Jump to ContentJump to Main Navigation
Symmetry of Crystals and Molecules$

Mark Ladd

Print publication date: 2014

Print ISBN-13: 9780199670888

Published to Oxford Scholarship Online: April 2014

DOI: 10.1093/acprof:oso/9780199670888.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 30 March 2017

(p.364) A9 The gamma function, Γ(n)

(p.364) A9 The gamma function, Γ(n)

Source:
Symmetry of Crystals and Molecules
Publisher:
Oxford University Press

The gamma function [1,2] may be defined by the equation

(A9.1)
Γ(n)=0xnexpax2dx

where a is a constant and n is a positive integer; the integrals occur in studying inter alia quantum chemistry and atomic scattering factors. The following results are useful:

  1. 1. For n > 0 and integral

    (A9.2)
    Γn=n1!

  2. 2. For n > 0

    (A9.3)
    Γn+1=nΓn

    and if n is also integral

    (A9.4)
    Γ(n+1)=n!

  3. 3.

    (A9.5)
    Γ1/2=π

Example A9.1

Consider the solution of the integral

I=0x4expx2/2dx

Let x2/2 = t, so that x = (2t)1/2 and dx = (2t)−1/2dt. Then

I=220t3/2exptdt=22Γ5/2=3π/21/2

Occasionally, the reduction formula hereunder is useful:

(A9.6)
xnexpaxdx=xnexpaxanaxn1expaxdx