Looman-Menchoff theorem - Revision history

Vipul: 3 revisions

2008-05-18T19:16:21Z

3 revisions

← Older revision	Revision as of 19:16, 18 May 2008
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Vipul at 21:08, 18 April 2008

2008-04-18T21:08:14Z

← Older revision		Revision as of 21:08, 18 April 2008
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	Note: It is ''not'' true that if <math>f</math> is continuous at a particular point <math>z \in U</math> and has partial derivatives satisfying the Cauchy-Riemann differential equations, then it is complex-differentiable at that point.		Note: It is ''not'' true that if <math>f</math> is continuous at a particular point <math>z \in U</math> and has partial derivatives satisfying the Cauchy-Riemann differential equations, then it is complex-differentiable at that point.

			==Related facts==

			* [[Menchoff's theorem]]: A generalization, that only requires continuity and differentiability along two lines at every point, with the complex derivatives matching up.

Vipul at 20:50, 18 April 2008

2008-04-18T20:50:13Z

← Older revision		Revision as of 20:50, 18 April 2008
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	==Statement==		==Statement==

			===Verbal statement===

			If a function on an open subset of the complex numbers is [[function continuous at a point\|continuous at every point]] and [[function satisfying Cauchy-Riemann differential equations at a point\|satisfies Cauchy-Riemann differential equations at every point]], then it is holomorphic.

			===Symbolic statement===

	Let <math>U</math> be a [[domain]] in the complex numbers, and <math>f:U \to \mathbb{C}</math> be a continuous function, such that:		Let <math>U</math> be a [[domain]] in the complex numbers, and <math>f:U \to \mathbb{C}</math> be a continuous function, such that:

Vipul: New page: ==Statement== Let $U$ be a domain in the complex numbers, and $f:U \to \mathbb{C}$ be a continuous function, such that: * For any point $z \in U$ , th...

2008-04-14T01:09:14Z

New page: ==Statement== Let <math>U</math> be a domain in the complex numbers, and <math>f:U \to \mathbb{C}</math> be a continuous function, such that: * For any point <math>z \in U</math>, th...

New page

==Statement==

Let <math>U</math> be a [[domain]] in the complex numbers, and <math>f:U \to \mathbb{C}</math> be a continuous function, such that:

* For any point <math>z \in U</math>, the real and imaginary parts of <math>f</math> have well-defined partial derivatives in the <math>x</math> and <math>y</math> directions
* These partial derivatives satisfy the [[Cauchy-Riemann differential equations]]

Then, <math>f</math> is a [[holomorphic function]].

Note: It is ''not'' true that if <math>f</math> is continuous at a particular point <math>z \in U</math> and has partial derivatives satisfying the Cauchy-Riemann differential equations, then it is complex-differentiable at that point.

Looman-Menchoff theorem - Revision history

Vipul: 3 revisions

Vipul at 21:08, 18 April 2008

Vipul at 20:50, 18 April 2008

Vipul: New page: ==Statement== Let U be a domain in the complex numbers, and f:U \to \mathbb{C} be a continuous function, such that: * For any point z \in U, th...

Vipul: New page: ==Statement== Let $U$ be a domain in the complex numbers, and $f:U \to \mathbb{C}$ be a continuous function, such that: * For any point $z \in U$ , th...