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In mathematics, 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. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions.
The latter property is the basis of the principle of analytic continuation which allows extending every real analytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed.
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space, that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space ), which the Mathematics Subject ...
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.
A holomorphic function is a complex function that is differentiable at every point of some open subset of the complex plane . It has been proved that holomorphic functions are analytic and analytic complex functions are complex-differentiable. In particular, holomorphic functions are infinitely complex-differentiable.
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers.It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, and electrical engineering.
Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.
The concept of zeros and poles extends naturally to functions on a complex curve, that is complex analytic manifold of dimension one (over the complex numbers). The simplest examples of such curves are the complex plane and the Riemann surface.