Fourier series with complex coefficients

Question

Hello i need to find the complex coffecients of the fouriers series of the $2\pi$ perriodic function: $$f(x)= \frac{\pi-x}{2}, \forall x\in]-\pi;\pi]$$ I stock in the computation of complex coefficients $$c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}{f(x)e^{-inx}}dx$$. Precisely in what doing with $e^{in\pi}$ in the result.

Answer 1

The integral with $\frac{\pi}{2}$ vanishes as you get $\frac{sin(n \pi)}{\pi n}$ with $n$ being an integer as result. For $\frac{-1}{2 \pi}\int_{-\pi}^{\pi}xe^{-inx}dx$, I get $-\frac{i(-1)^n}{n}$ as result using integration by parts. For $c_0$, I get $\frac{\pi}{2}$ as result. For $c_0$, it's the $-x$ which doesn't contribute to the result as you get $-\pi - (-\pi)=0$ for the variable $x$. So $c_0$ is $\frac{\pi}{2}$ and $c_n$ is $\frac{-i(-1)^n}{n}$ if I made no mistake. Maybe you were troubled to get $0$ for the $\frac{\pi}{2}$ of the $c_n$ ?

Hope this helps. If I made any mistake, please say it in the comments as soon as possible.