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For the complex numbers, the converse does hold, and in fact any function differentiable once on an open set is analytic on that set (see "analyticity and differentiability" below). For any open set Ω ⊆ C {\displaystyle \Omega \subseteq \mathbb {C} } , the set A (Ω) of all analytic functions u : Ω → C {\displaystyle u:\Omega \to \mathbb ...
That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. [1] Holomorphic functions are also sometimes referred to as regular functions. [2] A holomorphic function whose domain is the whole complex plane is called an entire function.
no bump function on the complex plane can be entire. In particular, on any connected open subset of the complex plane, there can be no bump function defined on that set which is holomorphic on the set. This has important ramifications for the study of complex manifolds, as it precludes the use of partitions of unity. In contrast the partition ...
In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844 [1]), states that every bounded entire function must be constant. That is, every holomorphic function f {\displaystyle f} for which there exists a positive number M {\displaystyle M} such that | f ( z ) | ≤ M ...
As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables z i. Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations.
In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions f and g analytic on a domain D (open and connected subset of or ), if f = g on some , where has an accumulation point in D, then f = g on D.
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following: Denoting by C the set of complex numbers, let K be a compact subset of C and let f be a function which is holomorphic on an open set containing K.
Great Picard's Theorem (meromorphic version): If M is a Riemann surface, w a point on M, P 1 (C) = C ∪ {∞} denotes the Riemann sphere and f : M\{w} → P 1 (C) is a holomorphic function with essential singularity at w, then on any open subset of M containing w, the function f(z) attains all but at most two points of P 1 (C) infinitely often.