Complex-analytic implies holomorphic: Difference between revisions

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(New page: ==Statement== Suppose <math>U \subset \mathbb{C}</math> is a domain and <math>f:U \to \mathbb{C}</math> is a complex-analytic function: for every point <math>z_0 \in U</math>, there e...)
 
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{{basic fact}}
==Statement==
==Statement==


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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.
==Facts used==
# [[uses::Power series is infinitely differentiable in disk of convergence]]


==Proof==
==Proof==


To execute this proof, we need to argue that
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).