# Math Help - singularities and poles

1. ## singularities and poles

http://www.math.ualberta.ca/~runde/files/ass411-6.pdf

If I could get some help on #1 it would be much appreciated as I'm not too sure on how to start
(unless you think the equation in the second part is a typo and it should be $\frac{g(z)}{(z-z_0)^n}$)
in which case I'll have another look since it would resemble my notes closer

2. $1\implies 2$*. Since $f$ is analytic on $D$ it means in the neighorhood of $z_0$ we have that $f(z) = \sum_{k\geq 0} \frac{f^{(k)}(z_0)}{k!}(z-z_0)^k$.
However, $f^{(k)}(z_0)=0$ for $0\leq k < n$. Thus, $f(z) = \sum_{k\geq n} \frac{f^{(k)}(z_0)}{k!}(z-z_0)^k = (z-z_0)^n \sum_{k\geq 0} \frac{f^{(n-k)}(z_0)}{(n-k)!}(z-z_0)^k$.
Therefore, $f(z) = (z-z_0)^n g(z)$ where $g(z) = \sum_{k\geq 0} \frac{f^{(n-k)}(z_0)}{(n-k)!}(z-z_0)^k$ and so $g(z_0) \not = 0$

*)There appears to be a mistake it should say $f^{(k)}(z_0) = 0$ for $k=0,1,2,...,n-1$.