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.
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Definition
Let be
- open subset in
and
- complex-valued function on
.
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
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Where
. Then the function
called holomorphic if
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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
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and partial derivatives
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.
Variations and generalizations
Multidimensional case
There is also a definition of the holomorphy of functions of several complex variables.
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For the definition used concepts
-differentiability and
-linearity of such functions
C-linearity [edit]
Function
called
-linear if conditions are satisfied:
.
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(for
-linear functions
).
- For any
-linear function
there are sequences
such that
.
- For any
-linear function
there is a sequence
such that
.
C-differentiability [edit]
Function
called
-differentiable at a point
if functions exist
and
such that in the neighborhood of a point 
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Where
-
-linear (for
-differentiability -
-linear function.
Holomorphy [edit]
Function
called holomorphic in the domain
If she
-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