Integrals in Plane Wave basis: $ \int_{-\infty}^{\infty} x^2 e^{ikx} \, dx $

Question

While I know

$$ \int_{-\infty}^{\infty} e^{ikx} \, dx = 2\pi \delta(k), $$

I am wondering if someone knows how to evaluate

$$ \int_{-\infty}^{\infty} x^2 e^{ikx} \, dx $$

? Any help would be much appreciated. Thanks.

Answer 1

Differentiating $$\int_{-\infty}^{\infty} e^{ikx} \, dx = 2\pi\, \delta(k)$$ twice with respece to $k$ we get $$- \int_{-\infty}^{\infty} x^2 e^{ikx} \, dx = 2\pi\, \delta''(k)$$