Poles (and order of) of $(\cosh(1/(z-\pi)))^2$ and Residue at $z=\pi$

Asked Mar 28 '14 at 18:33

Active Mar 29 '14 at 00:14

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Hi I need help finding the poles and the order of the poles of the following function:

$$\left(\cosh\frac1{z-\pi}\right)^2$$

and the residue at $z=\pi$.

I have tried a number of different methods including putting the function in exponential form but nothing seems to work, if someone could give some insight that would be great.

edited Mar 29 '14 at 00:14

Christoph

24,912

asked Mar 28 '14 at 18:33

Charles

Which exponential form did you get? It's not too hard to find the result in the right form. – Daniel Fischer Mar 28 '14 at 18:39
1

Is it $$\left(\cosh\left(\frac{1}{z-\pi}\right)\right)^2$$ or $$\cosh \left(\frac{1}{(z-\pi)^2}\right),$$ by the way? – Daniel Fischer Mar 28 '14 at 18:41
See related problems and techniques. – Mhenni Benghorbal Mar 28 '14 at 20:03
Look at each term in the sum individually. What are the singularities of $\exp\left(\frac{2}{z-\pi}\right)$? – Daniel Fischer Mar 28 '14 at 23:54
Show $z=\pi$ is an essential singularity. Does your textbook discuss residues at essential singularities? – GEdgar Mar 28 '14 at 23:59
@user138730 Why are you editing Charles question to reflect what you have tried? – Christoph Mar 29 '14 at 00:04
@user138730 Not being able to comment due to lack of reputation doesn't justify changing another persons question to reflect your own thoughts. Post an answer stating what you think could help to answer the question. That way you gain reputation and will soon be able to comment on questions. – Christoph Mar 29 '14 at 00:12

Poles (and order of) of $(\cosh(1/(z-\pi)))^2$ and Residue at $z=\pi$

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