Morera's Theorem states that a continuous complex-valued function on an open subset of the complex plane is holomorphic there provided its integral around the boundary of every closed triangle contained in that open set vanishes. Named for Giacinto Morera, it is essentially the converse of the Cauchy Integral Theorem and gives a convenient way to establish holomorphy from an integral condition without directly checking complex differentiability.
Facts
StatementA continuous, complex-valued function f defined on an open set D in the complex plane that satisfies the closed contour integral of f(z) dz equal to zero for every closed piecewise C1 curve in D must be holomorphic on D. 1 Classification
Statement FormCharacterization Theorem 1 Sources
1. Morera's theorem, Wikipedia
Statement section
a continuous, complex-valued function f defined on an open set D in the complex plane that satisfies the closed contour integral condition for every closed piecewise C1 curve gamma in D must be holomorphic on D.
Publication history section
Giacinto Morera published his foundational work titled Un teorema fondamentale nella teorica delle funzioni di una variabile complessa in the Rendiconti del Reale Instituto Lombardo di Scienze e Lettere in 1886.
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