# Complex Analysis. Showing absolute convergence of principal part of Laurent series for a holomorphic function.

I am given that $$f: D(P, r) \setminus \{ P \} \to \mathbb{C}$$ is holomorphic. I want to show that the principal part of the Laurent series expansion for $$f$$ converges absolutely on $$\mathbb{C} \setminus \{P\}$$. Confused on how to proceed.

I know that the convergence is absolute on $$D(P, r) \setminus \{ P \}$$. I don't see how to extend the principal part to $$\mathbb{C}\setminus\{P\}$$.

## 1 Answer

The principal part gets better as you get farther from the pole, since it has only negative powers of $$(z - P)$$. Apply a comparison test, using the fact that the series is convergent at some point $$z \ne P$$.

In particular, you can use the fact that

$$\left|\frac{a_n}{(z - P)^n}\right| \le \frac{|a_n|}{r^n}$$

for $$z \notin \mathbb{D}(P, r)$$.

• Thank you. I will try this out. – user623740 Dec 7 '18 at 0:36
• So basically take the limit of the right-hand side as $r \to \infty$? – user623740 Dec 7 '18 at 1:33
• No, because $r$ is just some fixed number. – user296602 Dec 7 '18 at 1:34