Holomorphic function

Holomorphy (from gr holos, "whole" and morphe, "form" ) is a property of certain complex-valued functions that are treated ( a branch of mathematics ) in function theory. A function of an open set is called holomorphic if it is complex differentiable at every point. Especially in older literature, such functions are called regular.

Although the definition is analogous to the real differentiability, is reflected in the theory of functions that holomorphy is a very strong property. Namely, they produced a variety of phenomena that have no counterpart in the real domain. For example, a holomorphic function always infinitely often ( ever ) is differentiable and can be locally at each point in a power series develop.

• 4.1 Entire Functions
• 4.2 Holomorphic, not all the functions
• 4.3 Nowhere holomorphic functions

• 5.1 Cauchy 's integral theorem
• 5.2 Cauchy's integral formula
• 5.3 holomorphy and analyticity
• 5.4 Identity Theorem
• 5.5 More

• 7.1 In the n-dimensional complex space
• 7.2 In the complex geometry

• 8.1 Standard Works

Definitions

It should be an open subset of the complex plane and a point of this subset. A function is called complex differentiable at the point, if the limit

With exists. In this case, this is referred to as a limit.

The function is called holomorphic in point, if a neighborhood exists, in which is complex differentiable. Is on all holomorphic, it is called holomorphic one. If further, it is called an entire function.

Notes

Relationship between complex and real differentiability

Is naturally a two-dimensional real vector space with the canonical basis and so you can examine a function on an open set on their total differentiability in the sense of multi-dimensional real analysis. As is known, is called (totally ) differentiable at if a linear map exists, so that

Holds, where a function with

Is. Now you can see that the function is exactly then in complex differentiable if it is differentiable and there totally is even - linear. The latter is a strong condition. It means that the image matrix of the mold with respect to the canonical base

Has.

Cauchy- Riemann differential equations

Now we decompose a function into its real and imaginary parts with real functions, so does the total derivative as a matrix representation of the Jacobian

Consequently, the function is complex differentiable if and only if it is real-differentiable and for the Cauchy- Riemann equations

Are fulfilled.

Equivalent properties of holomorphic functions of one variable

In an environment of a complex number following properties of complex functions are equivalent:

Examples

Entire functions

Entire functions are holomorphic on all. Examples are:

• Every polynomial with coefficients

• The exponential function

• The trigonometric functions and

• The hyperbolic functions and

Holomorphic, not all the functions

• Cracked rational functions are holomorphic except at the zeros of its denominator. There they have isolated poles and are therefore examples of meromorphic functions.
• The logarithm function can be in all aspects of in a power series development and thus is holomorphic on the set.

Nowhere holomorphic functions

The following functions are not complex differentiable and therefore nowhere holomorphic:

• The absolute value function

• The projections on the real part or the imaginary part on

• The complex conjugate

The function is complex differentiable only at the place, but not holomorphic there, because it is not on a whole area of complex differentiable.

Properties

Are complex differentiable in so well, and. If, as is also in complex differentiable. It shall also sum ​​, product, quotient rule and chain rule.

Following is a list of fundamental properties of holomorphic functions, all of which have no counterpart in the real theory. As a result, was a territory and holomorphic.

Cauchy 's integral theorem

Is simply connected and a cycle, then the Cauchy integral theorem holds

So the sentence is particularly true when a star field and a closed path is.

Cauchy's integral formula

Is the open circular disc of radius around the point. If the conclusion of yet in, then for all and the Cauchy integral formula

The feature value of a point in an area that is dependent only on the function values ​​at the edge of this area.

Holomorphy and analyticity

A consequence of the Cauchy integral formula is that in the complex plane is the notion of analyticity is equivalent to the holomorphy: Each in holomorphic function is analytic in. Conversely, in any analytic function to continue in holomorphic function.

Since power series are infinitely often differentiable complex (and by termwise differentiation ), we obtain in particular that holomorphic functions infinitely differentiable and all its derivatives, in turn, are holomorphic functions. And thus we can already see clear differences to the real differential calculus.

Identity theorem

It is shown that a holomorphic function is already uniquely determined by very little information. The identity theorem states that two holomorphic functions on an area already then are identical if they agree on a suitable subset. The match amount must not even be a continuous way, it is sufficient that a cluster point in having. Discrete subsets sufficient for this purpose, however, of not.

More

• The set of Liouville states that every bounded entire function is constant. A simple implication of this is the fundamental theorem of algebra.

• Where an area is not constant, then is an area again. ( Set by the territorial loyalty )

• One implication of the area fidelity is the maximum principle.

• Converges a sequence of holomorphic functions on compact against the limit function, it is again holomorphic, and you can limit the formation and differentiation of exchange, that is, the sequence converges compactly. ( Weierstrass ).

• If the sequence of holomorphic functions limited to local, then there exists a compact convergent subsequence. ( Set of Montel )

• Each on a simply connected domain twice continuously differentiable harmonic function is the real part of a complex- differentiable function. The real function also satisfies. You will be called conjugate harmonic and as a complex potential.

Biholomorphic functions

A function that is holomorphic, bijective and its inverse function is holomorphic, it is called biholomorphic. In the case of a complex variable is the equivalent to saying that the map is bijective and compliant. From the implicit function theorem it follows for holomorphic functions of variables already that a bijective holomorphic function always has a holomorphic inverse mapping. In the next section holomorphic functions of several variables are introduced. In this case, the set of Osgood guarantees this property. Thus, one can say that bijective, holomorphic map are biholomorphic.

From the perspective of category theory, a biholomorphic map is an isomorphism.

Holomorphy of several variables

In n-dimensional complex space

Let be a complex open subset. A mapping is called holomorphic if is holomorphic in every part and function of each variable. By Wirtinger calculus and a calculus is available with which you can manage the partial derivatives of a complex function easier. However holomorphic functions of several variables no longer have as many nice properties.

So not true for functions of Cauchy's integral theorem and the identity set is valid only in an attenuated version. For these functions, however, the integral formula of Cauchy can be generalized by induction to n dimensions. In 1944, Salomon Bochner could even prove a generalization of the -dimensional Cauchy integral formula. This is called Bochner - Martinelli formula.

In the complex geometry

Also in the complex geometry of holomorphic maps are considered. So you can define holomorphic maps between Riemann surfaces and between complex manifolds analogous to differentiable functions between smooth manifolds. There is also an important factor for integration theory counterpart to the smooth differential forms, which are called holomorphic differential forms.