I'm wondering if there's a function (mapping into the natural numbers) that computes the order of a pole of a meromorphic function ?
Put a little different, how does mathematical software finds this number? By looking at the exponents of the Lauent series? Or by approximation and then rounding?
At the end, what i would like to have, if possible, is a function similar to that computing the winding number at a point, where one does not have to "look" at the exponent to know the answer, if you see what I mean.
Thanks for your answers and thoughts.
You can use the argument principle to compute the orders of zeroes and poles. This is done, for example, to verify the Riemann hypothesis computationally.