Looman-Menchoff theorem: Difference between revisions
(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...) |
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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: | ||
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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. | |||
Latest revision as of 19:16, 18 May 2008
Statement
Verbal statement
If a function on an open subset of the complex numbers is continuous at every point and satisfies Cauchy-Riemann differential equations at every point, then it is holomorphic.
Symbolic statement
Let be a domain in the complex numbers, and be a continuous function, such that:
- For any point , the real and imaginary parts of have well-defined partial derivatives in the and directions
- These partial derivatives satisfy the Cauchy-Riemann differential equations
Then, is a holomorphic function.
Note: It is not true that if is continuous at a particular point 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.