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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. Holomorphic function 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 Holomorphic function and Holomorphic function - complex-valued function on Holomorphic function .

  • Function Holomorphic function called complex differentiable at the point Holomorphic function if there is a limit

    Holomorphic function

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

Other Definition [edit]

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

Holomorphic function

Holomorphic function

Where Holomorphic function . Then the function Holomorphic 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 Holomorphic function and having in all its particular points Holomorphic function pole.
  • Function Holomorphic function called holomorphic on a compact Holomorphic function if there is an open set Holomorphic function containing Holomorphic function such that Holomorphic function holomorphic in Holomorphic function .

Properties

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

    Holomorphic function

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 Holomorphic function . That is, if Holomorphic function Is a holomorphic function, then Holomorphic function and Holomorphic function - 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 Holomorphic function converges uniformly on any compact in Holomorphic function 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.

Variations and generalizations

Multidimensional case

There is also a definition of the holomorphy of functions of several complex variables.

Holomorphic function

For the definition used concepts Holomorphic function -differentiability and Holomorphic function -linearity of such functions

C-linearity [edit]

Function Holomorphic function called Holomorphic function -linear if conditions are satisfied:

  • Holomorphic function .
  • Holomorphic function

(for Holomorphic function -linear functions Holomorphic function ).

  • For any Holomorphic function -linear function Holomorphic function there are sequences Holomorphic function such that Holomorphic function .
  • For any Holomorphic function -linear function Holomorphic function there is a sequence Holomorphic function such that Holomorphic function .

C-differentiability [edit]

Function Holomorphic function called Holomorphic function -differentiable at a point Holomorphic function if functions exist Holomorphic function and Holomorphic function such that in the neighborhood of a point Holomorphic function

Holomorphic function

Where Holomorphic function - Holomorphic function -linear (for Holomorphic function -differentiability - Holomorphic function -linear function.

Holomorphy [edit]

Function Holomorphic function called holomorphic in the domain Holomorphic function If she Holomorphic function -differentiable in a neighborhood of every point of this area.

See also

  • Antiholomorphic function
  • Deduction

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Comprehensive analysis and operational calculus

Terms: Comprehensive analysis and operational calculus