an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions...

mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily...

branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds...

As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis...

analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function...

quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If f is an analytic function on...

holomorphic functions are analytic and vice versa. Among the corollaries of this theorem are the identity theorem that two holomorphic functions that agree...

global analytic function is a generalization of the notion of an analytic function which allows for functions to have multiple branches. Global analytic functions...

(3rd ed.). W. H. Freeman. ISBN 978-0-7167-2877-1. "Residue of an analytic function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric...

antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions. A function of...

The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic...

analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function f...

processing, an analytic signal is a complex-valued function that has no negative frequency components.  The real and imaginary parts of an analytic signal are...

example of a function that is differentiable but not locally Lipschitz continuous. The exponential function e x {\displaystyle e^{x}} is analytic, and hence...

"closed-form function" and a "closed-form number" in the discussion of a "closed-form solution", discussed in (Chow 1999) and below. A closed-form or analytic solution...

result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from here...

In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of...

In SQL, a window function or analytic function is a function which uses values from one or multiple rows to return a value for each row. (This contrasts...

complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation...

The Schwarz function of a curve in the complex plane is an analytic function which maps the points of the curve to their complex conjugates. It can be...

the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification...

L-function via analytic continuation. The Riemann zeta function is an example of an L-function, and some important conjectures involving L-functions are...

}{\frac {1}{1+e^{-2kx}}}.} There are many other smooth, analytic approximations to the step function. Among the possibilities are: H ( x ) = lim k → ∞ ( 1...

theorems about the range of an analytic function. They are named after Émile Picard. Little Picard Theorem: If a function f : C → C {\textstyle f:\mathbb...

of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some...

\operatorname {Re} (s)>1} , and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications...

analysis, the analytic capacity of a compact subset K of the complex plane is a number that denotes "how big" a bounded analytic function on C \ K can...

bounded analytic function can become Analytic continuation, a technique to extend the domain of definition of a given analytic function Analytic manifold...

sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging...

quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to ordered...