Don't understand an integral with complex numbers.

Question

I am studying the fourier transformation and I don't understand this integral (it is set to find the Fourier coefficients of the function $f(x)=x$: Specifically, I know it is integrating by parts, but I don't understand the last equality. I'd appreciate if someone could elaborate. Thank you

Answer 1

First of all, note that\begin{align}\int_{-\frac12}^\frac12e^{-2\pi inx}\,\mathrm dx&=\left[\frac{e^{-2\pi inx}}{2\pi in}\right]_{x=-\frac12}^{x=\frac12}=0,\end{align}since the exponential function is periodic with period $2\pi i$.

So, the expression after the second $=$ sign is just$$\left[\frac{-1}{2\pi in}e^{-2\pi inx}\right]_{x=-\frac12}^{x=\frac12},$$which is precisely $\dfrac{(-1)^{n+1}}{2\pi in}$.

Answer 2

Hint:

Expanding out the RHS gives us:

$$\frac{1}{2ni\pi}\bigg(\frac 12(e^{n\pi i}+e^{-n\pi i})\bigg)+\frac{1}{2ni\pi}\bigg(\frac{1}{2ni\pi}(e^{-n\pi i}-e^{n\pi i})\bigg)$$

Can you see this is:

$$\frac{\cos (n\pi)}{2ni\pi}+\frac{\sin(-n\pi)}{2n^2\pi^2i}$$