existence of power series

In this entry we shall demonstrate the logical equivalence of the holomorphic and analytic concepts. As is the case with so many basic results in complex analysis, the proof of these facts hinges on the Cauchy integral theorem, and the Cauchy integral formula.

Holomorphic implies analytic.

Theorem 1

Let $U\subset\mathbb{C}$ be an open domain that contains the origin, and let $f:U\rightarrow\mathbb{C},$ be a function such that the complex derivative

f^{\prime}(z)=\lim_{\zeta\rightarrow 0}\frac{f(z+\zeta)-f(z)}{\zeta}

exists for all $z\in U$ . Then, there exists a power series representation

f(z)=\sum_{k=0}^{\infty}a_{k}z^{k},\quad\|z\|<R,\quad a_{k}\in\mathbb{C}

for a sufficiently small radius of convergence $R>0$ .

Note: it is just as easy to show the existence of a power series representation around every basepoint in $z_{0}\in U$ ; one need only consider the holomorphic function $f(z-z_{0})$ .

Proof. Choose an $R>0$ sufficiently small so that the disk $\|z\|\leq R$ is contained in $U$ . By the Cauchy integral formula we have that

f(z)=\frac{1}{2\pi i}\oint_{\|\zeta\|=R}\frac{f(\zeta)}{\zeta-z}\,d\zeta,\quad% \|z\|<R,

where, as usual, the integration contour is oriented counterclockwise. For every $\zeta$ of modulus $R$ , we can expand the integrand as a geometric power series in $z$ , namely

\frac{f(\zeta)}{\zeta-z}=\frac{f(\zeta)/\zeta}{1-z/\zeta}=\sum_{k=0}^{\infty}% \,\frac{f(\zeta)}{\zeta^{k+1}}\,z^{k},\quad\|z\|<R.

The circle of radius $R$ is a compact set; hence $f(\zeta)$ is bounded on it; and hence, the power series above converges uniformly with respect to $\zeta$ . Consequently, the order of the infinite summation and the integration operations can be interchanged. Hence,

f(z)=\sum_{k=0}^{\infty}a_{k}z^{k},\quad\|z\|<R,

where

a_{k}=\frac{1}{2\pi i}\oint_{\|\zeta\|=R}\frac{f(\zeta)}{\zeta^{k+1}},

as desired. QED

Analytic implies holomorphic.

Theorem 2

Let

f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},\quad a_{n}\in\mathbb{C},\quad\|z\|<\epsilon

be a power series, converging in $D=D_{\epsilon}(0)$ , the open disk of radius $\epsilon>0$ about the origin. Then the complex derivative

f^{\prime}(z)=\lim_{\zeta\rightarrow 0}\frac{f(z+\zeta)-f(z)}{\zeta}

exists for all $z\in D$ , i.e. the function $f:D\rightarrow\mathbb{C}$ is holomorphic.

Note: this theorem generalizes immediately to shifted power series in $z-z_{0},\;z_{0}\in\mathbb{C}$ .

Proof. For every $z_{0}\in D$ , the function $f(z)$ can be recast as a power series centered at $z_{0}$ . Hence, without loss of generality it suffices to prove the theorem for $z=0$ . The power series

\sum_{n=0}^{\infty}a_{n+1}\zeta^{n},\quad\zeta\in D

converges, and equals $(f(\zeta)-f(0))/\zeta$ for $\zeta\neq 0$ . Consequently, the complex derivative $f^{\prime}(0)$ exists; indeed it is equal to $a_{1}$ . QED

Title	existence of power series
Canonical name	ExistenceOfPowerSeries
Date of creation	2013-03-22 12:56:27
Last modified on	2013-03-22 12:56:27
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	5
Author	rmilson (146)
Entry type	Result
Classification	msc 30B10