Cauchy–Riemann equations

The Cauchy -Riemann partial differential equations (also: Cauchy- Riemann differential equations or Cauchy -Riemann equations) in the mathematical branch of function theory are a system of two partial differential equations of two real-valued functions. They bridge the gap of the real - differentiable functions to the complex - differentiable of (complex) function theory.

For the first time they appear 1752 by d' Alembert. Euler joined the system in 1777 with the analytical functions. In a purely function-theoretic context they appear in 1814 near Cauchy and Riemann's 1851 dissertation.

• 3.1 Conformal Mapping
• 3.2 The Cauchy -Riemann operator
• 3.3 Physical Interpretation

• 4.1 Definition
• 4.2 Fundamental solution
• 4.3 Integral representation

• 5.1 Definition
• 5.2 Necessary condition
• 5.3 existential statement

Definition

The Cauchy- Riemann equations are the system of two differential equations of two real-valued functions in two real variables:

Main application

1:1 correspondence

Be an open subset, such as a field, and a complex-valued function of a complex variable. We make the canonical correlation as first argument on the side to the setting and by

Secondly on the side of the real-valued function values ​​by the two functions. Finally, it

The correspondence to.

Properties

• Theorem: A function is complex differentiable if and only if their corresponding features in partial derivatives, and these satisfy the Cauchy- Riemann equations.
• The theorem clarifies the relationship between the complex and differentiability of the (real) differentiable.
• The theorem helps in the proof of Cauchy's integral theorem.
• Overall, an equivalence of the terms resulting complex differentiable, holomorphic, analytic ( infinitely differentiable ) and can be integrated with independence of the integral by the integration path in the (complex) function theory.
• Using the Cauchy- Riemann equations, one can show that and harmonic functions, provided that is holomorphic.

Derivation

If is complex differentiable, then there exists

For each. We can use the partial derivatives with respect to using the chain rule attributed to derivatives with respect to:

With follows from these two relations:

The second line will be converted with:

Since the right-hand sides are divided into real and imaginary parts, we obtain the Cauchy- Riemann equations ( CRDG ):

Polar coordinates

Of course you can be represented as the Cartesian Cauchy Riemann differential equations in other coordinate. Hereinafter, the display will be described in polar coordinates. A representation of a complex number in polar form. This means that one has the partial derivatives of by or look at. For this

It follows with:

Since both brackets must vanish, the following applies:

And

These are the Cauchy- Riemann equations in polar coordinates.

Interpretation and alternative considerations

Conformal Mapping

The complex representation of the Cauchy- Riemann equations

The belonging to these matrices linear maps are provided and are not both zero rotation stretches in space, and is, the scaling factor and the rotation angle. This figure is thus conformal; that is, the angle between the two curves in the plane is maintained. Functions that satisfy the Cauchy- Riemann equations, that are compliant.

The Cauchy -Riemann operator

In this section a more compact notation of the Cauchy -Riemann differential equations is shown. It becomes evident that in holomorphic functions must be independent of the complex conjugate.

A complex number, and its complex conjugate are related to real and imaginary components by means of the equations

Together.

Given this context, it seems reasonable the differential operators

To define. The operator is called the Cauchy -Riemann operator and the calculus of these operators is called Wirtinger calculus. The complex representation of the Cauchy- Riemann equations in the previous section we obtain the equation

Is an alternative presentation of the Cauchy- Riemann equations and means that if is holomorphic, it must be independent of. Thus, analytic functions can be regarded as real functions of a complex variable rather than a complex function of two real variables.

Physical interpretation

This interpretation is not directly used complex variables. It should be a given function with. The scalar fields and to the Cauchy- Riemann equations satisfy (note different sign convention ):

Consider now the vector field as a real three-component vector:

Then the Cauchy- Riemann differential first equation describes the source of freedom:

And the second equation describes the rotational freedom:

Thus, source- free and has a potential. In hydrodynamics describes such a potential flow field.

Inhomogeneous Cauchy- Riemann differential equations in one variable

Definition

The inhomogeneous Cauchy- Riemann differential equation has the representation

Here is the Cauchy -Riemann operator, is a given function and is the desired solution. The fact that the above-defined homogeneous Cauchy -Riemann differential equations corresponds, is already mentioned earlier in the article. The theory of the inhomogeneous Cauchy -Riemann equation for solutions in different solutions in with and is scribed here in two different sections.

Fundamental solution

For dimension, the fundamental solution of the Cauchy -Riemann operator is given by. This means that the distribution generated by the function solves the equation, where the delta function is. Let be a smooth test function with compact support, then you can see the validity of the statement due

Integral representation

For having obtained with

A solution of the inhomogeneous Cauchy-Riemann differential equation.

Inhomogeneous Cauchy- Riemann differential equations in several variables

In the following, the dimension of the underlying space or the number of components of a function.

Definition

The inhomogeneous Cauchy- Riemann differential equations in several variables has also the representation

Here is the Dolbeault - cross - operator is a given complex - differential form with compact support and is the desired solution. Specifically, this means that the system

Must be solved by partial differential equations. The differential operator is the Cauchy -Riemann operator.

Necessary condition

For the condition is necessary. This can be seen when applying the cross - Dolbeault operator on both sides of the equation. Thus one obtains namely, as is true for the Dolbeault operator on differential forms, must apply. Since a (0,1) - form, does not mean that a holomorphic differential form is, because only (p, 0) - forms that satisfy this equation are called holomorphic.

Existence theorem

Be a (0,1) - form with and. Then there exists a function such that the Cauchy- Riemann differential equation is satisfied.