True or False? (Complex Analysis)

[Ted123]

$S$ is a star-shaped open subset of $\mathbb{C}$, $f$ is a holomorphic function from $S$ to $\mathbb{C}$, $z_0$ is an element of $S$.

I've just come out an exam and wondered whether the following 2 statements are true or false:

1 Let $g$ be a holomorphic function on $S \subseteq \mathbb{C}$, with the exception of a pole of order $N$ at $z_0$. If the Laurent Series of $g$ around $z_0$ is

$\displaystyle \sum_{n=-N}^{\infty} a_n ( z - z_0 )^n$

for $z \in D'(z_0, R)$ for some $R>0$ (and $D(z_0 , R) \subseteq S$) and constants $a_n \in \mathbb{C}$, then the residue of $g$ at $z_0$ is given by $a_{-1}$.

2 Suppose $S = D(z_0, R)$ for some $R>0$ and

$\displaystyle f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n$

for all $z \in S$ and some constants $a_n \in \mathbb{C}$. Then necessarily $a_0 = f(z_0)$ and $a_n = f^{(n)}(z_0)$ for all $n\geq 1$.

 

[Sethric]

1. How are the coefficients of the Laurent Series defined? (hint: it is related to the Cauchy Integral Formula)

2. Your function is holomorphic on S. So it is equal to its Taylor series on any point in its domain. What should the coefficients of a Taylor series centered at z_0 be?