Can the Fourier transform of a whole function $f:\mathbb R\mapsto\mathbb C$ be defined as integration over $\mathbb C$ instead of $\mathbb R$ as well, such that $\tilde f(k) = \frac{\mathcal N}{2\pi} \int_{\mathbb C}f(x) e^{-ikx}\,dx,$
and (if so) what is the correct value of the normalization $\mathcal N$ for consistency with the $\mathbb R$-integration?
Given a whole function $f:\mathbb R\mapsto\mathbb C$, its Fourier transform
$ \tilde f(k) = \frac1{2\pi}\int_{-\infty}^\infty f(x)e^{-ikx}\,dx$
can be determined by other integration paths as well by using Cauchy's Residue Theorem, for example by shifting $x$ by an imaginary constant $ic$. Assuming a sufficiently fast decaying function for $\Re(x)\to\pm\infty$ (and using the fact that a whole function has no residues), this results in simply adding that constant to the integration boundaries, i.e.
$ \tilde f(k) = \frac1{2\pi}\int_{-\infty+ic}^{\infty+ic} f(x)e^{-ikx}\,dx,$
which can be expressed by substitution as well:
$ \tilde f(k) = \frac1{2\pi}\int_{-\infty}^{\infty} f(x-ic)e^{-ik(x-ic)}\,dx.$
One can then average over different values of $c$ to obtain
$ \tilde f(k) = \frac1{2\pi\cdot 2T}\int_{-T}^{T}\int_{-\infty}^{\infty} f(x-ic)e^{-ik(x-ic)}\,dx\,dc.$
Now my question boils down to
1) Since infinity is involved, is this equivalent to
$ \tilde f(k) = \frac1{4\pi T}\int_{\mathbb R\times[-iT,iT]} f(x_1+x_2)e^{-ik(x_1+x_2)}\,d^2x,$
2) Can the limit $T\to\infty$ be taken such that $\mathbb R\times[-iT,iT]\to\mathbb C$
and 3) What would the correct normalization be?