Gamma function

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Graph of the gamma function in real terms
Complex gamma function: the brightness corresponds to the amount, the color to the argument of the function value. In addition, contour lines of constant magnitude are drawn.
Amount of the complex gamma function

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.

It then applies with the circle number . This relationship can be explained using the Gaussian normal distribution .


The earliest definition of the gamma function is that given in a letter from Daniel Bernoulli to Christian Goldbach on October 6, 1729:

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.

Global properties

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

Holder's theorem

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 .

Axiomatic characterization

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

added, which did not end the search for the most elementary or natural characterizing property possible. In 1931 Emil Artin discussed the possible characterization by functional equations .

Bohr-Mollerup's theorem

Bohr-Mollerup's theorem ( Harald Bohr and Johannes Mollerup 1922) allows a simple characterization of the gamma function:

A function in this area is exactly the same as the gamma function if:
  1. is logarithmically convex , that is, is a convex function .

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

Wielandt's theorem

Wielandt's theorem on the gamma function ( Helmut Wielandt 1939) characterizes the gamma function as a holomorphic function and states:

A holomorphic function , defined on a domain containing the stripe , is equal to the gamma function on if and only if:
  1. 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 ,

which was already given by Euler in 1729 for positive real numbers . The representation of as a Weierstrass product is derived from this :

with Euler's constant . The second product is commonly referred to as the Weierstrass representation, but Karl Weierstrass only used the first.

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.

Ernst Eduard Kummer gave the Fourier expansion of the logarithmic gamma function in 1847 :


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)


is obtained in particular (sequence A002161 in OEIS ) and

    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

(Follow A068466 in OEIS ).

The slope of the gamma function at point 1 is equal to the negative of the Euler-Mascheroni constant :

The gamma function has first order poles at the points . The residuals are obtained from the functional equation

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

Bernhard Riemann introduced the gamma function with the Riemann ζ function via the formula in 1859

and the following statement in relation: The term "remains unchanged when in turned is" so

Approximate calculation

Stirling's formula

Stirling's formula provides approximate values ​​of the gamma function for , among other things , it applies


Recursive approximation

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.

See also


  • Niels Nielsen : Handbook of the theory of the gamma function. BG Teubner, Leipzig 1906 ( the Internet Archive , ditto , ditto ).
  • ET Whittaker , GN Watson : The Gamma Function. Chapter 12 in A course of modern analysis. Cambridge University Press, 4th Edition 1927; New edition 1996, ISBN 0-521-58807-3 , pp. 235–264 (English; in the internet archive ).
  • Emil Artin : Introduction to the theory of the gamma function. BG Teubner, Leipzig 1931; The gamma function. Holt, Rinehart and Winston, New York 1964 (English translation by Michael Butler).
  • Friedrich Lösch, Fritz Schoblik: The faculty (gamma function) and related functions. With special consideration of their applications. BG Teubner, Leipzig 1951.
  • Philip J. Davis : Leonhard Euler's integral: A historical profile of the gamma function. The American Mathematical Monthly 66, 1959, pp. 849-869 (English; awarded the Chauvenet Prize in 1963 ; from MathDL ).
  • Konrad Königsberger : The gamma function. Chapter 17 in Analysis 1. Springer, Berlin 1990; 6th edition 2003, ISBN 3-540-40371-X , pp. 351-360.
  • Reinhold Remmert : The gamma function. Chapter 2 in Function Theory 2. Springer, Berlin 1991; with Georg Schumacher: 3rd edition 2007, ISBN 978-3-540-40432-3 , pp. 31-73 .
  • Eberhard Freitag , Rolf Busam: The gamma function. Chapter 4.1 in Function Theory 1. Springer, Berlin 1993; 4th edition 2006, ISBN 3-540-31764-3 , pp. 194-212 .

