Fourier Transform $\frac{e^{-ik\sqrt{(z-z_0)^2+a^2}}}{\sqrt{(z-z_0)^2+a^2}}$

Question

I am trying to compute the 1D Fourier transform (in $z$) of

$$\frac{e^{-ik\sqrt{(z-z_0)^2+a^2}}}{\sqrt{(z-z_0)^2+a^2}}.$$

I tried to use the fact in 3D,

$$\frac{e^{-ik\sqrt{(z_0-z)^2+(y_0-y)^2+(x_0-x)^2}}}{\sqrt{(z_0-z)^2+(y_0-y)^2+(x_0-x)^2}}$$

is the fundamental solution of the Helmholtz equation $\Delta f -k^2 =0$, but I'm stuck with the change of dimensions.

Does anyone have some hints? I am new to distribution theory.

Answer 1

This integral for Fourier transform can be written as derivative $$ -i\frac{\partial}{\partial k}\int_{R}e^{-i p z} \frac{e^{-ik \sqrt{z^2+a^2}}}{\sqrt{z^2+a^2}}dz=\int_{R}e^{-i p z} e^{-ik \sqrt{z^2+a^2}}dz. $$ May be this could help