Gamma function

1
Gamma Function
SOLO HERMELIN
Updated 28.10.12http://www.solohermelin.com

2
SOLO
TABLE OF CONTENT
Gamma Function
Gamma Function History
Gamma Function: Euler’s Second Integral
Properties of Gamma Function
Other Gamma Function Definitions: Gauss’ Formula
Some Special Values of Gamma Function:
Bohr-Mollerup-Artin Theorem
Other Gamma Function Definitions: Weierstrass’ Formula
Differentiation of Gamma Function
Beta Function: Euler’s First Integral
Euler Reflection Formula
Duplication and Multiplication Formula
Stirling Approximation Formula
References

3
SOLO
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
Leonhard Euler
)1707–1783(
( ) ( ) 0ln
1
0
1
>−=Γ ∫
=
=
−
xtdtz
t
t
x
Adrien-Marie Legendre
)1752–1833(
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,
∏
∞
=
+






+
=
1
1
1
1
!
k
n
k
n
k
n
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
Gamma Function

4
SOLO
Leonhard Euler
)1707–1783(
( ) ( ) 0ln
1
0
1
>−=Γ ∫
=
=
−
xtdtz
t
t
x
During the years 1729 and 1730, Euler introduced the following
analytic function,
By changing of variables we can obtain more known forms
( ) ( ) ( ) 0ln
0
10
1
1
0
1
>=−=−=Γ ∫∫∫
∞=
=
−=
∞=
−−
=
−=
=
=
−
−
−
xtd
e
t
tdetuduz
t
t
t
xt
t
tx
eu
dtedu
u
u
x
t
t
( ) ( ) ( )
( ) 022ln
0
12
0
12
2
1
0
1 22
2
2
>=−=−=Γ ∫∫∫
∞=
=
−−
=
∞=
−−
=
−=
=
=
−
−
−
xtdettdettuduz
t
t
tx
t
t
tx
eu
dtetdu
u
u
x
t
t
The notation Γ (x) is due to Legendre in 1809,
while Gauss used Π (x) = Γ (x+1)
Carl Friedrich
Gauss
(1777 – 1855)
)1752–1833(
Gamma Function

5
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Proof:
Gamma Function
0& >+= xyixz
∫∫∫
∞=
=
−=
+=
−∞=
+=
−
+=
t
t
t
zt
t
t
zt
t
t
z
td
e
t
td
e
t
td
e
t
1
11
0
1
0
1
For the first part:
x
t
xx
t
x
tdttd
e
t
td
e
t x
t
t
t
x
t
t
x
et
t
t
yixt
t
t
z t
1
lim
111
0
1
0
1
0
1
11
0
11
0
1
=−==≤=
+→
=
+=
=
+=
−
>=
+=
−+=
+=
−
∫∫∫
The first integral converges for any x ≥ δ > 0.
For the second integral, using integration by parts:
( ) ( )
( ) ( ) ( )( ) ∫
∫∫∫∫
∞=
=
−
∞=
=
−−
=
=
∞=
=
−
∞=
=
−−
=
=
∞=
=
−∞=
=
−+∞=
=
−
−−+−−+=
−+−===
−
−
−
−
t
t
t
x
e
t
t
tx
edv
tu
t
t
t
x
e
t
t
tx
edv
tu
t
t
t
xt
t
t
yixt
t
t
z
td
e
t
xxetx
e
td
e
t
xettd
e
t
td
e
t
td
e
t
t
x
t
x
1
3
/1
1
2
1
2
/1
1
1
1
1
1
1
1
1
211
1
1
2
1


Euler’s Second Integral
Gamma integral is defined, and
converges uniformly for x > 0.
Gamma Function

6
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Proof (continue):
Gamma Function
0& >+= xyixz
For the second integral, using integration by parts:
( ) ( )
( ) ( ) ( )( ) ∫
∫∫∫∫
∞=
=
−
∞=
=
−−
=
=
∞=
=
−
∞=
=
−−
=
=
∞=
=
−∞=
=
−+∞=
=
−
−−+−−+=
−+−===
−
−
−
−
t
t
t
x
e
t
t
tx
edv
tu
t
t
t
x
e
t
t
tx
edv
tu
t
t
t
xt
t
t
yixt
t
t
z
td
e
t
xxetx
e
td
e
t
xettd
e
t
td
e
t
td
e
t
t
x
t
x
1
3
/1
1
2
1
2
/1
1
1
1
1
1
1
1
1
211
1
1
2
1


