Holomorphic function

Lecture

The holomorphic function realizes a conformal mapping, transforming an orthogonal grid into an orthogonal grid (where the complex derivative does not vanish).

A holomorphic function , sometimes called a regular function, is a function of a complex variable, defined on an open subset of the complex plane. and complexly differentiable at every point.

In contrast to the real case, this condition means that the function is infinitely differentiable and can be represented by a Taylor series converging to it.

Holomorphic functions are also sometimes called analytic , although the second concept is much broader, since the analytic function need not be defined on a set of complex numbers. The fact that for complex-valued functions of a complex variable of the set of holomorphic and analytic functions coincide is a nontrivial and very remarkable result of a complex analysis.

Holomorphic function

Definition

Let be Holomorphic function - open subset in and - complex-valued function on .

Function called complex differentiable at the point if there is a limit
- In this expression, the limit is taken over all sequences of complex numbers converging to , for all such sequences, the expression must converge to the same number . Complex differentiation is in many respects similar to the real one: it is linear and satisfies the Leibniz identity.
Function called holomorphic in if it is complexly differentiable at each point .
Function called holomorphic in if it is holomorphic in some neighborhood .

Other Definition [edit]

The definition of a holomorphic function can be given a slightly different form, if you use the operators and determined by rule

Holomorphic function

Where Holomorphic function . Then the function called holomorphic if

Holomorphic function

which is equivalent to the Cauchy – Riemann conditions.

Related definitions

The entire function is a function that is holomorphic on the entire complex plane.
Meromorphic function - a function holomorphic in a domain and having in all its particular points pole.
Function called holomorphic on a compact if there is an open set containing such that holomorphic in .

Properties

Comprehensive function is holomorphic if and only if the Cauchy – Riemann conditions are satisfied

and partial derivatives Holomorphic function are continuous.

The sum and the product of holomorphic functions is a holomorphic function, which follows from the linearity of differentiation and the fulfillment of the Leibniz rule. The quotient of holomorphic functions is also holomorphic at all points where the denominator does not vanish.
The derivative of a holomorphic function is again holomorphic; therefore, holomorphic functions are infinitely differentiable in their domain of definition.
Holomorphic functions are analytic, that is, they can be represented as a Taylor series that converges in a certain neighborhood of each point. Thus, for complex functions of a complex variable, the set of holomorphic and analytic functions coincide.
From any holomorphic function, one can distinguish its real and imaginary parts, each of which will be a solution of the Laplace equation in . That is, if Is a holomorphic function, then and - harmonic functions.
If the absolute value of a holomorphic function reaches a local maximum at the inner point of its domain of definition, then the function is constant (it is assumed that the domain of definition is connected). It follows from this that a maximum (and a minimum, if it is not zero) of the absolute value of a holomorphic function can be reached only on the boundary of the region.
In the region where the first derivative of a holomorphic function does not turn into 0, but the function is univalent, it performs a conformal mapping.
The integral Cauchy formula associates the value of a function at an internal point of a region with its values at the boundary of this region.
From an algebraic point of view, the set of functions holomorphic on an open set is a commutative ring and a complex linear space. This is a locally convex topological vector space with a seminorm equal to the supremum on compact subsets.
According to the Weierstrass theorem, if a series of holomorphic functions in a domain converges uniformly on any compact in then its sum is also holomorphic, and its derivative is the limit of the partial sum of derivatives of the series [1] .

History

The term “holomorphic function” was introduced by two students of Cauchy, Brio (1817–1882) and Bouquet (1819–1895), and is derived from the Greek words őλος ( holos ), which means “whole”, and μoφφ ( morphe ) - a form, image . [2]

Today, many mathematicians prefer the term “holomorphic function” instead of “analytical function”, since the second concept is more general. In addition, one of the important results of the complex analysis is that any holomorphic function is analytic, which is not obvious from the definition. The term “analytical” is usually used for more general functions that are not necessarily defined on the complex plane.