Given a function $f(x)$ that you know has a pole in $x_0$, how do you find the order of that pole? Is there some generic method for doing this, or does the method you would use to determine the order of the pole vary from case to case?
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Is "use the argument principle" a satisfactory answer? Note that, generally, we have to know/find a contour containing the pole at $x_0$ and excluding all other poles and zeroes of the function, a step that isn't always mechanical (in the cases where it's possible). Does your $f$ have a derivative along that contour? (Not all functions are meromorphic everywhere.) We need that derivative for the argument principle. Of course, there are other methods, like finding its Laurent series (if it has one). – Eric Towers Oct 18 '21 at 23:49
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@EricTowers Oh, so $\lim_{x\to x_0} \frac{(x-x_0)f'(x)}{f(x)}$ gives the order of the pole at $x_0$? Very clever approach! Thanks! – HelloGoodbye Oct 20 '21 at 02:02