}+\sum _{{k=1}}^{\infty }a_{k}\cos(k\pi x)+b_{k}\sin(k\pi x)\,

a_{k}=\int _{{-1}}^{1}(z+1)(z-1)\cos(k\pi (z+2))\,dz={\frac  {4(-1)^{k}}{k^{2}\pi ^{2}}}\,

a_{0}=\int _{{-1}}^{1}(z+1)(z-1)dz={\frac  {-4}{3}}\,

b_{k}=\int _{{-1}}^{1}(z+1)(z-1)\sin(k\pi (z+2))\,dz=0\,

Since f(z)=f(x-2)=f(x)\,,

f(x)={\frac  {-2}{3}}+\sum _{{k=1}}^{\infty }{\frac  {4(-1)^{k}}{k^{2}\pi ^{2}}}\cos(k\pi x)\,

Fourier Series

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