Laurent expansion of cot(z)

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The Sleeping Tyrant
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Laurent expansion of cot(z)

Postby The Sleeping Tyrant » Fri Apr 23, 2010 4:53 am UTC

This week I had an exam for a class on complex numbers and functions and whatnot (not exactly a complex analysis course, so I'm not really sure what to call it). On it, the professor asked us to derive the Laurent expansion of [imath]cot(z)[/imath], expanded about [imath]z_0 = 0[/imath]. Completely stumped me.

Now, I've managed to find the expansion online, but not the derivation, and the expansion seems fairly unobvious. How is it derived?

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t0rajir0u
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Re: Laurent expansion of cot(z)

Postby t0rajir0u » Fri Apr 23, 2010 5:17 am UTC

Write cot(z) in terms of e^{iz} and try to relate it to the generating function for the Bernoulli numbers. Are you sure the professor didn't just want you to write down a few terms?

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Re: Laurent expansion of cot(z)

Postby The Sleeping Tyrant » Fri Apr 23, 2010 6:11 am UTC

t0rajir0u wrote:Write cot(z) in terms of e^{iz} and try to relate it to the generating function for the Bernoulli numbers. Are you sure the professor didn't just want you to write down a few terms?


I'm actually pretty sure he didn't want anyone to get it. As far as I know, he didn't ever mention Bernoulli numbers or do an example like this. He didn't exactly go very deep into the material.

But thanks, that makes sense.

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NathanielJ
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Re: Laurent expansion of cot(z)

Postby NathanielJ » Fri Apr 23, 2010 2:24 pm UTC

Do you know the Taylor series of sin and cos centered at z = 0? If so, you can do polynomial long division to derive the Laurent series you want, though it would be quite difficult I imagine (maybe not too bad if you were only expected to get a couple of terms in the series).
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