gamma function

gamma function

\PMlinkescapephrase

generated by

Introduction

The

gamma function

can be thought of as
the natural way to generalize the

concept

of the factorial
to non-integer

arguments

.

Leonhard Euler (

http://planetmath.org/EulerLeonhard

)
came up with a

formula

for such a

generalization

in 1729.
At around the same time,
James Stirling independently arrived at a different formula,
but was unable to show that it always converged.
In 1900, Charles Hermite showed that the formula given by Stirling does work,
and that it defines the same

function

as .

Definitions

Γ

⁢

(

z

)

=

lim

n

→

∞

⁡

n

z

⁢

n

!

∏

k

=

0

n

(

z

+

k

)

.

However, it is now more commonly defined by

Γ

⁢

(

z

)

=

∫

0

∞

e

-

t

⁢

t

z

-

1

⁢

𝑑

t

for

z

∈

ℂ

with

Re

⁡

(

z

)

>

0

,
and by

analytic continuation

for the rest of the complex plane,
except for the non-positive integers (where it has

simple poles

).

Another

equivalent

definition is

Γ

⁢

(

z

)

=

e

-

γ

⁢

z

z

⁢

∏

n

=

1

∞

(

1

+

z

n

)

-

1

⁢

e

z

/

n

,

where

γ

is Euler’s constant.

Functional equations

The gamma function

satisfies

the functional equation

Γ

⁢

(

z

+

1

)

=

z

⁢

Γ

⁢

(

z

)

except when

z

is a non-positive integer.
As

Γ

⁢

(

1

)

=

1

, it follows by

induction

that

Γ

⁢

(

n

)

=

(

n

-

1

)

!

for positive integer values of

n

.

Another functional equation satisfied by the gamma function is

Γ

⁢

(

z

)

⁢

Γ

⁢

(

1

-

z

)

=

π

sin

⁡

π

⁢

z

for non-integer values of

z

.

Approximate values

The gamma function for real

z

looks like this:

(generated by GNU Octave and gnuplot)

It can be shown that

Γ

⁢

(

1

/

2

)

=

π

.
Approximate values of

Γ

⁢

(

x

)

for some other

x

∈

(

0

,

1

)

are:

Γ

⁢

(

1

/

5

)

≈

4.5908

Γ

⁢

(

1

/

4

)

≈

3.6256

Γ

⁢

(

1

/

3

)

≈

2.6789

Γ

⁢

(

2

/

5

)

≈

2.2182

Γ

⁢

(

3

/

5

)

≈

1.4892

Γ

⁢

(

2

/

3

)

≈

1.3541

Γ

⁢

(

3

/

4

)

≈

1.2254

Γ

⁢

(

4

/

5

)

≈

1.1642

If the value of

Γ

⁢

(

x

)

is known for some

x

∈

(

0

,

1

)

,
then one may

calculate

the value of

Γ

⁢

(

n

+

x

)

for any integer

n

by making use of the formula

Γ

⁢

(

z

+

1

)

=

z

⁢

Γ

⁢

(

z

)

.
We have

Γ

⁢

(

n

+

x

)

=

(

n

+

x

-

1

)

⁢

Γ

⁢

(

n

+

x

-

1

)

=

(

n

+

x

-

1

)

⁢

(

n

+

x

-

2

)

⁢

Γ

⁢

(

n

+

x

-

2

)

⋮

=

(

n

+

x

-

1

)

⁢

(

n

+

x

-

2

)

⁢

⋯

⁢

(

x

)

⁢

Γ

⁢

(

x

)

which is easy to calculate if we know

Γ

⁢

(

x

)

.

References

1

Julian Havil,

Gamma: Exploring Euler’s Constant

,
Princeton University Press, 2003.
(Chapter 6 is about the gamma function.)

Title

gamma function

Canonical name

GammaFunction

Date of creation

2013-03-22 12:00:39

Last modified on

2013-03-22 12:00:39

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

44

Author

yark (2760)

Entry type

Definition

Classification

msc 30D30

Classification

msc 33B15

Synonym

gamma-function

Synonym

Γ

-function

Synonym

Euler’s gamma function

Synonym

Euler’s gamma-function

Synonym

Euler’s

Γ

-function

Related topic

BohrMollerupTheorem

Related topic

TableOfLaplaceTransforms

Generated on Fri Feb 9 20:18:39 2018 by

LaTeXML