Let $G\subset\mathbb{C}$ a domain, $f:G\to\mathbb{C}$ holomorphic, $z_0\in G$ and $f(z_0)=0$. Then $f^{(k)}(z_0)=0 \, \forall k\in\mathbb{N}_0$ or $\exists k>0$, an open surrounding $U=U(z_0)\subset G$ and a holomorphic function $g:U\to\mathbb{C}$, so that:
- $f(z)=(z-z_0)^k\cdot g(z)$ for $z\in U$.
- $g(z_0)\neq 0$
In the lecture we called this Thereom of local representation. Unfortunately I couldn't find anything about that. Do you know the common title? If so, do you know some references? Thank you!
I don't know of a name, but it is a simple calculation. Being analytic, $f$ is given by a power series in some neighborhood $U$ of $z_0$:
$$ f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^k $$
If the coeffiicents are not all zero, then if we set $k$ to be the index of the first nonzero $a_n$, then we can factor:
$$ f(z) = (z - z_0)^k \sum_{n=0}^{\infty} a_{n+k} (z - z_0)^n $$
and set
$$ g(z) = \sum_{n=0}^{\infty} a_{n+k} (z - z_0)^n $$
In fact, we can conclude something stronger: we can define $g$ on all of $G$ by setting
$$ g(z) = \begin{cases} (z - z_0)^{-k} f(z) & z \neq z_0 \\ a_k & z = z_0 \end{cases} $$