# Strange representation of Heaviside and Delta function

mhill

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

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$$\int \delta(x) f(x) \, dx = \int \frac{-1}{2i\pi} \frac{f(z)}{z} \, dz = (2i\pi) \operatorname{Res}_{z = 0} \frac{f(z)}{2 \pi i} = f(0)$$
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).