Fourier Series f(x)=Ax^2+Bx+C

Question

Gotta find the Fourier series of $f(x)=Ax^2+Bx+C$ on $(-\pi,\pi)$.

I'm kinda lost, I'm not used to find the Fourier series of that kind of function.

How do I proceed to solve and find the coefficients?

Answer 1

For a function (either periodic or not) $f(x)$ is defined on [-L,L]. The general expression of Fourier series is $$f(x)=\frac{A_0}{2} + \sum_{n=1}^{\infty} C_n \sin (n\pi x/L)+\sum_{n=1}^{\infty} D_n \cos(n \pi x/L) ~~~~~~(1),$$ where $$A_0 = \frac{1}{L} \int_{-L}^{L} f(x) dx, ~~C_n=\frac{1}{L} \int_{-L}^{L} f(x)~ \sin \left(\frac{n \pi x}{L}\right) dx,~~~ D_n=\frac{1}{L} \int_{-L}^{L} f(x)~ \cos \left(\frac{n \pi x}{L}\right) dx.$$

In your case $L=\pi$ and $f(x) =ax^2+bx+c$, so find $A_0, C_n, D_n.$ For instance, for $f(x)=x^2+x$ in $[\pi,\pi],$

we have for $x \in [-\pi, \pi],$ $$f(x)=\frac{\pi^2}{3}+\sum_{n=1}^{\infty} \left (\frac{2 (-1)^{n-1}}{n} \right) \sin \left(\frac {n \pi x}{L}\right)+\sum_{n=1}^{\infty} \left(\frac{4 (-1)^{n-1}}{n^2}\right) \cos \left(\frac{n \pi x}{L}\right) $$