Proving Improper Integral with Complex Analysis

[jusy1]

Hi everybody

I was trying to prove that $$\int_{-\infty}^{\infty}e^{\imath (k - k') x}dx = 2\pi\delta(k-k')$$ by solving $$\lim_{L\rightarrow \infty} \int_{-L}^{L}e^{\imath (k - k') x}dx$$

knowing that $$\delta(x)=\lim_{g\rightarrow \infty}\frac{\sin(gx)}{\pi x}$$

But is there a way of proving this result using complex analysis?

 

[Char. Limit]

Wouldn't it be easier to prove this considering piecewise, i.e. consider k≠k' and then consider k=k'?