I'm trying to solve this integral that appears in L.I. Deych's "Advanced Undergraduate Quantum Mechanics", which is supposed to be a self-teaching book but has no solution manual. The book offers a hint: "Use the representation of the delta-function as a Fourier integral to figure out the integral with respect to k."
$$\int ^{+\infty }_{-\infty }dxf\left( x\right) \int ^{+\infty }_{-\infty }dkke^{ik\left( x-x'\right) }$$
I've used integration by parts for the integral with respect to k, but to no avail. Any further hints?
Thanks in advance.
The book want you to use the fact that $$\int \mathrm{d} t \, e^{- i \omega t} e^{i a t} = 2 \pi \delta(\omega - a)$$ And we have that $$\int \mathrm{d}t \, t^n f(t) e^{-i \omega t}=i^{n}\frac{\mathrm{d}^nF(\omega)}{\mathrm{d}\omega^n}$$ So we have that $$\int \mathrm{d}k \, k e^{i k (x-x')}=i\frac{\mathrm{d}}{\mathrm{d}x'}\bigg(2 \pi \delta(x'-x)\bigg)=2i \pi \frac{\mathrm{d}\delta(x'-x)}{\mathrm{d}x'}$$