After [x] (the integer defined such that x-[x] < 1) such integration the power of t in
the integrand becomes x-[x]-1 < 0. and we have:
( )( ) [ ]( ) [ ]( ) ( )( ) [ ]( ) ∞<−−−<−−− ∫∫
∞=
=
∞=
=
−−
t
t
t
t
t
txx
td
e
xxxxtd
et
xxxx
11
1
1
21
1
21 
Therefore the Gamma integral is defined, and converges uniformly for x > 0.
Gamma integral is defined, and
q.e.d.
Gamma Function
Return to Table of Content

7
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Proof :
Gamma Function
0& >+= xyixz
( ) ( )zzz Γ=+Γ 1
( )  ( ) ( ) ( )zztdetztdtzeettdetz
t
t
tz
t
t ud
z
v
t
v
t
u
z
dtedvtu
partsby
t
t
tz
tz
Γ=+=−−−==+Γ ∫∫∫
∞=
=
−−
∞=
=
−−
∞
−
==∞=
=
−
−
0
1
0
1
0
,
nintegratio
0
01

Properties of Gamma Function : 1
Note that for the evaluation of Gamma Function for a Positive Real Number
we need to know only the value of Γ (x) for 0 < x < 1
( ) ( ) ( ) ( ) ( )xxxnxnxnx Γ+−+−+=+Γ 121 
( ) ( )
( ) ( ) ( )121 −+−++
+Γ
=Γ
nxnxxx
nx
x

For x < 0 with –n < x < -n+1 or 0 < x+n < 1, we define
We can see that for x = 0 or a negative integer the
denominator of the right side is zero, and so Γ (x) is
undefined (goes to infinity)
Gamma Function
( ) ,2,1,0!1 ==+Γ nnn

8
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Proof :
Gamma Function
( ) ( )
( )!1
1
Residue
1
1 −
−
=Γ
−
+−→ n
z
n
nz
Residues of Gamma Function at x = 0,-1, -2,---,-n,..:
( ) ( )
( ) ( ) ( )121 −+−++
+Γ
=Γ
nxnxxx
nx
x

q.e.d.
( ) ( ) ( )
( ) ( ) ( )
( )

( )( ) ( )
( )
( )!1
1
121
1
121
1limResidue
1
1
11
−
−
=
−+−+−
Γ
=
−+−++
+Γ
−+=Γ
−
+−→+−→
nnn
nxnxxx
nx
nxx
n
nxnx


Gamma Function

9
SOLO
Gamma Function Γ (x) and its Inverse 1/Γ (x)
Gamma Function

10
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Absolute value |Γ (z)|
Real value ReΓ (z)
Imaginary value ImΓ (z)
Gamma Function

11
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Absolute value |Γ (z)|
Gamma Function

12
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
( ) ( )zzz Γ=+Γ 1
Let compute
( ) 11
0
0
=−==Γ
∞−
∞=
=
−
∫
t
t
t
t
etde
Therefore for any n positive integer:
( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )!1122112111 −=Γ−−=−Γ−−=−Γ−=Γ nnnnnnnnn 
Properties of Gamma Function : 1
2
q.e.d.
Gamma Function

13
SOLO Primes
Second definition identical to First
( )[ ] ( ) ( ) ( ) ( ) ( )bayxallyfxfyxf ,,1,011 ∈∈−+≤−+ λλλλλ
Convex Function :
A Function f (x) is called Convex in an interval (a,b) if for every x,y (a,b) we haveϵ
A Function f (x), defined for x > 0, is called Convex, if the corresponding function
( ) ( ) ( )
y
xfyxf
y
−+
=φ
defined for all y > -x, y ≠ 0, is monotonic Increasing throughout the range of
definition.
If 0 < x1 < x < x2, are given by choosing y1 = x1 – x < 0, y2 = x2 – x > 0, we express
the condition of convexity as
( ) ( ) ( ) ( ) ( ) ( )
xx
xfxf
y
xx
xfxf
y
−
−
=≤
−
−
=
2
2
2
1
1
1 φφ
( ) ( )[ ] ( ) ( ) ( )[ ] ( )xxxfxfxxxfxf −−≥−− 1221
( ) ( ) ( )
( )
( ) ( )
( )
λλ −
−
−
+
−
−
≤
1
12
1
2
12
2
1
xx
xx
xf
xx
xx
xfxf
One other equivalent definition:

