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Good day,

I am looking for a help.

I should show that function $f(z)=(z^2+3z)/(e^z-1)$ can be expressed as a power series in some punctured disk around the origin. i.e. for some $r>0$ one has

$f(z)=\sum_{0}^{\infty}a_{k}z^{k}$ for $0<|z|<r$.

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Thank you a lot for any help.

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Here is a proof that works for the current example:

Note that $\lim_{z \to 0} f(z) = 3$. Define $g(z) = f(z)$ for $z \neq 0$ close to zero and $g(0) = 3$. It is clear that $g$ is analytic for $z \neq 0$ close to zero

and only zero is in question.

All we need to do is show that $g$ is differentiable at $z=0$.

We have ${g(z)-g(0) \over z} = { {z^2 + 3z \over e^z -1} – 3 \over z} = {z^2 + 3z -3 e^z +3\over z(e^z -1)} = {z^2-3 {z^2 \over 2!}-3{z^3 \over 3!}-\cdots \over z^2+{z^3 \over 2!} + \cdots} $, and we see that

$\lim_{z \to 0} {g(z)-g(0) \over z} =-{1 \over 2}$, hence

$g$ is differentiable at $z=0$ and $g'(0) = -{1 \over 2}$.

It follows that $g$ is analytic in a neighbourhood of $z=0$ hence it

has a power series expansion. Since $f(z) = g(z)$ for $z \neq 0$, it

follows that the same series is valid for $f$ in the corresponding

punctured neighbourhood.

This can always be done as long as the function is analytic in some punctured disc about the origin. In this case the singularities occur when $e^z-1=0$. So

- Solve this equation.
- Plot the solutions on a diagram (not essential but helpful).
- Find a disc about the origin which contains none of the solutions from step (1), except for the solution $z=0$.

Good luck!

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