Gamma function

The Euler gamma function, even for a short gamma function or Eulerian integral of the second kind, is one of the most important special functions and is being studied in the mathematical sciences of analysis and theory of functions. Today, it is denoted by a, the Greek capital letters gamma and is a transcendental meromorphic function with the property

For each natural number and is denoted by the faculty. The motivation for the definition of the gamma function is to extend the factorial function to real and complex arguments was. The Swiss mathematician Leonhard Euler in 1729 solved this issue and the gamma function defined by an infinite product. Today, the gamma function is often defined by means of an integral representation, which also goes back to Euler.

The gamma function is based on the Gamma probability distribution.

• 7.1 Stirling's formula
• 7.2 Recursive approximation

Definition

The gamma function for positive real numbers via the integral

Be defined.

History

As earliest definition of the gamma function applies those given in a letter from Daniel Bernoulli to Christian Goldbach on October 6, 1729:

Is infinite for A, in modern notation, or. A few days later, on 13 Oktoberjul. / October 24 1729greg. , Euler also described in a letter to Goldbach those similar, slightly simpler formula

Approaches with increasing n the true value, in today's notation, or the Gauss rediscovered in 1812 for the more general case of complex numbers ( the above-mentioned letters were published only in 1843 ). On January 8, 1730 Euler described in a letter to Goldbach following integral for interpolation of the factorial function, which he had presented to the St. Petersburg Academy on November 28, 1729:

This definition was used later by Euler preferably, it proceeds by the substitution in the form of

About. Euler discovered this integral in the investigation of the problem of the mechanism, in which the acceleration of a particle is considered.

Adrien -Marie Legendre in 1809 led the Greek majuscule (gamma ) as a function symbol. Gauss used 1812, the function symbol (Pi) such that and thus also applies to non-negative integer n. However, it did not sit through.

Other forms of presentation

Besides the representation of the gamma function from the definition there are some other equivalent representations of this particular function. A direct definition of all are the product representation of the gamma function by Gauss,

The real positive numbers has already been given by Euler in 1729. Derived from this is the representation of the Weierstrass product:

With the Euler constant. The second product is commonly referred to as the Weierstrass representation, however, Karl Weierstrass used only the first.

The integral representation of the definition also goes back to Euler in 1729, it holds more generally for complex numbers with positive real part:

By splitting this integral EF Prym concluded in 1876 a valid throughout presentation:

Another variant of the Euler integral representation for there with:

This representation can be, for example, in an elegant way the Fresnel integral formulas derived.

Ernst Eduard Kummer was 1847, the Fourier expansion of the logarithmic Gamma function to:

It is called the Kummer series. In 1846 was Carl Johan Malmstén a similar series:

Axiomatic characterization

Continuation of the Faculty

The conditions and describing the Faculty of natural numbers clearly be fulfilled by other analytical functions as the gamma function. For positive example performs the function

For the conditions characteristic of the gamma function. Weierstrass added in 1854, therefore, the necessary and sufficient condition

Support that opens but the search was not terminated after a possible elementary or natural characterizing property. Emil Artin in 1931 discussed the possible identification by functional equations.

The set of Hölder

The set of Hölder ( Hölder Otto 1886) is a negative result and indicates that the gamma function has no algebraic differential equation whose coefficients are rational functions fulfilled. That is, there is no differential equation of the form with a non- negative integer and a polynomial whose coefficients are rational functions, and to the solution.

The set of Bohr - Mollerup

The set of Bohr - Mollerup ( Harald Bohr and Johannes Mollerup 1922) allows a simple characterization of the gamma function:

These axioms are the starting point for the presentation of the theory of the gamma function at Nicolas Bourbaki.

The set of Wielandt

The set of Wielandt on the gamma function ( Helmut Wielandt 1939) characterizes the gamma function as a holomorphic function, and states:

More precisely, for all with.

Functional equations and special values

The gamma function satisfies the functional equation

With the addition rate of the gamma function ( Euler 1749)

Obtained in particular = 1.77245 38509 05516 02729 ... ( sequence A002161 in OEIS ) and

With general is selected from the last formula, the Legendre duplication formula ( Legendre 1809)

This is a special case of the Gauss multiplication formula ( Gauss 1812)

Applies with the lemniscate constant

The negative slope of the gamma function at the position 1 is the same γ Euler Mascheroni constant:

The gamma function has poles at the points of the first order. From the functional equation is obtained for the residuals

Since it additionally has no zeros, the function is therefore an entire function.

Connection with the Riemann ζ - function

Bernhard Riemann in 1859 brought the gamma function with the Riemann ζ - function via the formula

And the following statement in relation: The term " remains unchanged when it is turned into ," ie

Approximate calculation

Stirling's formula

Approximate gamma function for supplying inter alia the Stirling formula, it is

Recursive approximation

From the functional equation

Which involves a kind of periodicity, can be calculated recursively in 1 the values ​​in any other corresponding strips of known function values ​​in a strip of width. with

Can pass from one strip to the adjacent smaller the real part and the tray. Since there are large very good approximations for, so its accuracy can be transferred to areas in which direct application of any such approximation would not be advisable. After Rocktäschel recommended, as noted already by Carl Friedrich Gauss, which is derived from the Stirling formula asymptotic expansion in

While it has at close range with an irregularity, but is really useful for. With the correction term their error is reduced to the order of magnitude for growing.

The -fold application of this approximation leads to

The complex logarithm is computed over the polar representation of. For most applications, such as in the wave propagation, should be sufficient.

Incomplete Gamma Function

In the literature this term, in terms of limits of integration and normalization ( regularization), used inconsistently.

Common notations are:

When speaking of a regularized gamma function, this implies already that it is incomplete.

Stands for the generalized incomplete gamma function.