For the function $\frac{1}{x}$ on the real line, one can use a modified principal value integral to consider it as a distribution p.f.$(\frac{1}{x}),$ and one can do a similar construction to make $\frac{1}{x^m}$ into a distribution for $m>1.$ In the complex plane, the function $\frac{1}{z^m}$ is locally integrable for $m=1,$ but for larger $m$ some construction analogous to the one dimensional would have to be done to make it into a distribution.

More generally, given a meromorphic function on the plane (or torus), one should be able to consider it as a distribution by integrating against it and subtracting off some delta distributions or derivatives of delta distributions. Is this process explained in detail anywhere? Has anyone computed the Fourier series of such distributions, say for the Weierstrass $\mathfrak{p}$ function on the torus?

`$m-1$`

-th distributional derivatives of the locally integrable function`$1/z$`

one can extend`$z^{-m}$`

from the punctured complex plane to a distribution on the full complex plane. – Sönke Hansen Apr 18 '13 at 21:32