A question on the Dirac delta distribution

[user1139]

Homework Statement

Please refer below

Relevant Equations

Please refer below

Is it correct to say that

$$\int e^{-i(k+k’)x}\,\mathrm{d}x$$

is proportional to $\delta(k+k’)$?

 

[Orodruin]

In the distributional sense, yes.

Consider a test function $\hat \phi(k)$ being the Fourier transform of $\phi(x)$. It holds that
$$
\int \hat\phi(k) \int e^{-i(k+k')x} dx\, dk = \int e^{-ik'x} \int \hat\phi(k) e^{-ikx} dk\, dx
= \int e^{-ik'x} \phi(-x) dx = \int e^{ik'x} \phi(x) dx = 2\pi \hat \phi(-k') = \int 2\pi \delta(k+k') \hat\phi(k) dk.
$$