TITLE: Fourier series is to Fourier transform what Laurent series is to ...?
QUESTION [17 upvotes]: Since the coefficients
$$a_k = \frac1{2\pi i}\oint_C\frac{f(z)}{(z-c)^{k+1}}\,dz$$
for the Laurent series
$$f(z)\Big|_{r\le|z|\le R} = \sum_{k=-\infty}^{\infty}a_k\cdot(z-c)^k $$
of a function $f\in\mathcal H(B(r,R))$ (i.e. a function that is holomorphic on the Annulus of radii $r\le R$), evaluated on a circle of radius $\rho\in[r,R]$,
$$\tilde a_k := \rho^k a_k = \frac1{2\pi}\int\limits_{\phi=0}^{2\pi} f(c+\rho e^{i\phi}) e^{-ik\phi}\,d\phi
\\\Rightarrow f(c+\rho e^{i\phi}) = \sum_{k=-\infty}^\infty \tilde a_k\,e^{ik\phi},$$
are (up to the factor $\rho^k$) equivalent to the Fourier coefficients of a function 
$$\tilde f(x):=f\Big(c+\rho e^{i\tfrac x{2\pi\rho}}\Big),\quad x\in[-\pi\rho,+\pi\rho],$$
I was wondering if there is a meaningful limit $\rho\to\infty$ for the Laurent series. For the Fourier series, the limit becomes the Fourier transform with continous "coefficients", $\tilde a_k\to \tilde a(k)\equiv \mathcal F\{\tilde f(x)\}(k)$, such that
$$\tilde f(x) = \int_{-\infty}^\infty \tilde a(k) e^{ikx}\, dk,
\\\tilde a(k) = \frac1{2\pi}\int_{-\infty}^\infty \tilde f(x) e^{-ikx}\, dx.$$
So in a similar way, this limit would turn the Laurent series into a "Laurent transform",
$$f(z)\Big|_{r\le|z|} = \int_{-\infty}^\infty a(k)\cdot (z-c)^k,
\\ a(k) = \lim_{R\to\infty} \frac1{2\pi}\oint\limits_{|z|=R}\frac{f(z)}{(z-c)^{k+1}}\,dz$$
(up to some factors that I probably forgot, and assuming that $f$ remains holomorphic for $R\to\infty$), describing $f(z)$ by its values at $\mathbb C$-infinity.
So, does this make sense, has it been investigated upon before and/or any applications?

REPLY [4 votes]: You may want to look into the Mellin transform. The basic idea is similar to what you seem to have in mind, but the integration contours are different from what you guessed. Incidentally, one side of the Mellin transform applies even outside the context of analytic functions. Of course, the Fourier, Laplace and Mellin transforms are fundamentally related, through changes of integration variables.