Residue theorem
Simple poles

If c is a simple pole of f, the residue of f is given by:

If that limit does not exist, then f instead has an essential singularity at c. If the limit is 0, then f is either analytic at c or has a removable singularity there. If the limit is equal to infinity, then the order of the pole is higher than 1.

It may be that the function f can be expressed as a quotient of two functions, f (z) = g (z) h (z) {\displaystyle f(z)={\frac {g(z)}{h(z)}}}, where g and h are holomorphic functions in a neighbourhood of c, with h (c) = 0 and h(c) ≠ 0. In such a case, L'Hôpital's rule can be used to simplify the above formula to:
punctuation:no_comma