14
SOLO Primes
( )[ ] ( ) ( ) ( ) ( )1,0ln1ln1ln ∈−+≤−+ λλλλλ yfxfyxf
Logarithmic Convex Function :
A Function f (x)>0 is called logarithmic-convex or simply log-convex if ln (f (x) )
is convex or
This is equivalent to ( )[ ] ( ) ( )( )λλ
λλ
−
≤−+
1
ln1ln yfxfyxf
Since the logarithm is a momotonic increasing function we obtain
( )[ ] ( ) ( )( )
( ) yxyfxfyxf <∈≤−+
−
,1,01
1
λλλ
λλ

15
SOLO Primes
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Proof :
Gamma Function
0& >+= xyixz
( )[ ] ( ) ( ) ( ) ( )1,0ln1ln1ln ∈Γ−+Γ≤−+Γ λλλλλ baba
Properties of Gamma Function :
3
Gamma is a
Log Convex
Function
( )[ ] ( )
( ) ( )
( ) ( ) λλ
λλ
λλλλ
λλ
−
−∞
−−
∞
−−
∞
−−−−−
∞
−−−+
ΓΓ=















≤
==−+Γ
∫∫
∫∫
1
1
0
1
0
1
0
111
0
11
1
badtetdtet
dtetetdtetba
tbta
InequalityHolder
tbtatba
q.e.d.

16
SOLO Primes
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Proof :
Gamma Function
Other Gamma Function Definitions:
( )
( ) ( )nxxx
nn
x
x
n ++
=Γ
∞→ 1
!
limGauss’ Formula
Since the Gamma Function is monotonically increasing the logarithm of Gamma
Function is also monotonic increasing and for 0 < x < 1 and any n > 2 we have
( ) ( )
( ) nnx
nnx
−+
Γ−+Γ lnln
( )[ ] ( )[ ]
( )
( )
( ) ( )[ ] [ ] ( )[ ]
( )
    






−





−
−
−−
≤
−−+Γ
≤
−
−−−
!1
!
ln
!2
!1
ln
1
!1ln!ln!1lnln
1
!1ln!2ln
n
n
n
n
nn
x
nnxnn
( )
( )
( ) n
x
n
nx
n ln
!1
ln
1ln ≤
−
+Γ
≤−
( ) ( )
( )
1
1
ln1ln −=←
≤
−+−
Γ−+−Γ x
nn
nn ( ) ( )
( ) nn
nnx
−+
Γ−+Γ
≤
→=
1
ln1ln1
Carl Friedrich Gauss
(1777 – 1855)

17
SOLO Primes
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Proof (continue - 1) :
Gamma Function
Since the Gamma Function is monotonically increasing the logarithm of Gamma
Function is also monotonic increasing and for 0 < x < 1 and any n > 2 we have
( )
( )
( ) n
x
n
nx
n ln
!1
ln
1ln ≤
−
+Γ
≤− ( ) ( )
( )
xx
n
n
nx
n ln
!1
ln1ln ≤
−
+Γ
≤−
10 << x
( ) ( ) ( ) ( )!1!11 −≤+Γ≤−− nnnxnn xx
Use ( ) ( ) ( ) ( ) ( )xxxnxnxnx Γ+−+−+=+Γ
>
  

0
121
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) xxnxnx
nn
x
xxnxnx
nn xx
121
!1
121
!11
+−+−+
−
≤Γ≤
+−+−+
−−

