Laurent series for f(z) = 1/(exp(z)-1)^2 ?

[quasar_4]

Laurent series for f(z) = 1/(exp(z)-1)^2 ??

Homework Statement

Determine the Laurent series and residue for f(z) = \frac{1}{(e^{z} - 1)^{2}}.

Homework Equations

We know that the Taylor series expansion of e^{z} is = 1 + z + (z^2)/2! + ...

The Attempt at a Solution

I am soooo confused. It's almost like a geometric series, except the denominator is squared. I know I need to use the Taylor series expansion, but I don't know where. Do I just invert the Taylor series expansion? How do I deal with the fact that it's squared?

I almost thought of writing it into the geometric series anyway, then squaring the terms (obviously this isn't really accurate, as I'd be missing tons of cross-terms, but if we only need a few terms of the series anyways...). Can anyone help to explain what's going on?

I understood the example in our book (there's only 1 example ) but it had two different poles, and I was able to expand the Taylor series around the "other" pole for each Laurent series. Here our only pole is 0 (though of order 2) and I don't know how to proceed.
Any help would be so appreciated!

 

[Dick]

You expand around z=0. exp(z)-1=z+z^2/2!+...=z(1+z/2!+...). Write that as z(1+a) where a=z/2!+z^2/3!+... So 1 over that squared is (1/z^2)*(1/(1+a)^2). The series expansion for 1/(1+a)^2 is 1-2a+3a^2+... As you said, you only need the first few terms. Now start throwing away terms that you know won't contribute to the terms you need. E.g. a^2 starts with a z^2 term.

 

[quasar_4]

oh,oh, that is SOOO Nice!

IT WORKED!

I think this has helped me a LOT. You use the Taylor series to simplify your expression into a form that you can expand in geometric/other series, because it's a lot simpler than doing the nasty line integral for the coefficients? Basically we just put this into a different looking form, and then it all stood out.

I even got the same answer as Maple, up to the first 5 terms!

Thank you Dick!