MathLiveLiouville's theorem (bounded entire functions)
Every entire function (holomorphic on the whole complex plane) that is bounded must be constant, so its derivative is zero. It is a standard tool in complex analysis for proving results such as the fundamental theorem of algebra.
\forall f\in\mathcal O(\mathbb C,\mathbb C)\colon\left(\sup_{z\in\mathbb C}|f(z)|<\infty\implies f'=0\right)What the symbols mean
| an entire function from the complex plane to the complex numbers | |
| the set of holomorphic (entire) functions from the complex plane to the complex numbers | |
| a complex number in the domain of f | |
| the derivative of f | |
| the supremum of the absolute value of f over the whole complex plane |
About this formula
- FieldMathematics
- TopicComplex analysis
- LevelUniversity
- Also calledLiouville's boundedness theorem, bounded entire function is constant, entire function bounded implies constant, Liouville theorem complex analysis
- SourceWikidata Q576478
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