I have just recently started revising for my complex analysis module at university and come across and interesting exercise in a textbook while reading. I vaguely understand the concept but am struggling with extending a function. This is the question:
Consider the following complex differentiable function: $$f(z)=\frac{z}{e^z-1}, 0<|z|<2\pi $$ (a) show that there exists a complex differentiable extension of $f$ onto the set $D=\{{z\in\mathbb{C} : |z|<2\pi\}}$. Denote this extension by $g$.
This is the part I am struggling with primarily as I feel I could do the other parts once I have this but really don't understand how to extend the function to include zero, all I can see is that $f(0)$ must be $1$.
(b) The power series expansion for $f$ around $0\in\mathbb{C}$ can be written as: $$f(z)=\sum^\infty_{n=0}\frac{B_n}{n!}z^n $$ The coefficients $B_n$ are called Bernoulli numbers. Determine $B_0$ and $B_1$.
(c) Show that $$limsup_{n\to\infty}|B_n|^\frac{1}{n}=\infty$$
I have never come across Bernoulli numbers before this question and have looked at some more material on them and can see how to do part (b) once part (a) is complete. But I haven't found much material on any functions on how to extend them to include undefined points differentiable. Also I cannot really see how part (c) would follow from (b) or (a). Please help! Any hints or solutions are greatly appreciated.