Morera'S Theorem

(Theory)

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What is Morera's theorem?

In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic. Morera's theorem states that a continuous, complex-valued function f defined on an open set D in the complex plane that satisfies for every closed piecewise C1 curve in D must be holomorphic on D. The assumption of Morera's theorem is equivalent to f having an antiderivative on D. The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. The converse does hold e.g. if the domain is simply connected; this is Cauchy's integral theorem, stating that the line integral of a holomorphic function along a closed curve is zero. The standard counterexample is the function f(z) = 1/z, which is holomorphic on ℂ − {0}. On any simply connected neighborhood U in ℂ − {0}, 1/z has an antiderivative defined by L(z) = ln(r) + iθ, where z = reiθ. Because of the ambiguity of θ up to the addition of any integer multiple of 2π, any continuous choice of θ on U will suffice to define an antiderivative of 1/z on U. (It is the fact that θ cannot be defined continuously on a simple closed curve containing the origin in its interior that is the root of why 1/z has no antiderivative on its entire domain ℂ − {0}.) And because the derivative of an additive constant is 0, any constant may be added to the antiderivative and it's still an antiderivative of 1/z. In a certain sense, the 1/z counterexample is universal: For every analytic function that has no antiderivative on its domain, the reason for this is that 1/z itself does not have an antiderivative on ℂ − {0}.

Technology Types

mathematical theorempropositionstatementtheoremtheorems in complex analysistheory

Synonyms

Morera theorem

Translations

Morerova věta (cs)Satz von Morera (de)Teorema de Morera (ca)Teorema de Morera (es)Teorema di Morera (it)Théorème de Morera (fr)Twierdzenie Morery (pl)Теорема Морери (uk)Теорема Мореры (ru)모레라 정리 (ko)モレラの定理 (ja)莫雷拉定理 (zh)

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