The Residue Theorem

Maximoff · 2015-10-25 20:05:37

Hello everyone. Could anybody help me with this? Its about residue theorem.

Find the residue at z=i of the function

Thanks in advance.

bobbym · 2015-10-26 18:23:32

Hi;

is what I am getting.

Maximoff · 2015-10-26 22:09:52

Hi bobbym..

Could you show the working step on how you get it? I got a less bit of knowledge on this theorem.
Thanks.

bobbym · 2015-10-27 00:19:21

I used a CAS on that.

What methods have you been taught for simple poles?

zetafunc · 2015-10-27 05:07:39

Firstly, note that

so there is indeed a pole at z = i, and it has multiplicity two. Recall that, if a is a pole of multiplicity n, then the residue of f around the pole z = a is given by the following formula:

Notice that for a simple pole (i.e. n = 1) we can drop the differential operator. However, in this case, the pole is not simple.

Now we just put n = 2, and a = i. We obtain the following:

where the last equality is just algebra, from plugging in z = i.

zetafunc · 2015-10-27 05:13:23

You also asked a question on the LearnMathsFree Facebook page, which I will respond to here:

To deal with , we can first multiply the top and bottom by , to get:

, which you can then turn into the form z = x + iy by multiplying by the conjugate of the denominator in the usual way.

Math Is Fun Forum

#1 2015-10-25 20:05:37

The Residue Theorem

#2 2015-10-26 18:23:32

Re: The Residue Theorem

#3 2015-10-26 22:09:52

Re: The Residue Theorem

#4 2015-10-27 00:19:21

Re: The Residue Theorem

#5 2015-10-27 05:07:39

Re: The Residue Theorem

#6 2015-10-27 05:13:23

Re: The Residue Theorem

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