Ојлеров интеграл

Из Википедије, слободне енциклопедије

У математици, постоје два типа Ојлеровог интеграла:

  1. Ојлеров интеграл прве врсте: Бета-функција
     \Beta(x,y)= \int_0^1t^{x-1}(1-t)^{y-1}\,dt =\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
  2. Ојлеров интеграл друге врсте: Гама-функција
    
\Gamma(z) = \int_0^\infty  t^{z-1}\,e^{-t}\,dt

За позитивне цијеле бројеве m и n

\Beta(n,m)= {(n-1)!(m-1)! \over (n+m-1)!}={n+m \over nm{n+m \choose n}}
\Gamma(n) = (n-1)! \,

Види још[уреди]

Литература[уреди]

  • Milton Abramowitz and Irene Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.
  • I.S. Gradshteyn (И. С. Градштейн), I.M. Ryzhik (И. М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)
  • A.P. Prudnikov (А. П. Прудников), Yu.A. Brychkov (Ю. А. Брычков), O.I. Marichev (О. И. Маричев). Integrals and Series. First edition (Russian), volume 1–5, Nauka, 1981−1986. First edition (English, translated from the Russian by N.M. Queen), volume 1–5, Gordon & Breach Science Publishers/CRC Press, 1988–1992, ISBN 2-88124-097-6. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
  • Yu.A. Brychkov (Ю. А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 1-58488-956-X.
  • Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. (Many earlier editions as well.)
  • Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln]
  • Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899)