Let $f(z)$ be a complex-valued meromorphic function.

If we say that $k$ is "the order of zero or pole" of $f(z)$ the the point $z=z_0$. What does this exactly mean?

As far as I understand it, if the function has no zeros or poles at $z_0$, we should say that $k=0$, and if it has a zero or order $r$, that $k=r$. Is this correct?

And if $f$ has a pole of order $s$, should it be $k=s$ or $k=-r$?0

Thanks in advance for any information

  • $\begingroup$ Can you indicate where you see that phrase used? I think it may depend a bit on the context. $\endgroup$ – Willie Wong Sep 14 '16 at 6:00
  • $\begingroup$ @WillieWong see the complex-analysis tag, and I don't think it has a different meaning in other context $\endgroup$ – reuns Sep 14 '16 at 7:57
  • $\begingroup$ Well, the phrases "order of zero" and "order of pole" are both entirely clear. But your question is whether the number $k$ being "order of zero or pole" should be set equal to the "order of pole" or its negative when you have a pole. For example, if I were to write something about the Laurent expansion of a function, I may say that the summation starts at "$n$" being the order of zero or pole of the function, and in this case obviously I would mean that negative of the usual "order of pole". But there may also be contexts where the text is written in such a way that the usual sense is used. $\endgroup$ – Willie Wong Sep 14 '16 at 13:35

Let $f\colon \mathbb{C}\to \mathbb{C}$ be a meromorphic function. Suppose $f$ has a pole at $z = a$. Then there exists a postive integer $m$ and an analytic function $g$ such that $g(a)\neq 0$ and $$ f(z) = \frac{g(z)}{(z - a)^m} $$ We say that $f$ has a pole of order $m$ at $a$.

The definition for the order of a zero is analagous. The reference is Conway's Functions of One Complex Variable I

  • $\begingroup$ I think you might need to add that $g$ is non zero at $a$. $\endgroup$ – Joel Cohen Sep 14 '16 at 6:36
  • $\begingroup$ @JoelCohen Thanks $\endgroup$ – user259242 Sep 14 '16 at 6:42

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