( )
( ) ( )nxxx
nn
x
x
n ++
=Γ
∞→ 1
!
limGauss’ Formula
Euler 1729
Gauss 1811

18
SOLO Primes
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) xxnxnx
nn
x
xxnxnx
nn xx
121
!1
121
!11
+−+−+
−
≤Γ≤
+−+−+
−−

( ) ( ) ( )
( ) ( )
( ) ( ) ( ) xxnxnx
nn
x
xxnxnx
nn
xx
11
!1
11
!
+−++
+
≤Γ≤
+−++ 
Take the limit n → ∞
( ) ( ) ( )
( )
( ) ( ) ( ) xxnxnx
nn
n
x
xxnxnx
nn x
n
x
n
x
n 11
!
lim
1
1lim
11
!
lim
1
+−++






+≤Γ≤
+−++ ∞→∞→∞→ 


( )
( ) ( ) ( )
( )1,0
11
!
lim ∈
+−++
=Γ
∞→
x
xxnxnx
nn
x
x
n 
Substitute n+1 for n
( )
( ) ( )nxxx
nn
x
x
n ++
=Γ
∞→ 1
!
limGauss’ Formula

19
SOLO Primes
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Let substitute x + 1 for x
Gamma Function
( )
( ) ( ) ( )
( )
( )1,0
11
!
lim ∈
+−++
=Γ
Γ
∞→
x
xxnxnx
nn
x
x
x
n
n
  

q.e.d
( )
( ) ( )nxxx
nn
x
x
n ++
=Γ
∞→ 1
!
limGauss’ Formula
( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( )1,0
11
!
lim
1
lim
11
!
lim1
1
1
∈Γ=
+−++++
=
++++
=+Γ
Γ
∞→∞→
+
∞→
xxx
xxnxnx
nn
nx
n
x
xnxnx
nn
x
x
x
nn
x
n
  

  

The right side is defined for 0 < x <1. The left side extend the definition for
(1 , 2). Therefore the result is true for all x , but 0 and negative integers.

20
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Definitios:
Start from Gauss Formula ( ) ( )xx n
n
Γ=Γ
∞→
lim
q.e.d
( )
constantMascheroni-Euler57721566.0ln
1
2
1
1lim
11
≈





−+++=
+
=Γ
∞→
∞
=
−
∏
n
n
k
x
e
x
e
x
n
k
k
x
x
γ
γ
Weierstrass’ Formula
Proof :
( )
( ) ( ) ( )






+





−
+





+
=






+





−
+





+
=
+−++
=Γ






−−−−
n
x
n
xx
x
eee
e
x
x
n
x
n
x
n
xxnxnx
nn
x
n
xxx
n
nx
xx
n
1
1
1
1
1
1
1
1
11
11
!
:
21
1
2
1
1ln





( ) ( ) ∏∏
∞
=
−
=






−−−−
∞→∞→
+
=
+
=Γ=Γ
11
1
2
1
1ln
11
1
limlim
k
k
x
xn
k
k
x
n
nx
n
n
n
k
x
e
x
e
k
x
e
x
exx
γ
Karl Theodor Wilhelm
Weierstrass
(1815 – 11897)
Gamma Function

21
SOLO Primes
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Some Special Values of Gamma Function:
q.e.d
( ) π
π
====Γ ∫∫
∞=
=
−
=
=
∞=
=
−
2
222/1
0
2
0
2
2 t
t
u
ut
duudt
t
t
t
udetd
t
e
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) πn
n
nnnnn
2
12531
2/12/112/32/12/12/12/1
−⋅⋅
=Γ+−−=−Γ−=+Γ


( ) ( )
( )
( )
( )( ) ( )
( )
( )
π
12531
21
2/12/32/1
2/1
2/1
2/3
2/1
−⋅⋅
−
=
−+−+−
Γ
=
+−
+−Γ
=+−Γ
nnnn
n
n
nn

( ) π=Γ 2/1
( ) ( ) πn
n
n
2
12531
2/1
−⋅⋅
=+Γ

( ) ( )
( )
π
12531
21
2/1
−⋅⋅
−
=+−Γ
n
n
nn

Proof:

