Strange representation of Heaviside and Delta function

[mhill]

in the .pdf article http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6027/1/jfs080104.pdf

i have found the strange representation

\delta (x) = -\frac{1}{2i \pi} [z^{-1}]_{z=x}

and a similar formula for Heaviside function replacing 1/z by log(-z) , what is the meaning ? of this formula

 

[CompuChip]

Probably the idea is, that we make it a contour integration. Then the only pole is at z = 0 and the result of an integration would be e.g.
\int \delta(x) f(x) \, dx = \int \frac{-1}{2i\pi} \frac{f(z)}{z} \, dz = (2i\pi) \operatorname{Res}_{z = 0} \frac{f(z)}{2 \pi i} = f(0)
if f(x) does not have any poles in the upper half plane.
but I actually doubt how valid this is (even if it works, one would need requirements on f(x) for x \to \pm i\infty to close the countour; and I wonder what happens if f(x) itself has poles).

So I think that is the idea, but I wonder if it works and under what conditions.