I would like to know under which circumstances the two following procedures for calculating the residue of a function $f$ at a pole of order $N>1$ are equivalent.
i) evaluating the residue using the standard formula which requires taking $N-1$ derivatives of $f$;
ii) considering that the multiple pole is a set of $N$ simple poles $z_1,...,z_N$ very close to each other. Then you compute the sum of the residues of $f$ at these simple poles and pass to the limit $z_2,z_3,...z_N$ tending to $z_1$.
I have looked for any result like this but I could not find this second method as a form of calculating residues at higher order poles. Under which circumstances it can be applied? Is there a general result like this in complex analysis?