Fourier transform of $\mathrm{e}^{ik|x|}$?

Question

What is the Fourier transform of $\mathrm{e}^{ik|x|}$? Here, $k > 0$ is real.

I use the definition $$ F(\omega) = \int_{-\infty}^\infty \mathrm{e}^{-i\omega x} f(x) \mathrm{d}x.$$

Thanks!

Answer 1

You can't use that definition in order to compute that Fourier transform, because that formula is valid for elements in $L^1(\mathbb{R})$ and the function $x\mapsto e^{ik|x|}$ is not in $L^1(\mathbb{R})$.

Answer 2

We have $$\mathcal F[H] = -\frac i w + \pi \delta(w), \\ \mathcal F[e^{i k x} H(x) + e^{-i k x} H(-x)] = \\ \delta(w - k) * \mathcal F[H] + \delta(w + k) * \mathcal F[H](-w) = \\ -\frac i {w-k} + \pi \delta(w-k) + \frac i {w+k} + \pi \delta(w+k) = \\ -\frac {2 i k} {w^2 - k^2} + 2 \pi k \delta(w^2-k^2),$$ where $1/(w-w_0)$ and $1/(w^2-w_0^2)$ are p.v. functionals.