I am looking for the Fourier series of a monomial restricted to the inteval $(0,2\pi)$.
Let $n\in\mathbb{N}$ and
$$\forall x\in (0, 2\pi), \ f(x)=x^n.$$
By definition, the Fourier coefficients are
$$c_k = \frac{1}{2\pi}\int_0^{2\pi} x^n e^{-ikx} dx,$$
and we know that
$$f(x) =_{\text{a.e.}} \sum_{k\in\mathbb{Z}} c_k e^{ikx}.$$
This can be written in terms of gamma incomplete function, but there might be a closed form for this particular definite integral.
What is the exact value of $c_k$, the Fourier coefficient of the monomial $x^n$?
Best Answer
To make the dependence on $n$ explicit, let's write $$c(k,n) = \dfrac{1}{2\pi} \int_0^{2\pi} x^n e^{-ikx}\; dx$$ For $k = 0$, we have $$ c(0,n) = \frac{(2 \pi)^n}{n+1} $$ For $n=0$ and $k \ne 0$, $c(k,0) = 0$, while by an integration by parts $$ c(k,n+1) = \frac{(2\pi)^n i}{k} - i (n+1)\frac{c(k,n)}{k}$$ and by induction we can prove (for $k \ne 0$) $$ c(k,n) = - \sum_{j=1}^n \frac{n!}{(n+1-j)!} \frac{(-i)^j (2\pi)^{n-j}}{k^j} $$