Function holomorphic at a point: Difference between revisions

Revision as of 21:19, 16 April 2008

This article defines a property that can be evaluated for a function on a (particular kind of) set, and a point in that set. A function satisfying the property at every point, it is termed a holomorphic function
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Definition

In one dimension

Let $U$ be an open subset (without loss of generality, an open connected subset, i.e. a domain) in $\mathbb {C}$ . Let $f:U\to \mathbb {C}$ be a function, and let $z_{0}\in U$ . We say that $f$ is holomorphic at $z_{0}$ if there exists an open subset $V\ni z_{0}$ such that $f$ is complex-differentiable for every $z\in V$ , and further, if the complex differential $f'$ is a continuous function on $V$ .

It turns out that this is equivalent to a function complex-analytic at a point.

Revision as of 21:13, 16 April 2008 (view source) Vipul (talk \| contribs) (New page: {{at-point function property\|holomorphic function}} ==Definition== ===In one dimension=== Let <math>U</math> be an open subset (without loss of generality, an open connected subset, i.e...)		Revision as of 21:19, 16 April 2008 (view source) Vipul (talk \| contribs) No edit summary Newer edit →
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	Let <math>U</math> be an open subset (without loss of generality, an open connected subset, i.e. a [[domain]]) in <math>\mathbb{C}</math>. Let <math>f:U \to \mathbb{C}</math> be a function, and let <math>z_0 \in U</math>. We say that <math>f</math> is '''holomorphic''' at <math>z_0</math> if there exists an open subset <math>V \ni z_0</math> such that <math>f</math> is [[function complex-differentiable at a point\|complex-differentiable]] for every <math>z \in V</math>, and further, if the complex differential <math>f'</math> is a continuous function on <math>V</math>.		Let <math>U</math> be an open subset (without loss of generality, an open connected subset, i.e. a [[domain]]) in <math>\mathbb{C}</math>. Let <math>f:U \to \mathbb{C}</math> be a function, and let <math>z_0 \in U</math>. We say that <math>f</math> is '''holomorphic''' at <math>z_0</math> if there exists an open subset <math>V \ni z_0</math> such that <math>f</math> is [[function complex-differentiable at a point\|complex-differentiable]] for every <math>z \in V</math>, and further, if the complex differential <math>f'</math> is a continuous function on <math>V</math>.

			It turns out that this is equivalent to a [[function complex-analytic at a point]].