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Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. This is a much stronger condition than real differentiability and implies that the function is infinitely often differentiable and can be described by its Taylor series. The term analytic function is often used interchangeably with "holomorphic function", although note that the former term has several other meanings. A function that is holomorphic on the whole complex plane is called an entire function. The phrase "holomorphic at a point a" means not just differentiable at a, but differentiable everywhere within some open disk centered at a in the complex plane. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...
Partial plot of a function f. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ...
In mathematics, a smooth function is one that is infinitely differentiable, i. ...
As the degree of the Taylor series rises, it approaches the correct function. ...
In mathematics, an analytic function is a function that is locally given by a convergent power series. ...
In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. ...
Definition
If U is an open subset of C and f : U → C is a function, we say that f is complex differentiable at the point z0 of U if the limit In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...
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The limit here is taken over all sequences of complex numbers approaching z0, and for all such sequences the difference quotient has to approach the same number f '(z0). Intuitively, if f is complex differentiable at z0 and we approach the point z0 from the direction r, then the images will approach the point f(z0) from the direction f '(z0) r, where the last product is the multiplication of complex numbers. This concept of differentiability shares several properties with real differentiability: it is linear and obeys the product, quotient and chain rules. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
If f is complex differentiable at every point z0 in U, we say that f is holomorphic on U. We say that f is holomorphic in the point z0 if it is holomorphic on some neighborhood of z0. We say that f is holomorphic on some non-open set A if it is holomorphic in an open set containing A.
An equivalent definition is the following. A complex function f(x + iy) = u + iv is holomorphic if and only if it satisfies the Cauchy-Riemann equations and u and v have continuous first partial derivatives with respect to x and y. IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...
In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. ...
Examples All polynomial functions in z with complex coefficients are holomorphic on C, and so are sine, cosine and the exponential function. (The trigonometric functions are in fact closely related to and can be defined via the exponential function using Euler's formula). The principal branch of the logarithm function is holomorphic on the set C - {z ∈ R : z ≤ 0}. The square root function can be defined as In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...
In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
The exponential function is one of the most important functions in mathematics. ...
This article is about the Eulers formula in complex analysis. ...
Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ...
and is therefore holomorphic wherever the logarithm ln(z) is. The function 1/z is holomorphic on {z : z ≠ 0}.
Typical examples of functions which are not holomorphic are complex conjugation and taking the real part. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...
In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ...
Properties Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is non-zero.
Every holomorphic function is infinitely often complex differentiable at every point. It coincides with its own Taylor series and the Taylor series converges on every open disk that lies completely inside the domain U. The Taylor series may converge on a larger disk; for instance, the Taylor series for the logarithm converges on every disk that does not contain 0, even in the vicinity of the negative real line. See holomorphic functions are analytic for a proof. As the degree of the Taylor series rises, it approaches the correct function. ...
In complex analysis, a complex-valued function f of a complex variable is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a, and is analytic at a if in some open disk centered at a it can be expanded as...
If one identifies C with R2, then the holomorphic functions coincide with those functions of two real variables which solve the Cauchy-Riemann equations, a set of two partial differential equations. In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. ...
An illustration of a differential equation. ...
Close to points with non-zero derivative, holomorphic functions are conformal in the sense that they preserve angles and the shape (but not size) of small figures. In mathematics, a conformal map is a function which preserves angles. ...
Cauchy's integral formula states that every holomorphic function inside a disk is completely determined by its values on the disk's boundary. In mathematics, Cauchys integral formula, named after Augustin Louis Cauchy, is a central statement in complex analysis. ...
From an algebraic point of view the set of holomorphic functions on an open set is a commutative ring and a complex vector space. In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...
Vector space - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...
In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ...
In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ...
Several variables A complex analytic function of several complex variables is defined to be analytic and holomorphic at a point if it is locally expandable (within a polydisk, a cartesian product of disks, centered at that point) as a convergent power series in the variables. This condition is stronger than the Cauchy-Riemann equations; in fact it can be stated as follows: The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ...
In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs. ...
In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes...
In geometry, a disk is the region in a plane contained inside of a circle. ...
In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. ...
A function of several complex variables is holomorphic if and only if it satisfies the Cauchy-Riemann equations and is locally square-integrable. It has been suggested that this article or section be merged with Logical biconditional. ...
In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ...
Extension to functional analysis -
The concept of a holomorphic function can be extended to the infinite-dimensional spaces of functional analysis. For instance, the Fréchet or Gâteaux derivative can be used to define a notion of a holomorphic function on a Banach space over the field of complex numbers. In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. ...
In mathematics, the Gâteaux derivative is a generalisation in functional analysis of the standard derivative of the differential calculus. ...
In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
Terminology Today, many mathematicians prefer the term "holomorphic function" to "analytic function", as the latter is a more general concept. This is also because an important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow directly from the definitions. The term "analytic" is however also in wide use.
The word "holomorphic" derives from the Greek holos meaning "whole" and morphe meaning "form" or "appearance".
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