Let $P(z)= \Sigma^{\infty}_{n=0} a_{n}(z-z_{0})^{n}$ be a complex power series centered at $z_{0} \in \mathbb{C}$ with radius of convergence $\rho > 0$. Then for every $r< \rho$ show directly, without using Cauchy's theorem, that
$\int_{\partial D_{r}(z_{0})} P(z) dz = 0$,
where $D_{r}(z_{0})$ is the open disc of radius $0 \leq r \leq +\infty$, centered at $z_{0} \in \mathbb{C}$.
I'm not really sure how to prove this without using Cauchy's theorem and I'm stuck on how I should even start this. Any help would be appreciated!
Edit: I know $P(z)$ is analytic so should I be using something like this? Term-by-term integration of complex power series
Best Answer
Let $Q(z)=\sum_{n=0}^\infty\frac{a_n}{n+1}(z-z_0)^{n+1}$. Then $Q$ is an antiderivative of $P$. Since $P$ has an antiderivative, its integral along any closed path is $0$.