- #1

- 14

- 0

[tex]\int_{-\infty}^{\infty}e^{-ikx}e^{ik'x} = \delta(k-k')[/tex]

I understand conceptually why this must be true, since you are taking the fourier transform of a plane wave with a single frequency element.

I have also seen it sort of derived by looking at the formula for the fourier series and tracking its components in the limit that it becomes a continuous fourier transorm (letting the period go to infinity and [tex]\Delta\omega[/tex] go to 0)

But I really want to come up with some explicit expression, from doing the integral that behaves like a delta function. I have tried messing around with it, by sticking it inside of another integral and multiplying it by a test function etc. Is there a way to do this?