FS7

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 $-1\le z\le 1\,$

$f(z) = \frac{a_0}{2} + \sum_{k=1}^\infty a_k \cos(k\pi x) + b_k \sin(k\pi x)\,$

$a_k = \int_{-1}^1(z+1)(z-1)\cos(k\pi(z+2))\,dz = \frac{4(-1)^k}{k^2\pi^2}\,$

$a_0 = \int_{-1}^1(z+1)(z-1)dz = \frac{-4}{3}\,$

$b_k = \int_{-1}^1(z+1)(z-1)\sin(k\pi(z+2))\,dz = 0\,$

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

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