Complex analysis

The function theory is a branch of mathematics. It deals with the theory of differentiable complex-valued functions with complex variables. Since in particular makes the theory of functions of a complex variable ample use of methods from real analysis, is called the sub-region also complex analysis.

One of the main founders of the theory of functions include Augustin- Louis Cauchy, Bernhard Riemann, and Karl Weierstrass.

  • 5.1 Important Phrases
  • 5.2 Entire Functions
  • 5.3 Meromorphic functions

Theory of functions of a complex variable in

Complex Functions

A complex function assigns a complex number to a more complex number. Because any complex number can be written by two real numbers, in the form can be a general form of a complex function by

Represent. Here are real and functions of two real variables and dependent. is the real part and the imaginary part of the function. In this respect, a complex function of a special real image of after (that is a figure that two real numbers again, to two real numbers ). Indeed, one could treat the entire function theory with real analysis. The difference with the real analysis is only apparent when one considers complex - differentiable functions. Now maps to that in the theory of functions of the, makes the plotting of functions a little more complicated than usual, since now four dimensions must be reproduced. For this reason, one makes do with shades or saturations.

Holomorphic function

The Differenzierbarkeitsbegriff the one-dimensional real analysis is extended in the theory of functions of complex differentiability. Similar to the real case we define: A function of a complex variable is called complex - differentiable if the limit

Exists. It must be defined in a neighborhood of. While the complex notion of distance must be used for the definition of the limit.

Thus, two different Differenzierbarkeitsbegriffe are defined for complex-valued functions of a complex variable: the complex differentiability and differentiability of the two-dimensional real analysis (real differentiability ). Complex - differentiable functions are real - differentiable, the converse is not true without additional requirements.

Functions that are complex - differentiable in a neighborhood of a point is called holomorphic or analytic. These have a number of excellent properties which justify that his own theory mainly busy - just the function theory. For example, a function that is complex - differentiable once, automatically as often as complex - differentiable, which of course does not apply in the real case.

A different approach to the theory of functions provide the system of Cauchy- Riemann equations

A function is namely if and complex differentiable, ie holomorphic if it is real-differentiable and the Cauchy -Riemann system satisfies differential equations. Therefore, one could understand the function theory as a branch of the theory of partial differential equations. However, the theory has become too extensive, than that they are embedded today in the context of partial differential equations.

Cauchy's integral formula

With an integration path that wraps around any singularities of and applies to the winding number to that

Apply the Cauchy integral formula:

This says that the value of a complex - differentiable function in a field depends only on the function values ​​on the edge of the area.

Functions with singularities

Since the set of holomorphic functions is quite small, you look in the theory of functions and functions that are holomorphic everywhere except at isolated points. These isolated points are called isolated singularities. If a function in an environment around a singularity limited, so you can continue to function in the holomorphic singularity. This statement is called Riemannian Hebbarkeitssatz. Is not liftable a singularity of a function, however, has the function in a removable singularity, then one speaks of a pole of order k, where k is chosen minimal. Has a function insulated pole and is otherwise holomorphic, so you call the function meromorphic. Is the singularity neither be raised nor a pole, then one speaks of an essential singularity. By the theorem of Picard functions are characterized by an essential singularity that there is at most one exception value a, so that they take on any complex number with a maximum value except a in any arbitrarily small neighborhood of the singularity.

Since you can develop any holomorphic function in a power series, one can also develop functions with singularities in elevating power series. Meromorphic functions can be expanded in a Laurent series that only a finite number of terms with negative power, and the Laurent series of functions with essential singularities have a non-terminating development of the powers with negative exponents. The coefficient of the Laurent expansion is called residual. After the residue theorem one can determine integrals of meromorphic functions and functions with essential singularities only with the help of this value. This sentence is not only in the theory of functions of importance, because one can determine with the help of this statement also integrals of real analysis that have no closed-form representation of the parent function as the Gaussian error integral.

Other important topics and results

Important results are also still the Riemann mapping theorem and the fundamental theorem of algebra. The latter states that a polynomial in the field of complex numbers can be completely decomposed into linear factors. For polynomials in the field of real numbers, this is generally ( with real linear factors ) is not possible.

Other major research areas are the analytic continuation of holomorphic and meromorphic functions on the boundaries of its domain of definition and beyond.

Function theory in several complex variables

There are also complex-valued functions of several complex variables. Compared to real analysis there are in complex analysis are fundamental differences between functions of one and several variables. In the theory of holomorphic functions of several variables, there is no analogue of the Cauchy integral theorem. Also, the identity theorem applies only in a reduced form for holomorphic functions of several variables. The Cauchy integral formula, however, can be quite analogous to generalize to several variables. In this general form it is also called Bochner - Martinelli formula. In addition meromorphic functions of several variables are not isolated singularities, which follows from the so-called ball set of Hartogs, and as a consequence no isolated zeros. Also, the Riemann mapping theorem - a highlight of the theory of functions in one variable - has no equivalent in higher dimensions. Not even the two natural generalizations of the one-dimensional circular disc, the unit sphere and the polycylinder, are in several dimensions biholomorphic equivalent. A large part of the theory of functions of several variables deals with continuation phenomena ( Riemannian Hebbarkeitssätze, ball set of Hartogs, set of Bochner on tube areas, Cartan Thullen theory ). The theory of functions of several complex variables is used, for example, in quantum field theory.

Complex Geometry

The complex geometry is a branch of differential geometry, which draws on methods of function theory. In other fields of differential geometry, such as differential topology and Riemannian geometry smooth manifolds with techniques from real analysis are investigated. In contrast, the complex geometry of manifolds are examined with complex structures. In contrast to the smooth manifolds, it is possible on complex manifolds, to define with the help of the Dolbeault operator holomorphic maps. These manifolds are then analyzed with methods of function theory and algebraic geometry. In the previous section it was explained that there are large differences between the theory of functions of one variable and the theory of functions of several variables. These differences are also reflected in the complex geometry. The theory of Riemann surfaces is a subfield of the complex geometry and only deals with surfaces with a complex structure, ie with one-dimensional complex manifolds. This theory is richer than the theory of n- dimensional complex manifolds.

Function theoretical methods in other mathematical sciences

A classic application of the theory of functions is in number theory. If one uses there function-theoretic methods, called this area then analytic number theory. One important result is, for example, the prime number theorem.

Real functions, which can be developed in a power series, are also real parts of holomorphic functions. This means that these functions can be extended to the complex plane. With this addition, you can often find relationships and properties of functions that are hidden in the real case, for example, Euler's identity. Here About him numerous applications open up in physics ( for example, in quantum mechanics, the representation of the wave functions, as well as two-dimensional in the electrical current-voltage diagrams). This identity is also the basis for the complex form of the Fourier series and the Fourier transformation. In many cases, these can be calculated with the methods of function theory.

For holomorphic functions that real and imaginary parts are harmonic functions, that satisfy the Laplace equation. This links the theory of functions with partial differential equations, both areas have regularly affected each other.

The path integral of a holomorphic function is independent of the path. This was historically the first example of a homotopy. For this aspect of the theory of functions, many ideas of algebraic topology emerged, starting with Bernhard Riemann.

In the theory of complex Banach algebras function-theoretic agents play an important role, a typical example is the set of Gelfand - Mazur. The holomorphic functional calculus allows for the application of holomorphic functions on elements of a Banach algebra, a holomorphic functional calculus of several variables is possible.