Morera's Theorem and Uniform Convergence of Holomorphic Functions
Автор: Mike, the Mathematician
Загружено: 2025-03-28
Просмотров: 546
Описание:
We have already proved the Cauchy-Goursat Theorem, which states that a holomorphic function will integrate to zero along any triangle ( • Goursat's Theorem ). We now prove a converse of this theorem; namely, if there is a continuous function defined on a region in the complex plane and it integrates to zero over very triangle, then the function is holomorphic in that region! While the proof of this theorem follows directly from previous work on the existence of primitives, it is still a very powerful theorem! We will also show that a sequence of holomorphic functions which converges uniformly on compact subsets of the of a region converges to a holomorphic function!
#mikethemathematician, #mikedabkowski, #profdabkowski, #complexanalysis, #morera
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