# Function theory

Function graph of f (z) = (z 2 -1) (z-2-i) 2 / (z 2 + 2 + 2i) in polar coordinates . The hue indicates the angle, the lightness the absolute value of the complex number.

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

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

## Function theory in a complex variable

### Complex functions

A complex function assigns a complex number , a further complex number to. Since any complex number can be written by two real numbers in the form , a general form of a complex function can be passed through ${\ displaystyle x + iy}$

${\ displaystyle x + iy \ mapsto f (x + iy) = u (x, y) + iv (x, y)}$

represent. Where and are real functions that depend on two real variables and . is called the real part and the imaginary part of the function. In this respect, a complex function is nothing more than a mapping from to (i.e. a mapping that assigns two real numbers to two real numbers). In fact, one could also build function theory using methods from real analysis. The difference to real analysis only becomes clearer when one considers complex-differentiable functions and brings into play the multiplicative structure of the field of complex numbers, which the vector space lacks. The graphical representation of complex functions is a bit more complicated than usual, since four dimensions now have to be represented. For this reason, one makes do with color tones or saturations. ${\ displaystyle \, u (x, y)}$${\ displaystyle \, v (x, y)}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle \, u (x, y)}$${\ displaystyle \, v (x, y)}$${\ displaystyle \ mathbb {R} ^ {2}}$${\ displaystyle \ mathbb {R} ^ {2}}$${\ displaystyle \ mathbb {R} ^ {2}}$

### Holomorphic function

The concept of differentiability in one-dimensional real analysis is expanded to include complex differentiability in function theory . Analogous to the real case, one defines: A function of a complex variable is called complex-differentiable (in the point ) if the limit value ${\ displaystyle a}$

${\ displaystyle f '(a): = \ lim _ {w \ to 0} {\ frac {f (a + w) -f (a)} {w}}}$

exists. Must be defined in an environment of . The complex concept of distance must be used for the definition of the limit value . ${\ displaystyle f}$${\ displaystyle a}$

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

Functions that are complex-differentiable in the vicinity of a point are called holomorphic or analytical . These have a number of excellent properties that justify the fact that a theory of their own is mainly concerned with them - the theory of functions. For example, a function that is complex-differentiable once can automatically be complex-differentiable as often as required, which of course does not apply in the real case.

The system of Cauchy-Riemann differential equations offers a different approach to function theory

{\ displaystyle {\ begin {aligned} \ partial _ {x} u (x, y) & = \ partial _ {y} v (x, y), \\\ partial _ {y} u (x, y) & = - \ partial _ {x} v (x, y). \ end {aligned}}}

A function is complex differentiable at a point if and only if it is real differentiable there and satisfies the system of Cauchy-Riemann differential equations. Therefore, one could understand function theory as a branch of the theory of partial differential equations . However, the theory is now too extensive and too versatile networked with other sub-areas of analysis to be embedded in the context of the partial differential equations.

The complex differentiability can be interpreted geometrically as (local) approximability through orientation-true affine maps, more precisely through the concatenation of rotations, stretches and translations. Correspondingly, the validity of the Cauchy-Riemann differential equations is equivalent to the fact that the associated Jacobi matrix is the representation matrix of a rotational extension. Holomorphic mappings therefore prove to be locally conformal (apart from the derivation zeros), that is, true to angle and orientation.

### Cauchy's integral formula

With an integration path that does not revolve around any singularities and for whose number of revolutions um it applies that ${\ displaystyle f}$${\ displaystyle z}$

${\ displaystyle 1 = {\ frac {1} {2 \ pi i}} \ oint {\ frac {1} {wz}} \ mathrm {d} w,}$

Cauchy's integral formula applies:

${\ displaystyle f (z) = {\ frac {1} {2 \ pi i}} \ oint {\ frac {f (w)} {wz}} \ mathrm {d} w.}$

This states that the value of a complex-differentiable function in a domain depends only on the function values ​​on the boundary of the domain.

