How to determine the order of poles in a quick way? How do you determine the order of poles in a quick way?. For example, if you have the complex function $\frac{ln(x)}{(x^2+1)^2}$, you notice that the singularities are at $x=i,-i$. Assume that there is a contour that encloses them. How do you determine the order of these poles without using power series expansion. My professor talked about the vanishing points. Like when the denominator vanishes, but I did not understand his argument. It would also be nice if you can provide other examples in which we can calculate the order of the poles by analyzing the function. Besides that, how can you tell right away the singularities are poles?

The $\ln(x)$ is analytic and nonzero in a neighbourhood of the poles, so it doesn't change anything: the orders of the poles are the same as those of 
$1/((x^2+1)^2)$.   The denominator is $(x+i)^2 (x-i)^2$: it has zeros of order $2$ at $\pm i$, so the orders of the poles are $2$.  
