# Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation:

$\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)-q^{2}\hat{p}(\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0

here $\bf{q}$=$(q_{x},q_{y})$ and $q^{2}$=$q_{x}^{2}$+$q_{y}^{2}$

$\hat{p}$($\bf{q}$$,z) is the fourier transform of the real function p(\bf{s}$$,z)$:

$\hat{p}$($\bf{q}$$,z)= \int$$p$$(\bf{s}$$,z)$$e^{-i\bf{q}\cdot\bf{s}}$$d\bf{s}$

$z_{2}$ is a parameter in the differential equatin , $i$ is the imaginary unit.

the autors report the following solution: