Complex Analysis Ⅲ
1.Power Series
Definition 1.1
In general, a power series is the expansion of where .
Next, we will give a theorem on the convergence of the power series.
Theorem 1.2 (Cauchy-Hadmard Theorem)
Given a power series , there exists a such that (1) when , the series converges absolutely; (2) when , the series diverges.If we use the convention , .
Proof: When , . So for all , there exists infinitely many . So which means the series diverges.
When , . So for all , when . Hence . Since , the series converges absolutely.
Otherwise, when , choose a small such that . Since , there exists infinitely many . So . Since converges, the series converges absolutely. When , for any , there exists . If , and hence the series diverges.
Q.E.D.
2.Analytic Function
Theorem 2.1
Let and . and have the same convergence disc. is holomorphic on its convergence disc and .
Proof: Because , since . This shows that and have the same convergence disc.
Let where and . Hence,
Obviously, there exists a and a such that and when .
Next,
Assume that and . So (1) is less than . Since converges when , there exists a such that (1) is less than . Therefore, . So .
Q.E.D.
Corollary 2.2
A power series is infinitely differentiable on its convergence disc.Definition 2.3
A function is analytic at if there is a power series centered at with positive radius of convergence and for all in a neighbourhood of . If is analytic at all , then is analytic on .
From the definition, we have:
Fact 2.4
If a function is analytic, then it is holomorphic.
In fact, the converse of Fact 2.4 is also true. So analytic and holomorphic are equivalent.
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