Holomorphic function

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. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives The term analytic function is often used interchangeably with "holomorphic function", although the former term is also used in the broader sense of a function (real, complex, or of more general type) that is equal to its Taylor series in a neighborhood of each point in its domain. This article is about both real and complex analytic functions The fact that the class of analytic functions coincides with the class of holomorphic functions is a major theorem in complex analysis. Holomorphic functions are sometimes said to be regular functions. ^[1] A function that is holomorphic on the whole complex plane is called an entire function. In Complex analysis, an entire function, also called an integral function is a complex-valued function that is holomorphic everywhere on the 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.

1 Definition
2 Terminology
3 Properties
4 Examples
5 Several variables
6 Extension to functional analysis
7 References
8 See also

Definition

If U is an open subset of C and f : U → C is a complex function on U, we say that f is complex differentiable at a point z₀ of U if the limit

$f'(z_0) = \lim_{z \rightarrow z_0} {f(z) - f(z_0) \over z - z_0 }$

exists. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close"

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). In Mathematics, a sequence is an ordered list of objects (or events 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 Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

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 at 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.

The relationship between real differentiability and complex differentiability is the following. If a complex function f(x + iy) = u(x, y) + iv(x, y) is holomorphic, then u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy-Riemann equations:

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad \mbox{and} \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

The converse is not necessarily true. In Mathematics, the Cauchy-Riemann differential equations in Complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two A simple converse is that if u and v have continuous first partial derivatives and satisfy the Cauchy-Riemann equations, then f is holomorphic. In Mathematics, the Cauchy-Riemann differential equations in Complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two A more satisfying converse, which is much harder to prove, is the Looman-Menchoff theorem: if f is continuous, u and v have first partial derivatives, and they satisfy the Cauchy-Riemann equations, then f is holomorphic. In the mathematical field of Complex analysis, the Looman–Menchoff theorem states that a continuous complex -valued function defined in an In Mathematics, the Cauchy-Riemann differential equations in Complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two

Terminology

The word "holomorphic" was introduced by two of Cauchy's students, Briot (1817 - 1882) and Bouquet (1819 - 1895), and derives from the Greek őλoς (holos) meaning "entire", and μoρφń (morphe) meaning "form" or "appearance". ^[2]

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.

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 not zero.

If one identifies C with R², then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives 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 In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R². In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties In other words, if we express a holomorphic function f(z) as u(x, y) + i v(x, y) both u and v are harmonic functions. In Mathematics, Mathematical physics and the theory of Stochastic processes a harmonic function is a twice continuously differentiable function

In regions where the first derivative is not zero, 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 In the most common case the function is between domains in the Complex plane

Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary. In Mathematics, Cauchy's integral formula, named after Augustin Louis Cauchy, is a central statement in Complex analysis.

Every holomorphic function is analytic. In Complex analysis, a field of Mathematics, a complex -valued function f of a complex variable z is holomorphic That is, a holomorphic function f has derivatives of every order at each point a in its domain, and it coincides with its own Taylor series at a in a neighborhood of a. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In fact, f coincides with its Taylor series at a in any disk centered at that point and lying within the domain of the function.

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 Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets. In Functional analysis and related areas of Mathematics, locally convex topological vector spaces or locally convex spaces are examples of Topological In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

Examples

All polynomial functions in z with complex coefficients are holomorphic on C, and so are sine, cosine and the exponential function. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a coefficient is a Constant multiplicative factor of a certain object The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) (The trigonometric functions are in fact closely related to and can be defined via the exponential function using Euler's formula). This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic The principal branch of the complex logarithm function is holomorphic on the set C \ {z ∈ R : z ≤ 0}. In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers The square root function can be defined as

$\sqrt{z} = e^{\frac{1}{2}\log z}$

and is therefore holomorphic wherever the logarithm log(z) is. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose The function 1/z is holomorphic on {z : z ≠ 0}.

Typical examples of continuous 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 z is the first element of the Ordered pair of Real numbers representing z

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. The theory of functions of several complex variables is the branch of Mathematics dealing with functions f ( z1 z2 In the theory of functions of Several complex variables, a branch of Mathematics, a polydisc is a Cartesian product of discs More specifically Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. This condition is stronger than the Cauchy-Riemann equations; in fact it can be stated as follows:

A function of several complex variables is holomorphic if and only if it satisfies the Cauchy-Riemann equations and is locally square-integrable. In Mathematics, the Cauchy-Riemann differential equations in Complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two ↔ In Mathematics, an integrable function is a function whose Integral exists

Extension to functional analysis

Main article: infinite-dimensional holomorphy

The concept of a holomorphic function can be extended to the infinite-dimensional spaces of functional analysis. In Mathematics, infinite-dimensional holomorphy is a branch of Functional analysis. For functional analysis as used in psychology see the Functional analysis (psychology article 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, the Fréchet derivative is a Derivative defined on Banach spaces Named after Maurice Fréchet, it is commonly used to formalize In Mathematics, the Gâteaux differential is a generalisation of the concept of Directional derivative in Differential calculus. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis

References

^ [1][2]
^ Markushevich, A. I. ; Silverman, Richard A. (ed. ) [1977] (2005). Theory of functions of a Complex Variable, 2nd ed. , New York: American Mathematical Society, 112. The American Mathematical Society (AMS is an association of professional Mathematicians dedicated to the interests of mathematical research and scholarship which ISBN 0-8218-3780-X.

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