**Question:** Is there a simple method for calculating the Fourier transform of a holomorphic complex function ${f{{\left({z}\right)}}}:\Omega\to{\mathbb{{{C}}}}$?

In order for my question to be well-posed I define a holomorphic function ${f}:\Omega\to{\mathbb{{{C}}}}$ to posses continuous first partial derivatives and satisfy the Cauchy-Riemann equations in a simple connected domain ${\Omega}\subseteq{\mathbb{{{C}}}}$ without any singularities. I am quite familiar with a Fourier transform for a real, periodic function ${f}:{\mathbb{{{R}}}}\to{\mathbb{{{R}}}}$ that uses complex exponentials as a basis of eigenfunctions to generate an expansion ${f{{\left({x}\right)}}}={\sum_{{{n}=-\infty}}^{{\infty}}}{A}_{{{n}}}{e}^{{in{x}}}$.

Given that all functions satisfying the Cauchy-Riemann equations are harmonic, I wondered if the Laplace PDE with homogeneous Dirichlet boundary conditions ${\Delta}{f{{\left({z}\right)}}}={0}\forall{z}\in{\Omega}$ and ${f{{\left({z}\right)}}}{\mid}_{{\partial{\Omega}}}={0}$ could be used to generate a class of harmonic functions in ${\mathbb{{{R}}}}^{{{2}}}$. Admittedly, none would be guaranteed to correspond with analytic functions let alone approximate a desired function ${f{{\left({z}\right)}}}$ within a sufficiently small error bound.

Next, I considered the viability of taking a Fourier decomposition of the real and imaginary components separately, which could be superposed to recover the original function. While this approach merits consideration for sufficiently simple functions, I noticed that it would fail for cases where separability is more enigmatic. For an example, I turn to the Schwarz-Christoffel transform.

$${f{{\left({z}\right)}}}={\int_{{{z}_{{{0}}}}}^{{{z}}}}\frac{{{A}{\left.{d}{z}\right.}}}{{{\prod_{{{j}={1}}}^{{{n}}}}{\left({z}-{x}_{{{j}}}\right)}^{{{k}_{{{j}}}}}}}+{B}$$

In the above, ${A},{B}\in{\mathbb{{{C}}}}$ are both taken to be constants. Given the integral representation of the formula, I find that it would present a particular challenge to separate the components for an arbitrary choice of ${x}_{{{j}}}$.