### Functions with singularities

Since the set of holomorphic functions is quite small, one also considers functions in function theory that are holomorphic everywhere except in isolated points . These isolated points are called isolated singularities . If a function is bounded by a singularity in a neighborhood, the function can be continued holomorphically in the singularity. This statement is called Riemann's theorem of deductibility . Is a singularity of a function not liftable, however, has the function in a removable singularity, then one speaks of a pole order k-th, where k is minimal selected. If a function has isolated poles and is otherwise holomorphic, the function is called meromorphic . If the singularity is neither liftable nor a pole, one speaks of an essential singularity. According to Picard's theorem , functions with an essential singularity are characterized by the fact that there is at most one exception value a, so that in any small neighborhood of the singularity they take on any complex numerical value with at most the exception a. ${\ displaystyle z_ {0}}$${\ displaystyle f}$${\ displaystyle (z-z_ {0}) ^ {k} f (z)}$${\ displaystyle z_ {0}}$

Since one can develop every holomorphic function into a power series , one can also develop functions with removable singularities in power series. Meromorphic functions can be expanded into a Laurent series which only have finitely many terms with negative exponents, and the Laurent series of functions with essential singularity have a non-terminating expansion of the powers with negative exponents. The coefficient of the Laurent expansion is called the residual . According to the residual theorem , integrals over meromorphic functions and over functions with significant singularities can only be determined with the help of this value. This theorem is not only important in function theory, because with the help of this statement one can also determine integrals from real analysis which, like the Gaussian error integral, do not have a closed representation of the antiderivative. ${\ displaystyle a _ {- 1}}$${\ displaystyle (z-z_ {0}) ^ {- 1}}$

### Other important topics and results

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

Other important research focuses are the analytical continuation of holomorphic and meromorphic functions to the limits of their domain and beyond.

## Function theory in several complex variables

There are also complex-valued functions of several complex variables. Compared to real analysis, there are fundamental differences in complex analysis between functions of one and more variables. In the theory of holomorphic functions of several variables there is no analogue to Cauchy's integral theorem . The identity theorem also only applies in a weakened form to holomorphic functions of several variables. However, Cauchy's integral formula can be generalized to several variables in a very analogous manner. In this more general form it is also called the Bochner-Martinelli formula . In addition, meromorphic functions of several variables have no isolated singularities , which follows from Hartogs' so-called ball theorem, and as a consequence also no isolated zeros . Even the Riemannian mapping theorem - a high point of function theory in one variable - has no equivalent in higher dimensions . Not even the two natural generalizations of the one-dimensional circular disk , the unit sphere and the poly-cylinder , are biholomorphically equivalent in several dimensions . A large part of the function theory of several variables deals with continuation phenomena (Riemann's theorem of levitation, Hartogs' ball theorem, Bochner's theorem on tube regions, Cartan-Thullen theory). The function theory of several complex variables is used, for example, in quantum field theory .

## Complex geometry

Complex geometry is a branch of differential geometry that makes use of methods from function theory. In other sub-areas of differential geometry such as differential topology or Riemannian geometry , smooth manifolds are examined using techniques from real analysis. In the complex geometry, however, manifolds with complex structures are examined. In contrast to the smooth manifolds, on complex manifolds it is possible to define holomorphic maps with the help of the Dolbeault operator . These manifolds are then investigated using methods of function theory and algebraic geometry . In the previous section it was explained that there are big differences between the function theory of one variable and the function theory of several variables. These differences are also reflected in the complex geometry. The theory of Riemann surfaces is a branch of complex geometry and deals exclusively with surfaces with a complex structure, i.e. with one-dimensional complex manifolds. This theory is richer than the theory of n-dimensional complex manifolds.

## Function theory methods in other mathematical sub-areas

A classic application of function theory is in number theory . If one uses function theory methods there, one calls this area analytical number theory . An important result, for example, is the prime number theorem .

Real functions that can be expanded into a power series are also real parts of holomorphic functions. This allows these functions to be extended to the complex level. Through this extension one can often find connections and properties of functions that remain hidden in the real, for example Euler's identity . Here About to open up numerous applications in physics (for example, in quantum mechanics the representation of wave functions , as well as in the electrical two-dimensional current - voltage - diagrams ). This identity is also the basis for the complex form of the Fourier series and for the Fourier transform . In many cases, these can be calculated using methods of function theory.

For holomorphic functions, the real and imaginary parts are harmonic functions , i.e. they satisfy the Laplace equation . This links the function theory with the partial differential equations , both areas have regularly influenced each other.

The path integral of a holomorphic function is independent of the path. This was historically the first example of homotopy invariance . Many ideas of algebraic topology arose from this aspect of function theory , starting with Bernhard Riemann.

Functional means play an important role in the theory of complex Banach algebras , a typical example being the Gelfand-Mazur theorem . The holomorphic functional calculus allows the application of holomorphic functions to elements of a Banach algebra, a holomorphic functional calculus of several variables is also possible.

## Web links

Wikibooks: Introduction to Function  Theory - Learning and Teaching Materials