I am currently taking an introductory, undergraduate PDEs course and became interested in the applicability of Fourier decompositions to functions defined on the regions of the complex plane. Classically, I consider a function $f:[-\pi, \pi] \to \mathbb{R}$ to yield a Fourier transform where the basis of the infinite-dimensional vector space is either comprised of sines, cosines, or complex exponentials. The coefficients are then computed by normalizing the inner product in the Hilbert space.
$$a_k = \frac{2}{L} \int_{-\pi}^{\pi}f(x)\overline{e^{ikx}}dx$$
I currently think about the inner product between the original function $f(x)$ and the basis function $e^{ikx}$ as an open contour integral in $\mathbb{R} \subset \mathbb{C}$. In the event that $f$ were a holomorphic function of class $C^{\infty}$, I believe one could trivially decompose its real and imaginary components individually to develop a Fourier series for the function.
Question: Is there a more canonical method to produce a Fourier transform of a holomorphic function?
Considering that $e^{ikx}$ is entire, I would think that the Cauchy-Goursat theorem would guarantee that the integral of $f(x)\overline{e^{ikx}}$ over a closed contour $C$ would strictly equal $0$. If there was to be a two-dimensional planar equivalent of the Fourier transform on the finite interval, would the contour defining the inner product for functions (which from a measure-theoretic perspective are no more than continuous distributions) still have to remain open?