Function theory

Apart from the function theory described on this side the function theory of the music sciences co-exists to mathematics. These two terms are sometimes confounded with one another.

The function theory is a subsection of mathematics. It is concerned with the differentiable komplexwertigen functions of complex variables.Common also the designation is complex analysis.

Table of contents

of complexes functions

a complex function arrangesa complex number a further complex number too. There each complex number by two real numbers in the form <math> x + iy< /math> to be written, sees a general form of a complex function knows so out

< math> x + iy \ tons of f (x+iy) =u (x, y) + i v (x, y)< /math>.

Here math <u> (x, y) are </math> and <math> v (x, y)< /math> real functions, those of two real variables<math> x< /math> and <math> y< /math> depend. <math> u (x, y)< /math> that is called real part and <math> v (x, y)< /math> the imaginary part of the function. To that extent a complex function does not differ from a realIllustration of <math> \ mathbb {R} ^2< /math> after <math> \ mathbb {R} ^2< /math> (thus an illustration, which assigns two real numbers again two real numbers). Actually one could treat the whole function theory also with real analysis. The difference to the real analysis becomes only clearer if one complex differentiableFunctions regards.

functions of several complex variables

gives also komplexwertige functions of several complex variables. The study of these functions is substantially more complicated than that of the simple complex functions and in this article is not continued to treat. In particular applymost results of the simple function theory only more with restrictions. The function theory of several complex variables is used for example in the Quantenfeldtheorie.

complexes differentiability

the differentiability term of the linear real analysis becomes in the function theory the complex differentiabilityextended. Similarly to the real case one defines: A function is called complex differentiable, if the following limit value exists:

<math> f'(a) = \ lim_ {w \ tons of 0} \ frac {f (a+w) - f (A)}{w}< /math>.

For an accurate definition math <f> /math< must > in an environment of <math> A< /math> defined its.For the limit value thereby the complex spacer term must be used:

<math> |(x_1 + i y_1) - (x_2 + i y_2) | = \ {(x_1-x_2) ^2 sqrt +

(y_1-y_2) ^2} </math>.

Thus two different differentiability terms are defined for komplexwertige functions of a complex variable: thosecomplex differentiability and the differentiability of the two-dimensional real analysis (real differentiability). Complex differentiable functions are generally not differentiable also really, the reversal apply.

Cauchy Riemann differential equations

major items: Cauchy Riemann differential equations

equivalent to the complex differentiability are the demands on thosereal partial derivatives of the function:


\ partial_x u (x+iy) = \ partial_y v (x+iy) </math>


\ partial_y u (x+iy) = - \ partial_x v (x+iy) </math>

Cauchy formula

major item: Cauchy integral formula

by choice of a suitable path of integration one can find the Cauchy formula:


f (z) = \ frac1 {2 \ pi i} \ oint \ frac {f (w)}{CR} dw </math>

This means that the value of a complex differentiable function in an area depends only on the function values on the edge of the area.

Holomorphe functions

functions, those in an environment of one point complex differentiablyare, one calls holomorphe or analytic functions. These have a number of outstanding characteristics, which justify it that its own theory is occupied mainly with it - evenly the function theory. For example a function, which is complex differentiable once, is automaticarbitrarily often complex differentiably! (Contrary to the real case).

equivalents definitions of holomorpher functions of a variable

in an environment of a complex numberthe following characteristics of complex functions are equivalent:

  1. A function is once complex differentiably
  2. a function is arbitraryoften complex differentiably
  3. real and imaginary part fulfill the Cauchy Riemann differential equations and are at least once really constantly differentiably
  4. the function can into a complex power series be developed
  5. the way integral of the function over any closed pull togetherable way disappear.
  6. The function values inAn internal circular disk can be determined from the function values at the edge with the help of the Cauchy integral formula.
  7. It applies <math> \ frac {\ partial for f} {\ partial \ without z} =0< /math> with <math> \ frac \ partial {\ partial \ without z} = \ frac12 \ bend (\ frac \ partial {\ partial x} + \ mathrm i \ frac \ partial {\ partial y} \ bend). </math>

Holomorphe functions are thus much “pleasant” functions: They are arbitrarily often differentiable,can be developed into a power series (Taylor series) and much more besides. Nearly all functions, which admits from school mathematics is, are real or imaginary part of a complex function (at least on a part of the complex level): In particular that applies for polynomials, rational functions, trigonometric functions (Sine, cosine), exponential function, logarithm, and root functions.

Meromorphe functions

Meromorphe functions are up to isolated pole places holomorph. They can be developed in Laurent series, however onlyfinally many row members possess, with which powers with negative exponents occur.

there are functions with

substantial singularities apart from holomorphen and meromorphen functions in the function theory functions with substantial singularities. They are characterized by the fact that a functionin the environment of a substantial singularity to assume knows any complex numerical value with at the most one exception (sentence of spades pool of broadcasting corporations). Functions with substantial singularities have a not breaking off Laurent expansion for powers with negative exponents.

function-theoretic methods in othersmathematical subsections

real functions, which can be developed into a power series, are also real part of a holomorphen function. Thus these functions can be extended to the complex level. By this extension one can often find connections and characteristics of functions, thosein the real one remain hidden, for example the Euler identity. Concerning this various ranges of application in physics are opened (for example in quantum mechanics the representation of wave functions, as well as in electro-technology two-dimensional river - tension - of diagrams). This identityis also the basis for the complex form of the fourier transformation. In many cases this can be computed simply by complex analysis.

To holomorphe functions applies that real and imaginary part are harmonious functions, thus the Laplace equation fulfill. This links thoseFunction theory with the partial differential equations, both areas regularly mutually affected each other.

The way integral of a holomorphen Funktione is unahängig from the way. This was historically the first example of a Homotopieinvarianz. From this aspect of the function theory many ideas developed for that algebraic topology.

In addition one can use track independence, around real integrals to compute, by accomplishing the integration in the complex level (see residue theorem).

further ones important topics and results

important results are the Cauchy integral set, the residue theorem and the Riemann mapping theorem. A further important result is the fundamental principle of algebra. It means that a polynomial can be divided in the range of the complex numbers completely into linear factors. For polynomials within the range of the realThis is generally not possible for numbers.

Further important main points of research are the analytic continuation barness of holomorphen and meromorphen functions on the borders of their definition range and beyond that.

see also

to bases

important sentences

whole functions

Meromorphe functions



  > German to English > (Machine translated into English)