Complex-analytic implies holomorphic: Difference between revisions

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Then, <math>f</math> is a [[holomorphic function]]: it is complex-differentiable, and the complex differential is a continuous function. In fact, <math>f</math> is differentiable infinitely often.
Then, <math>f</math> is a [[holomorphic function]]: it is complex-differentiable, and the complex differential is a continuous function. In fact, <math>f</math> is differentiable infinitely often.


==Proof==
==Facts used==


We will show the following somewhat stronger fact: if <math>f</math> is a complex-analytic function at <math>z_0</math> with a power series:
# [[uses::Power series is infinitely differentiable in disk of convergence]]


<math>f(z) := \sum_{n=0}^\infty a_n(z - z_0)^n</math>
==Proof==
 
By basic facts of power series, <math>f</math> converges absolutely, and uniformly on every disc of radius strictly smaller than its radius of convergence. We'll show that <math>f'</math> exists on the open disc of the radius of convergence, and is given by the following absolutely convergent power series:
 
<math>f'(z) := \sum_{n=0}^\infty na_n(z - z_0)^{n-1}</math>


Note that the power series on the right clearly has the same radius of convergence as the power series for <math>f</math>, so we can concentrate on showing that it equals <math>f</math>. Observe that if <math>s_n</math> is the partial sum till the <math>n^{th}</math> term of <math>f</math>, then <math>s_n'(z)</math> indeed equals the sum, till the <math>n^{th}</math> term, for <math>f'(z)</math>. Thus, what we really need to show is that the error term goes to zero.
The proof follows directly from fact (1).

Latest revision as of 16:08, 12 September 2008

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Suppose is a domain and is a complex-analytic function: for every point , there exists a real number and a power series such that the power series converges and agrees with in the ball of radius .

Then, is a holomorphic function: it is complex-differentiable, and the complex differential is a continuous function. In fact, is differentiable infinitely often.

Facts used

  1. Power series is infinitely differentiable in disk of convergence

Proof

The proof follows directly from fact (1).