22
SOLO
Harald August Bohr
( 1887 – 1951)
Proof:
Choose n > 2, and 0 < x < 1 and let 11 +≤+<<− nxnnn
By logarithmic convexity of f (x), we get
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) nn
nfnf
nxn
nfxnf
nn
nfnf
−+
−+
≤
−+
−+
≤
−−
−−
1
ln1lnlnln
1
ln1ln
( ) ( ) ( ) ( ) ( )
1
!1ln!ln!1lnln
1
!1ln!2ln −−
≤
−−+
≤
−
−−− nn
x
nxnfnn
By the second property ( ) ( ) ( ) ( ) ( ) !1,!1,!21 nnfnnfnnf =+−=−=−
( ) ( )( ) ( ) ( )xfxxxnxnxnf 121 +−+−+=+ 
( ) ( )
( )
xx
n
n
xnf
n ln
!1
ln1ln ≤
−
+
≤−
Emil Artin
(1898 – 1962)
Hamburg University
Johannes Mollerup
(1872 – 1937)
Gamma Function
Bohr-Mollerup-Artin Theorem:
The theorem characterizes the Gamma Function, defined for x > 0 by
as the only function f (x) on the interval x > 0 that simultaneously has
the three properties
• f (1) = 1
• f (1+x) = x f (x) for x > 0
• f is logarithmically convex
or Gauss Formula( ) ∫
∞=
=
−−
=Γ
t
t
tz
tdetz
0
1
( )
( ) ( )nxxx
nn
z
x
n ++
=Γ
∞→ 1
!
lim

23
SOLO
Harald August Bohr
( 1887 – 1951)
• f (1) = 1
• f (1+x) = x f (x) for x > 0
Proof (continue-1):
By the second property ( ) ( )( ) ( ) ( )xfxxxnxnxnf 121 +−+−+=+ 
( ) ( )( ) ( ) ( )
( )
xx
n
n
xfxxxnxn
n ln
!1
121
ln1ln ≤
−
+−+−+
≤−

We found
Since lan is a monotonic increasing function, we have
( ) ( )
( ) ( )( )
( ) ( )
( ) ( )( )121
!1
121
!11
−+−++
−
≤≤
−+−++
−−
xnxnxx
nn
xf
xnxnxx
nn xx

( ) ( )( )
( )
( ) ( )( )
( )
x
xxx
n
n
xnxnxx
nn
xf
xnxnxx
nn 1
11
!
11
! +
+−++
≤≤
+−++ 
n
n
↓
−1
( ) ∫
∞=
=
−−
=Γ
t
t
tz
tdetz
0
1
( )
( ) ( )nxxx
nn
x
x
n ++
=Γ
∞→ 1
!
limor Gauss Formula
Emil Artin
(1898 – 1962)
Hamburg University
Johannes Mollerup
(1872 – 1937)
Gamma Function

24
SOLO
q.e.d.
Harald August Bohr
( 1887 – 1951)
• f (1) = 1
• f (1+x) = x f (x) for x > 0
Johannes Mollerup
(1872 – 1937)
Proof (continue - 2):
( ) ( )( )
( )
( ) ( )( )
xxx
nxnxnxx
nn
xf
xnxnxx
nn






+
+−++
≤≤
+−++
1
1
11
!
11
!

( ) ∫
∞=
=
−−
=Γ
t
t
tz
tdetz
0
1
( )
( ) ( )nxxx
nn
x
x
n ++
=Γ
∞→ 1
!
limor Gauss Formula
By taking n → ∞ we obtain
( ) ( )( )
( )
( )
( ) ( )( )
( )
  

  

1
1
1lim
11
!
lim
11
!
lim
x
n
x
x
n
x
x
n nxnxnxx
nn
xf
xnxnxx
nn






+
+−++
≤≤
+−++ ∞→
Γ
∞→
Γ
∞→
But this is possible only if
( ) ( )xxf Γ=
Emil Artin
(1898 – 1962)
Hamburg University
Gamma Function

25
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function Gamma integral is defined, and
Differentiation of Gamma Function:
q.e.d
( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( )
( ) ( )
( )
0,2
!11'
ln
0
1'''
ln
constantMascheroni-Euler57721566.0
111'
ln
1
1
1
1
22
2
2
2
1
>≥
+
−−
=
Γ
Γ
=Γ
>
+
=
Γ
Γ−ΓΓ
=Γ
≈





