Holomorphic implies complex-analytic - Revision history

Vipul: /* Facts used */

2008-09-12T16:12:50Z

Facts used

← Older revision		Revision as of 16:12, 12 September 2008
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	==Facts used==		==Facts used==

	# [[Holomorphic function admits power series ~~about center for any disk in domain~~]]		# [[Holomorphic function on open disk admits globally convergent power series]]

	==Converse==		==Converse==

Vipul at 16:11, 12 September 2008

2008-09-12T16:11:22Z

← Older revision		Revision as of 16:11, 12 September 2008
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	==Facts used==		==Facts used==

	~~===Cauchy integral formula===~~		# [[Holomorphic function admits power series about center for any disk in domain]]
	~~{{further\|~~[[~~Cauchy integral formula~~]]}}

	~~We use here the version for a circle and its center. The general version, involving winding numbers, is not necessary for this proof.~~

	==Converse==		==Converse==

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	==Proof==		==Proof==

	~~{{fillin}}~~		The proof follows directly from fact (1).

Vipul: 1 revision

2008-05-18T19:13:32Z

1 revision

← Older revision	Revision as of 19:13, 18 May 2008
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Vipul: New page: {{definition equivalence|holomorphic function}} ==Statement== Suppose $U$ is a domain in $\mathbb{C}$ , and $f: U \to \mathbb{C}$ is holomorphic: it i...

2008-04-14T00:52:09Z

New page: {{definition equivalence|holomorphic function}} ==Statement== Suppose <math>U</math> is a domain in <math>\mathbb{C}</math>, and <math>f: U \to \mathbb{C}</math> is holomorphic: it i...

New page

{{definition equivalence|holomorphic function}}

==Statement==

Suppose <math>U</math> is a [[domain]] in <math>\mathbb{C}</math>, and <math>f: U \to \mathbb{C}</math> is holomorphic: it is complex-differentiable, and its differential is a continuous function. Then, <math>f</math> is a complex-analytic function: for every point in <math>U</math>, we can find a disc about that point, and a power series that agrees with <math>f</math> on that disc.

In fact, something stronger is true: given any point <math>z_0 \in U</math> and any disc of radius <math>r</math> about <math>z_0</math> contained completely inside <math>U</math>, we can find a power series that converges on that disc, and agrees with <math>f</math> on the disc. In other words, we can make the radius of convergence of the power series as large as we can make a disc completely contained inside the open subset.

==Definitions used==

==Facts used==

===Cauchy integral formula===
{{further|[[Cauchy integral formula]]}}

We use here the version for a circle and its center. The general version, involving winding numbers, is not necessary for this proof.

==Converse==

The converse is also true. {{further|[[Complex-analytic implies holomorphic]]}}

==Proof==

{{fillin}}

Holomorphic implies complex-analytic - Revision history

Vipul: /* Facts used */

Vipul at 16:11, 12 September 2008

Vipul: 1 revision

Vipul: New page: {{definition equivalence|holomorphic function}} ==Statement== Suppose U is a domain in \mathbb{C}, and f: U \to \mathbb{C} is holomorphic: it i...

Vipul: New page: {{definition equivalence|holomorphic function}} ==Statement== Suppose $U$ is a domain in $\mathbb{C}$ , and $f: U \to \mathbb{C}$ is holomorphic: it i...