Interpretation of $\int_{-\infty}^{\infty}\exp(i\,k\,x)\,\mathrm{dx}$

Question

According to another solution here the solution should sound like $2\,\pi\,\delta(k)$

I ask myself: what's the matter of including that factor $2\pi$ ? The reason being $\delta(k)$ is infinite for $0$ and $0$ else, that $2\,\pi$ actually has no effect at all?

Answer 1

Don't think of the local value at a point. Here we are dealing with distributions.

The Dirac delta distribution $\delta$ satisfies $\int \delta(k) \, f(k) \, dk = f(0)$ and so, multiplication by $2\pi$ gives another value: $$\int 2 \pi\delta(k) \, f(k) \, dk = 2\pi f(0).$$