Say, we define Fourier transform:
$$\tilde f(p)= \int_{-\infty}^\infty \mathrm{d}z f(z)e^{-ipz}$$
and the inverse transform
$$f(z)=\frac{1}{2\pi}\int_{-\infty}^\infty \mathrm{d}p~ \tilde f (p) e^{ipz}$$
Are these only defined for $f\in L^1(\mathbb{R})$ or can it also be defined for $f\in L^1(\mathbb{C})$?
Is there a way to define an equivalent transform for complex functions by defining the integral over some contours instead of the real lines?