I am a bit confused about laurent series. I know the definitions where the coefficients are expressed as integrals.(adsbygoogle = window.adsbygoogle || []).push({});

However, I am confused about how to actually find the laurent series in practice, for analytic functions.

The information I can find online is just terrible, some of them do solve them but they don't use integration at all, they all seem to have some different but similar tricks to find the series and none of them actually explain the steps.

For example a problem I am looking at says to find all taylor and Laurent expansion about the origin, of the function:

[tex]\frac{1}{z^{2}+1}[/tex]

There are two poles at [itex]z=\pm i[/itex]

So by my limited understanding of how to expand this function, I think the idea is to just do a taylor series aboutz=0, which is only valid in the region|z|<1

Then I think the next step is to do a Laurent series which is valid in the region|z|>1

but I am unsure how to do this.

Some sources I found seem to suggest expanding about the poles but which one do I pick?

Just to be clear I am not really asking for a solution to the function I posed above, what I would like is an understanding of how to find laurent expansions of functions such as the one I wrote above. Nothing too complicated, this doesn't seem like too complex of a subject, but I just can't find any decent coherent source explaining how to perform the expansion online.

So thank you for any help or general advice on performing laurent expansions you can give me.

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# Confused about Laurent series of analytic functions

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