+
−+−−=
Γ
Γ
=Γ
∑
∑
∑
∞
=
−
−
∞
=
∞
=
xn
kx
n
x
x
xd
d
x
xd
d
kxx
xxx
x
xd
d
kxkxx
x
x
xd
d
k
n
n
n
n
n
n
k
k
γγ
Proof :
Start from Weierstrass Formula ( ) ∏
∞
=
−
+
=Γ
1
1k
k
x
x
k
x
e
x
e
x
γ
( ) ∑∑
∞
=
∞
=






+−+−−=Γ
11
1lnlnln
kk k
x
k
x
xxx γ ( ) ∑∑
∞
=
∞
=
+
−+−−=Γ
11
1
1
11
ln
kk
k
x
k
kx
x
xd
d
γ
( )
( ) ( )
0
111111
ln
0
2
1
22
1
2
2
>
+
=
+
+=











+
−+−−=Γ ∑∑∑
∞
=
∞
=
∞
= kkk kxkxxkxkxxd
d
x
xd
d
γ
( ) ( )
( )
( ) ( )
( )∑
∞
=
−
−
+
−−
=
Γ
Γ
=Γ
0
1
1
!11'
ln
k
n
n
n
n
n
n
kx
n
x
x
xd
d
x
xd
d
Gamma Function
We can see that ( ) ( )
( )
γγ −=





+
−+−−=
Γ
Γ
==Γ
+
−
=
∞→
∑
  
1
1
1
1 1
11
lim
1
1
1
1'
1ln
n
n
k
n kk
x
xd
d

26
SOLO
( ) ( )∫
=
=
−−
−=
1
0
11
1,
s
s
zy
sdsszyBBeta Function
Beta Function is related to Gamma Function:
( ) ∫∫
∞=
=
−−
=
=∞=
=
−−
==Γ
u
u
uy
duudt
utt
t
ty
udeutdety
0
12
2
0
1 2
2
2
( ) ( ) ( )
( )zy
zy
zyB
+Γ
ΓΓ
=,
Proof:
In the same way:
( ) ∫
∞=
=
−−
=Γ
v
v
vz
vdevz
0
12 2
2
( ) ( ) ( )
∫ ∫
∞=
=
∞=
=
+−−−
=ΓΓ
u
u
v
v
vuuzy
vdudevuzy
0 0
1212 22
4
Use polar coordinates:
ϕϕ
ϕϕ
ϕϕ
ϕ
ϕ
ϕ
ϕ
ϕ
drdrdrd
r
r
drd
vrv
uru
vdud
rv
ru
=
−
=
∂∂∂∂
∂∂∂∂
=



=
=
cossin
sincos
//
//
sin
cos
( ) ( ) ( )
( ) ( )
( )
( )
( ) ( ) 



















=
=ΓΓ
∫∫
∫ ∫
=
=
−−
+Γ
∞=
=
−−+
∞=
=
=
=
−−−−+
2/
0
1212
0
12
0
2/
0
121212
sincos22
sincos4
2
2
πϕ
ϕ
πϕ
ϕ
ϕϕϕ
ϕϕϕ
drder
drderzy
zy
zy
r
r
rzy
r
r
rzyzy
  
Euler’s First Integral
Gamma Function

27
SOLO
( ) ( )∫
=
=
−−
−=
1
0
11
1,
s
s
zy
sdsszyBBeta Function Euler’s First Integral
Beta Function is related to Gamma Function: ( ) ( ) ( )
( )zy
zy
zyB
+Γ
ΓΓ
=,
Proof (continue):
( ) ( ) ( ) ( ) ( ) 







