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Is there a systematic way to determine the order of a pole of a function? For instance, I want to calculate the residues of $\frac{1}{\sin z}$, for the poles at $k\pi$.

If the poles are simple, and apparently they are, it's easy to see the residues to be $(-1)^k$. However, how can one tell the poles are simple?

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2 Answers 2

You are asking if the zeros of $\sin z$ are simple. It suffices to check that at each zero $z_0$, the derivative $\cos z_0 \neq 0$, which is easy to check. (Reason: look at the Taylor expansion around $z_0$. If it is not a simple zero, the derivative would be 0 at $z_0$)

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So in general, is the order of the pole the same as the order of the zero of the denominator? –  Cye Mar 21 '12 at 7:48
    
Yes, by definition of order of pole/zero. –  user27126 Mar 21 '12 at 7:49
    
@Cye I am sure you know that, you should just think it over... –  AD. Mar 21 '12 at 7:50
    
Guess I had a bad misunderstanding! Thanks both. –  Cye Mar 21 '12 at 7:51
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To be bit picky: In general it's not the order of the zero of the denominator but the order of the zero of the reciprocal $1/f$. –  Dirk Mar 21 '12 at 7:51

Hint

  1. A pole of $f$ is a zero of $1/f$ and vice verse.

  2. What can you say about $\sin z/z$ as $z\to0$?

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