Gamma function brief review

1. Gamma Function
&
its applications
Fatemeh Habibi

2. CONTENTS OF THIS PRESENTATION
-Introduction
-History
-The Gamma function
-Other Gamma function definitions
-Properties of the Gamma function
-The Beta function
-Applications
-References
2

3. The Gamma function may
be regarded as
generalization of n!, where
n is any positive integer to
x!, where x is any real
number.
The common method for
determining the value of n!
is naturally recursive, found
by multiplying
1∗2∗3∗...∗(n−2)∗(n−1)∗n,
though this is terribly
inefficient for large n.
3
Introduction

4. The Gamma Function was first introduced by the Swiss
mathematician Leonhard Euler (1707 – 1783). His goal was
to generalize the factorial to non-integer values. Later, it
was studied by Adrien-Marie Legendre (1752-1833), Carl
Friedrich Gauss (1777-1855), Christoph Gudermann (1798-
1852), Joseph Liouville (1809 – 1882), Karl Weierstrass
(1815- 1897), Charles Hermite (1822-1901),…and others.
4
History
LEONARDO EULER
ADRIEN
LEGENDRE

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The problem of extending the factorial to non-integer
arguments was apparently first considered by Daniel
Bernoulli and Christian Goldbach in the 1720s, and was
solved at the end of the same decade by Leonhard Euler.
Euler gave two different definitions: the first was not his
integral but an infinite product,
of which he informed Goldbach in a letter dated October
13, 1729. He wrote to Goldbach again on January 8, 1730,
to announce his discovery of the integral representation

6. Some important characteristics of the gamma function:
1) For x ∈{N},Γ(x) = x!
2) Γ(x + 1) = xΓ(x)
3) ln(Γ(x)) is convex
6
The Gamma
function
Table of values
Graph of the Gamma function

7. The Laplace transform of the Gamma function:
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Euler’s second Integral:
Functional equation:
These equations showed that why the gamma function can be seen as an
extension of the factorial function to real non null positive numbers.

8. Gauss’s formula:
Other Gamma
function definitions
Friedrich Gauss
Weierstrass’s formula:
Weierstrass

9. The complement
formula
There is an important
identity connecting
the gamma function
at the complementary
values x and 1−x.
Duplication and
Multiplication formula
Stirling’s formula
It’s of interest to
study how the gamma
function behaves
when the argument x
becomes large. If we
restrict the argument
x to integral values n,
the following result,
due to James Stirling
(1692-1730) and
Abraham de Moivre
(1667-1754) is
famous and of great
importance :
9
Properties of the
Gamma function

10. “
Let us now consider the useful and related function to the
gamma function which occurs in the computation of many
definite integrals. It’s defined, for x>0 and y>0 by the two
equivalent identities:
This definition is also valid for complex numbers x and y such as R(x) >
0 and R(y) > 0 and Euler in 1730. The name beta function was
introduced for the first time by Jacques Binet (1786-1856) in 1839 and
he made various contributions on the subject. The beta function is
symmetric and may be computed by mean of the gamma function
thanks to the important property :
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The Beta function
Jacques Binet

11. Volume Of The N-Dimensional Ball:
The Largest Unit Ball in Any Euclidean Space, Jeffrey Nunemacher lays down the basis for one interesting application of the
gamma function, though he never explicitly uses the gamma function [3]. He first defines the open ball of radius r of
dimension n, Bn(r), to be the set of points such that, for 1 ≤ j ≤ n,
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Applications
Its volume will be referred to as Vn(r). In an argument that he describes as being “accessible to a multivariable calculus
class”, Nunemacher uses iterated integrals to derive his formula. He notes that, by definition:
By applying equation 1 to the limits of the iterated integral in and performing trigonometric substitutions, he gets the
following - more relevant - identity, specific to the unit ball, where r = 1:

12. Psi And Polygamma Functions:
￮ Definition : The psi or digamma function denoted Ψ(x) is
defined for any non null or negative integer by the logarithmic
derivative of Γ(x), that is :
￮ For polygamma function :
￮ Use In The Computation Of Infinite Sums. on some
analytical applications of the gamma, digamma, and
polygamma functions, van der Laan and Temme state: “An
infinite series whose general term is a rational function in the
index may always be reduced to a finite series of psi and
polygamma functions.
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13. References
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G.B. Arfken, H.J. Weber, “Mathematical Methods for
Physicists”, Academic Press, Fifth Ed., 2001
M.AbramowitzandI.Stegun, Handbook of Mathematical
Functions, Dover, New York, (1964)
G.E. Andrews, R. Askey and R. Roy, Special functions,
Cambridge University Press, Cambridge, (1999)
E.W. Barnes, The theory of the gamma function,
Messenger Math. (2), (1900), vol. 29, p. 64-128.
M. Godefroy, La fonction Gamma ; Th´eorie, Histoire,
Bibliographie, Gauthier-Villars, Paris, (1901)

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Thanks!
Any questions?
Fatemeh.habibi75@ut.ac.ir
Fatemeh Habibi, School of ECE ,University of TEHRAN