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Bonding, Structure and Solid-State Chemistry$

Mark Ladd

Print publication date: 2016

Print ISBN-13: 9780198729945

Published to Oxford Scholarship Online: May 2016

DOI: 10.1093/acprof:oso/9780198729945.001.0001

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(p.401) A7 Gamma Function Γ‎‎(n)

(p.401) A7 Gamma Function Γ‎‎(n)

Source:
Bonding, Structure and Solid-State Chemistry
Author(s):

Mark Ladd

Publisher:
Oxford University Press

The gamma function [1; 2] is useful in handling integrals of the type:

(A7.1)
0xnexpax2dx

where a is a constant and n is a positive number. The integrals occur in studying inter alia quantum chemistry and atomic scattering factors. The gamma function may be defined by the equation:

(A7.2)
Γ(n)=0xn1expax2dx

The following results are useful.

  1. 1. For n > 0 and integral:

    (A7.3)
    Γ(n)=(n1)!

  2. 2. For n > 0 and not necessarily integral:

    (A7.4)
    Γ(n+1)=nΓ(n)

    but if n is also integral, then:

    (A7.5)
    Γ(n+1)=n!(which follows also from 1)
    (A7.6)
    3.Γ(1/2)=π

(p.402) The reduction formula can also be useful:

(A7.7)
xnexpaxdx=xnexpaxanaxn1exp(ax)dx

References A7

Bibliography references:

[1] Margenau H and Murphy GM. The Mathematics of Physics and Chemistry. van Nostrand, 1943.

[2] eFunda Inc. Γ‎‎-function finder, 2015. http://www.efunda.com/math/gamma/findgamma.cfm