Is it possible to locate a pole of a function given the residue?

Question

It is straightforward to find a residue of a function, given the location of its poles, but can we go the other way? Given the value of the residue and a function, can we find the location of the poles? Do we need to know the number and order of the poles? Would this only work if we knew the function had one pole?

Answer 1

No, with a same function, knowing the value of the residue, you cannot deduce necessarily the location of the pole.

Here is a counterexample :

$$f(z)=\dfrac{z}{z^2-1}$$

has a same residue $1/2$ in its two poles $1$ and $-1$.

Remark: I have used formula $Res(f,p)=\dfrac{N(p)}{D'(p)}$ for a single pole $p$ of a rational function $f=\dfrac{N}{D}$.

Edit: Another facet :

Let us consider the family of rational functions $R(z)=N(z)/D(z)$ ($N,D$ being polynomials) with simple poles, residues $r_k$ appear like this:

$$R(z)=\sum_{k=1}^n \frac{r_k}{z-p_k}$$

For values of $z$ in a close vicinity of, say, $p_1$, fraction $R(z) \approx \frac{r_1}{z-p_1}$ (this term becomes dominant). Therefore, testing whether or not $(z-p_1)R(z) \to r_1$ when $z \to p_1$ could establish that $p_1$ is the pole associated with this residue.