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The converse is not true for real functions; in fact, in a certain sense, the real analytic functions are sparse compared to all real infinitely differentiable functions. 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 ...
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.
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 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 ...
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.
Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, extended appropriately to complex arguments as functions , are holomorphic over the entire complex plane, making them entire functions, while rational functions /, where p and q are polynomials, are holomorphic on domains ...
Because X is a complex variety, its set of complex points X(C) can be given the structure of a compact complex analytic space. This analytic space is denoted X an . Similarly, if F {\displaystyle {\mathcal {F}}} is a sheaf on X , then there is a corresponding sheaf F an {\displaystyle {\mathcal {F}}^{\text{an}}} on X an .
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.