So given is the definition:
$ f(x):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{ikx}dk $
I'm supposed to show that this is a representation of the Dirac delta "function" ($f(x) = \delta(x)$) using complex path integrals. That is I need to show that:
$f(x) = 0\quad \text{for}\quad x\ne 0$ $I = \int_{-\infty}^{+\infty} f(x) dx = 1$
So my idea for the first equation was to say that in complex space:
$ 0 = \oint e^{izx}dz = \underbrace{\int_{-\infty}^{+\infty} e^{izx}dz}_{=f(x)} + \int_{γ_\text{arc}}e^{izx}dz$
Where $\gamma_\text{arc}$ is a half circle around 0 of radius $\infty$ in the complex plane. I suspect the integral $\int_{γ_\text{arc}}e^{izx}dz$ over this path $\gamma_\text{arc}$ is zero for $x\ne 0$ which would make f(x) = 0. But I have no idea how to actually do that.
Also I have no idea for the second identity $\int_{-\infty}^{+\infty} f(x) dx = 1$.
Please help me. It's not homework but could pop up in a future test. The requirements explicitly state that it has to be solved with complex path integrations.
Edit: Here is an integral similar to my idea.