Guide alphabétique, de la thermodynamique, amplification de chaleur, transfert de masse, et dynamique des fluides
Français English Русский 中文 Português Español Deutsch À propos Comité de rédaction Contactez-nous Accès Begell House
View in A-Z Index

The gamma function, one of the special functions introduced by L. Euler (1927), is the extension of the factorial into fractional and complex values of the argument and can be obtained as a solution of the equation

The function was first difined (Euler) by the integral

for the values of a complex argument z with positive real part (Re z > 0). It is widely used in analytical solutions of equations of mathematical physics by the integral transformation method, in particular, when applying the Laplace transform to the function written (approximated) as a power series in time.

For a whole value of an argument

For a fractional value of an argument 0 ≤ x ≤ 1

where a0 = 1; a1 = –.57486; a2 = –.95124; a3 = –.69986; a4 = –.42455; a5 = –.10107; |ε(x)| ≤ 5×10−5.

Other useful properties of the Gamma function, which reflect various expansions into series, asymptotic and approximating expressions, etc. are given in handbooks on special functions.

Out of the related special functions we shall note a polygamma function

and an incomplete gamma function

which have a number of known asymptotic and other properties.

REFERENCES

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions, National Bureau of Standards, Appl. Math. Series-55.

References

  1. Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions, National Bureau of Standards, Appl. Math. Series-55. DOI: 10.1119/1.1972842
Nombre de vues : 17508 Article ajouté : 2 February 2011 Dernière modification de l'article : 14 February 2011 © Copyright 2010-2021 Retour en haut de page
Index A-Z Auteur / Rédacteurs Carte sémantique Galerie visuelle Contribuez Guest