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. | ||
== | ==Facts used== | ||
# [[uses::Power series is infinitely differentiable in disk of convergence]] | |||
==Proof== | |||
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
Proof
The proof follows directly from fact (1).