**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 ${x}\_{{{j}}}$.



Fourier transform of a holomorphic function
It isn't clear what you mean by the Fourier transform of a holomorphic function. You would need to supply a definition of this concept before you can get coherent answers. Perhaps your question is about an attempt to define such a transform?


It seems like you are interested in expressing functions in some orthonormal basis. For this, you might be interested in "Bergman spaces". One example of a Bergman space is the set of all holomorphic and square integrable functions on an open set $\Omega \subset \mathbb{C}$. One can prove that these are complete Hilbert spaces, and possess an orthonormal basis. One could view this as a kind of "Fourier transform" like situation. In fact, these spaces are even nicer since they are "reproducing kernel Hilbert spaces", and have a kernel ("the Bergman Kernel") such that one can recover point values of $f$ by integrating against the kernel.


Most information about Bergman Kernels is in the Several Complex Variables literature. Krantz's SVC textbook has a good treatment.


Another direction you might be interested in exploring are all the various flavors of "Paley-Wiener" theorems. One which might be of interest to you is the following:


Let $\mathbb{H}$ be the lower half plane. Let $f: \mathbb{H} \to \mathbb{C}$ be a holomorphic function. Define $f\_y: \mathbb{R} \to \mathbb{C}$ by $f\_y(x) = f(x+iy)$. If the integral


$$
\int\_{-\infty}^{\infty} |f\_y(x)|^2 \textrm{d}x
$$


is bounded above by a constant independent of $y$ then there is a function $g \in L^2(0,\infty)$ so that the Fourier transform of $g$


$$
\mathcal{F}(g)(z) = \int\_0^\infty g(t)e^{-2\pi itz} \textrm{d}t
$$


is equal to $f$. This gives you an inverse Fourier transform for holomorphic functions of this type defined on the lower halfplane.


For a quick reference


[https://www.ndsu.edu/pubweb/~littmann/Topics15/class10-7.pdf](https://www.ndsu.edu/pubweb/%7Elittmann/Topics15/class10-7.pdf)


(beware several typos involving negative signs and factors of $2\pi$).


More in depth treatments can be found in the ``Logarithmic Integral" tomes of Koosis.


Note: We need $z$ in the lower half plane since if $z = a+bi$, we have $|e^{-2\pi itz}| = e^{2\pi tb}$. We want this to decay to $0$ as $t \to \infty$ so that the integral is well defined, so we want $b < 0$.