Web links

Individual evidence

  1. ^ Letter ( JPG file, 136 kB) from Daniel Bernoulli to Christian Goldbach dated October 6, 1729, printed in Paul Heinrich Fuss (ed.): Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle. (Volume 2), St.-Pétersbourg 1843, pp. 324-325 (French).
  2. Peter Luschny: Interpolating the natural factorial n! or The birth of the real factorial function (1729-1826). (English).
  3. a b Letter ( PDF file, 118 kB) from Leonhard Euler to Christian Goldbach of October 13, 1729, printed in Paul Heinrich Fuss (ed.): Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle. (Volume 1), St.-Pétersbourg 1843, pp. 3-7 (Latin).
  4. ^ A b Carl Friedrich Gauß : Disquisitiones generales circa seriem infinitam 1 +… Pars I. (January 30, 1812), Commentationes Societatis Regiae Scientiarum Gottingensis recentiores 2 (classis mathematicae), 1813, p. 26 (Latin; also in Gauß: Werke Volume 3. p. 145 ).
  5. ^ Letter ( PDF file, 211 kB) from Leonhard Euler to Christian Goldbach dated January 8, 1730, printed in Paul Heinrich Fuss (ed.): Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle. Volume 1, St.-Pétersbourg 1843, pp. 11-18 (Latin).
  6. ^ A b Leonhard Euler : De progressionibus transcendentibus, seu quarum termini generales algebraice dari nequeunt. (November 28, 1729), Commentarii academiae scientiarum imperialis Petropolitanae 5, 1738, pp. 36-57 (Latin).
  7. ^ Leonhard Euler : De evolutione integralium per producta infinita. ( PDF file, 1.2 MB), Chapter 9 in Part 1 of the first volume by Euler: Institutionum calculi integralis. 1768, pp. 225-250 (Latin).
  8. ^ Adrien-Marie Legendre : Research on diverse sortes d'intégrales définies. (November 13, 1809), Mémoires de la classe des sciences mathématiques et physiques de l'Institut de France 10, 1809, p. 477 (French).
  9. ^ Adrien-Marie Legendre : Traité des fonctions elliptiques et des intégrales Eulériennes. (Volume 2), Huzard-Courcier, Paris 1826, p. 365 (French).
  10. O. Holder : About the property of the gamma function not to satisfy any algebraic differential equation. June 26, 1886, Mathematische Annalen 28, 1887, pp. 1-13 .
  11. Steven B. Bank, Robert P. Kaufman: A note on Holder's theorem concerning the Gamma function. Mathematische Annalen 232, 1978, pp. 115-120 (English).
  12. Karl Weierstrasse : About the theory of the analytical faculties. (May 20, 1854), Journal for Pure and Applied Mathematics 51, 1856, p. 36 .
  13. Nielsen: Handbook of the theory of the gamma function. 1906, p. 3 .
  14. ^ Davis: Leonhard Euler's integral: A historical profile of the gamma function. 1959, p. 867.
  15. Artin: Introduction to the theory of the gamma function. 1931, pp. 31-35.
  16. Harald Bohr , Johannes Mollerup : Lærebog i matematisk Analyze III. (Textbook of Mathematical Analysis III), Jul. Gjellerups Forlag, København (Copenhagen) 1922 (Danish).
  17. Artin: Introduction to the theory of the gamma function. 1931, pp. 12-13.
  18. ^ N. Bourbaki : Éléments de mathématique IV. Fonctions d'une variable réelle. Hermann, Paris 1951 (French).
  19. Konrad Knopp : Function Theory II. (5th edition), de Gruyter, Berlin 1941, pp. 47–49.
  20. ^ Reinhold Remmert : Wielandt's theorem about the-function. The American Mathematical Monthly 103, 1996, pp. 214-220 (English).
  21. ^ Letter from Carl Friedrich Gauß to Friedrich Wilhelm Bessel dated November 21, 1811, printed in Arthur Auwers (ed.): Correspondence between Gauss and Bessel, Wilhelm Engelmann, Leipzig 1880, pp. 151–155 (excerpt in Gauß: Werke. Volume 10.1. Pp. 362-365).
  22. O. Schlömilch : Some about the Eulerian integrals of the second kind. Archive of Mathematics and Physics 4, 1844, p. 171 .
  23. Remmert: The gamma function. Chapter 2 in Function Theory 2. 2007, p. 39 .
  24. ^ E. Freitag, R. Busam: Function theory 1. Springer-Verlag, ISBN 3-540-31764-3 , page 225.
  25. See Remmert: Function Theory 2. Chapter 2, p. 51.
  26. ^ EE Kummer : Contribution to the theory of function . Journal for pure and applied mathematics 35, 1847, p. 4 .
  27. ^ CJ Malmstén : De integralibus quibusdam definitis, seriebusque infinitis. (May 1, 1846), Journal for pure and applied mathematics 38, 1849, p. 25 (Latin).
  28. Ia. V. Blagouchine: Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results. 2014, Ramanujan J., 35 (1), 21-110, doi : 10.1007 / s11139-013-9528-5 . Erratum-Addendum doi : 10.1007 / s11139-015-9763-z
  29. L. Euler : Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques. (1749), Histoire de l'Académie Royale des Sciences et Belles-Lettres 17 (1761), 1768, pp. 96/97 (French).
  30. L. Euler: Evolutio formulae integralis integratione a valore x = 0 ad x = 1 extensa. July 4, 1771, Novi commentarii academiae scientiarum imperialis Petropolitanae 16, 1772, p. 121 (Latin).
  31. ^ Adrien-Marie Legendre : Research on diverse sortes d'intégrales définies. (November 13, 1809), Mémoires de la classe des sciences mathématiques et physiques de l'Institut de France 10, 1809, p. 485 (French).
  32. Carl Friedrich Gauß : Disquisitiones generales circa seriem infinitam 1 +… Pars I. January 30, 1812, Commentationes Societatis Regiae Scientiarum Gottingensis recentiores 2 (classis mathematicae), 1813, p. 30 (Latin; also in Gauß: Werke. Volume 3, P. 150 ).
  33. ^ Steven R. Finch: Mathematical constants. Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , p. 33 (English).
  34. Gregory V. Chudnovsky : Contributions to the theory of transcendental numbers. American Mathematical Society, 1984, ISBN 0-8218-1500-8 , p. 8 (English).
  35. Bernhard Riemann : About the number of prime numbers under a given size. (October 19, 1859), monthly reports of the Royal Prussians. Academy of Sciences in Berlin, 1860, pp. 671–680 .
  36. ^ Paul Eugen Böhmer: Difference equations and certain integrals. KF Koehler, Leipzig 1939, p. 108.
  37. Otto Rudolf Rocktäschel: Methods for calculating the gamma function for complex argument. Dissertation, Dresden 1922, p. 14.
  38. ^ Karl Rawer : Wave Propagation in the Ionosphere. Kluwer Academic Publishers, Dordrecht 1993, ISBN 0-7923-0775-5 (English).