I am trying to find new particular solution for the Helmholtz equation and I was wondering if it is possible to use the complex function theory in this context.
The two dimensional Helmholtz equation is given by
$$\frac{\partial^2 u(x,y)}{\partial x^2}+\frac{\partial^2 u(x,y)}{\partial y^2}+\lambda^2\,u(x,y)=p(x,y)$$ with $x\in\Bbb R$ and $y\in\Bbb R$.
If I use the variable transformation $\eta=x+\mathrm{i}\,y$ and $\zeta=x-\mathrm{i}\,y$ we get $$4\,\frac{\partial^2 u(\eta,\zeta)}{\partial \eta\,\partial\zeta}+\lambda^2\,u(\eta,\zeta)=p(\eta,\zeta)$$ with $\eta\in\Bbb C$ and $\zeta\in\Bbb C$.
I am wondering now, if it is possible to derive the solution for $p(x,y)=\delta(x)\,\delta(y)$ ($\delta()$ the dirac delta function) also for the transformed equation.
Solving the problem in the cartesian coordinate system $(x,y)$ is straight forward using double Fourier Transform. Is this also possible for the transformed equation in $(\eta,\zeta)$? Is the Fourier Transform also valid for functions of complex variables?
Thank you for any hints.