We briefly touched on Complex Fourier Transforms in my Complex Analysis class, which is extremely interesting to me since the Real Fourier Transform has so many applications in solving PDE's. I am wondering what is an example where you explicitly need to use the Complex Fourier Transform in an application?
Note that the complex Fourier Transform is given by: Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function with $f(x)\in L^2(\mathbb{R})$ and suppose there exists $a,C>0$ constants such that
$$ |f(z)|\le Ce^{a|z|} $$
Then for all $z\in \mathbb{C}$, there exists $\hat{f}(x)$ in $L^2([-a,a])$ such that
$$ f(z)=\int_{-a}^a \hat{f}(t)e^{itz}dt $$