# Find the function $f(x)$ by using its fourier coefficient

It is easy to find the fourier coefficient and fourier expansion of $f(x)$ function.

But I want solve the inverse problem

How to find the function $f(x)$, if I know its fourier coefficient (or fourier expantion)?

$$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx=\frac{1}{\pi^2n^2}$$

$$b_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)dx=0$$

$$a_0=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)dx=\frac{1}{6}$$ therefore $$f(x)=\frac{1}{6}-\sum_{n=1}^{\infty}\frac{\cos2xn\pi}{(n\pi)^2}$$

• You can find a closed form for the series by using the polylogarithm function. – Mhenni Benghorbal Apr 22 '16 at 23:15
• The summand should be $\cos nx/(n\pi) ^2$! – Mhenni Benghorbal Apr 22 '16 at 23:36
• You should convince yourself that the function is a polynomial of degree two! – Mhenni Benghorbal Apr 22 '16 at 23:51