- #1

FeDeX_LaTeX

Gold Member

- 437

- 13

**Homework Statement**

Determine the Laurent series expansion of

[tex]\frac{1}{e^z - 1}[/tex]

**The attempt at a solution**

I've spotted that

[tex]\frac{1}{e^z - 1} = \frac{1}{2}\left( \coth{\frac{z}{2}} - 1\right)[/tex]

but I don't know what to do next. WolframAlpha gives the series centred at 0 as:

[tex]\frac{1}{z} -\frac{1}{2} + \frac{z}{12} - \frac{z^3}{720} + \frac{z^5}{30240} + ...[/tex]

but I don't know how they arrived at this. How are they evaluating f(0), f'(0), etc.? I'm getting an undefined answer for f(0) and f'(0) too.

I'm defining f(z) as

[tex]f(z) = \frac{1}{2}\left( \coth\frac{z}{2} - 1 \right)[/tex]

Any help?