The Euler gamma function, also briefly gamma function or Euler integral of the second kind, is one of the most important special functions and is used in the mathematical branches of analysis and the theory of functions studied. Today it is denoted by a , the Greek capital letter gamma , and is a transcendent meromorphic function with the property
for every natural number , where the factorial is denoted. The motivation for defining the gamma function was to be able to extend the factorial function to real and complex arguments. The Swiss mathematician Leonhard Euler solved this problem in 1729 and defined the gamma function using an infinite product . Today the gamma function is often defined using an integral representation , which also goes back to Euler.
The gamma function is the basis of the gamma probability distribution .
Classification without prior mathematical knowledge
A math function is basically like a calculating machine . You enter a value into the function, and it then delivers a result depending on the input value, at least in theory. What this means is that the function does not count as such, but mostly just an arithmetic rule formulaic holds. A simple example of a function is the quadratic function , which multiplies the input by itself. In formulaic terms, this is written as . Thus, the quadratic function assigns the value to the number, for example . If you calculate this, the result is , so .
The gamma function is based on a rule also known as the faculty . This assigns the product of all natural numbers up to this number to a natural number. The factorial is denoted by the symbol of the exclamation mark . So, for example
In mathematics, the problem was whether this rule could also be extended to numbers of other types. In concrete terms this means:
- Can faculties also be calculated for any rational , real , complex numbers? How roughly could you imagine?
- If such “universal” rules are found, what mathematical properties can they be given? Is one of these rules particularly natural and structural? Is this particular regulation clearly defined, ie "the one" generalized faculty?
The gamma function provides the answer to these questions. For any values provides , so For example , the shift of one of the above mentioned faculty is due to a convention of the 19th century. The strategy of generalization is based on the observation that a further faculty is obtained from a previous faculty by adding a further factor. Roughly applies and very generally . Accordingly, all values of the gamma function should be in relation with. If one sets further important conditions, such as differentiability , this can finally be clearly defined, with which “the” generalized faculty is found.
for infinitely large , according to today's notation or . A few days later, on October 13th, July. / October 24th 1729 greg. , Euler also described the similar, somewhat simpler formula in a letter to Goldbach
which Gauss rediscovered in 1812 for the more general case of complex numbers (the letters mentioned were not published until 1843). As it grows, it approaches the true value for or . On January 8, 1730, Euler described the following integral for the interpolation of the faculty function in a letter to Goldbach, which he had presented to the St. Petersburg Academy on November 28, 1729 :
- in today's notation:
This definition was later preferred by Euler and goes into the form through the substitution
over. Euler discovered this integral while investigating a problem in mechanics , in which the acceleration of a particle is considered.
Adrien-Marie Legendre introduced the Greek capital letters ( gamma ) as a functional symbol in 1809 . In 1812 Gauss used the function symbol ( Pi ) in such a way that and therefore also applies to non-negative integers . However, it did not prevail; today it is used as a symbol for a product (analogous to a sum).
Definition and elementary forms of representation
There is no standard definition for the gamma function in the literature.
The Euler's integral of the second kind is often given. A disadvantage is that this integral does not converge everywhere. A global calculation using this definition is therefore only possible indirectly. For complex numbers with a positive real part , the gamma function is the improper integral
The function thus defined is holomorphic , since the integral (due to the rapid decay of the exponential function) converges uniformly on compact sets. This enables the use of Weierstrass' convergence theorem . By meromorphic continuation can eventually for all values calculated.
Another representation by means of a product motivates the generalization of the faculty in a direct way. It is given by:
In his book Number Theory. Analytic and modern tools. Henri Cohen gives a definition using the Hurwitz zeta function . The reasons given for this are a “simple possibility of generalization” and the “emphasis on important formulas”. It therefore applies to complex numbers with a positive real part
where the derivative is formed with respect to the first variable.
Functional equation and meromorphism
The gamma function satisfies in its domain for all the functional equation
An inductive continuation (for example of Euler's integral) is possible by means of this relation . It applies to everyone :
It can be concluded that an in is entirely holomorphic and has simple poles in all points . For the residual there applies
Hölder's theorem ( Otto Hölder 1886) is a negative result and says that the gamma function does not satisfy an algebraic differential equation whose coefficients are rational functions . That is, there is no differential equation of the form with a nonnegative integer and a polynomial in whose coefficients are rational functions of and the solution .
Continuation of the faculty
The conditions and , which uniquely describe the faculty for natural numbers, are also fulfilled by other analytical functions than the gamma function. For example, it fulfills the function for positive
for the characteristic conditions of the gamma function. Weierstrass therefore added the necessary and sufficient condition in 1854
- A function in this area is exactly the same as the gamma function if:
These axioms are the starting point for Nicolas Bourbaki's presentation of the theory of the gamma function.
- A holomorphic function , defined on a domain containing the stripe , is equal to the gamma function on if and only if:
- is on the strip is limited, that is, there exists a so for all of .
More precisely applies to everyone with .
Other forms of representation
In addition to the representation of the gamma function from the definition, there are other equivalent representations. A direct definition of for all is given by the product representation of the gamma function according to Gauss ,
The integral representation from the definition also goes back to Euler 1729, it applies more generally to complex numbers with a positive real part:
By decomposing this integral, EF Prym in 1876 concluded a completely valid representation:
Another variant of Euler's integral representation is available for with :
From this representation, for example, the Fresnel integral formulas can be derived in an elegant way.
It is also called the Kummer series. Carl Johan Malmstén found a similar series as early as 1846 :
Functional equations and special values
The gamma function satisfies the functional equation
With the supplementary theorem of the gamma function (Euler 1749)
- and for
With more generally chosen , the last formula becomes Legendre's doubling formula (Legendre 1809)
This is a special case of the Gaussian multiplication formula (Gauß 1812)
- for and
Gregory Chudnovsky showed in 1975, that each of the numbers , , , , and transcendental and algebraically independent of is. In contrast, it is not even known whether the function value (sequence A175380 in OEIS ) is irrational .
With the lemniscate constant applies
The slope of the gamma function at point 1 is equal to the negative of the Euler-Mascheroni constant :
Alternatively, they can be added directly to the formula
read off. Since it has no zeros, it is a whole function .
Connection with the Riemann schen function
and the following statement in relation: The term "remains unchanged when in turned is" so
Stirling's formula provides approximate values of the gamma function for , among other things , it applies
From the functional equation
which includes some kind of periodicity can be recursively calculated from known function values in a strip of width 1 into the values in every other corresponding strip. With
one can get from one strip to the neighboring one with a smaller real part, and the -fold. Since there are very good approximations for large , their accuracy can be transferred to areas in which direct application of the relevant approximation would not be advisable. According to Rocktäschel, as already noted by Carl Friedrich Gauß , the asymptotic expansion derived from the Stirling formula in
This has an irregularity in the close range , but is already usable. With the correction term , your error is reduced to the order of magnitude for unlimited growth .
The -fold application of this approximation leads to
The complex logarithm is calculated using the polar representation of . For most applications, such as wave propagation, should be sufficient.
Incomplete gamma function
In the literature, this term is not used uniformly with regard to integration limits and normalization (regularization).
Common notations are:
- Incomplete gamma function of the upper limit
- incomplete gamma function of the lower limit
- regularized (incomplete) gamma function of the upper limit
- regularized (incomplete) gamma function of the lower limit
If one speaks of a regularized gamma function, this already implies that it is incomplete.
stands for the generalized incomplete gamma function.
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