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?