Pole

A Complex function $f$ has a pole of order $m$ at $z_0$ if, in the Laurent Series, $a_n=0$ for $n<-m$ and $a_m\not= 0$. Equivalently, $f$ has a pole of order $n$ at $z_0$ if $n$ is the smallest Positive Integer for which $(z-z_0)^nf(z)$ is differentiable at $z_0$. If $f(\pm \infty)\not=\pm\infty$, there is no pole at $\pm\infty$. Otherwise, the order of the pole is the greatest Positive Coefficient in the Laurent Series.

This is equivalent to finding the smallest $n$ such that

$${(z-z_0)^n\over f(z)}$$

is differentiable at 0.

See also Laurent Series, Residue (Complex Analysis)

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 396-397, 1985.