+Γ=ΓΓ ∫
=
=
−−
2/
0
1212
sincos2
πϕ
ϕ
ϕϕϕ dzyzy zy
Change variables in the integral using ϕϕϕϕ dsds cossin2sin2
==
( ) ( ) ( ) ( )zyBsdssd
s
s
yzzy
,1sincos2
1
0
11
2/
0
1212
=−= ∫∫
=
=
−−
=
=
−−
πϕ
ϕ
ϕϕϕ
( ) ( ) ( ) ( )zyBzyzy ,+Γ=ΓΓTherefore q.e.d.
Use z→y and y → 1 - z
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ∫∫
∫
∞=
=
−∞=
=
−
−
−+
=
+
=
=
=
−−
+
=
+






+
−
+
=
−=−Γ=−ΓΓ
u
u
zu
u
z
z
zu
u
s
u
ud
sd
s
s
zz
ud
u
u
u
ud
u
u
u
u
dssszzBzz
0
1
0
21
11
1
1
0
1
111
1
1
11,11
2
q.e.d.
Gamma Function

28
SOLO
Proof
( ) ( ) ( ) ( )
( ) ( )
( ) ( )yzBzyzyBzyyz
yzBzyB
,,
,,
+Γ=+Γ=ΓΓ
=
Use y → 1 - z
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ∫∫
∫
∞=
=
−∞=
=
−
−
−+
=
+
=
=
=
−−
+
=
+






+
−
+
=
−=−Γ=−ΓΓ
u
u
zu
u
z
z
zu
u
s
u
ud
sd
s
s
zz
ud
u
u
u
ud
u
u
u
u
dssszzBzz
0
1
0
21
11
1
1
0
1
111
1
1
11,11
2
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Properties: ( ) ( )
( )z
zz
π
π
sin
1 =−ΓΓ Euler
Reflection Formula
Gamma Function

29
SOLO
Proof (continue - 1)
( ) ( ) ∫
∞=
=
−
+
=−ΓΓ
u
u
x
ud
u
u
xx
0
1
1
1
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Properties:
Replace the path from 0 to ∞ by the Hankel contour Hε
in the Figure, described by four paths, traveled in
counterclockwise direction:
1. going counterclockwise above the real axis, (u = |u|)
2. along the circular path CR,
3. bellow the real axis, (u= |u|e -2πi
)
4. along the circular path Cε.
∫∫∫∫ +
−
+
−
+
+
+
−−
−
−−
εε
π
ε C
yR y
yi
C
yR y
ud
u
u
ud
u
u
eud
u
u
ud
u
u
R
1111
2
Define y = 1 – x, and assume x,y (0,1)ϵ
( ) ( )
( )z
zz
π
π
sin
1 =−ΓΓ Euler
Reflection Formula
Gamma Function

30
SOLO
( ) ( ) ∫
∞=
=
−
+
=−ΓΓ
u
u
x
ud
u
u
xx
0
1
1
1
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
This path encloses the pole u=-1 of that has the residue
1+
−
u
u y
yi
eu
y
y
eu
u
u
i
π
π
−
=−=
−
−
==





+ 11
Residue
By the Residue Theorem
For z ≠ 0 we have
( ) yzyzyzyy
zeeez
−−−−−
====
lnlnReln
( ) ( )
( )z
zz
π
π
sin
1 =−ΓΓ Euler
Reflection Formula
( ) yi
y
eu
y
C
yR y
iy
C
yR y
ei
u
u
ui
u
u
izd
z
z
ud
u
u
ezd
z
z
ud
u
u
i
R
π
ε
π
ε
ππ
π
π
ε
−
−
=−→
−−−
−
−−
=











+
+=






+
=
+
−
+
−
+
+
+
−
∑∫∫∫∫
2
1
1lim2
1
Residue2
1111
1
2
Gamma Function

31
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
yi
C
yR y
iy
C
yR y
eizd
z
z
ud
u
u
ezd
z
z
ud
u
u
R
π
ε
π
ε
π
ε
−
−−
−
−−
=
+
−
+
−
+
+
+ ∫∫∫∫ 2
1111
2
For the second and forth integral we have
( )
0
lnlnReln
≠====
−−−−−
zzeeez
yzyzyzyy
z
z
z
z
z
z
yyy
−
≤
+
≤
+
−−−
111
Hence for small ε we have:
and for large R we have:
0
1
2
1
01 →−−
→
−
≤
+∫
ε
ε
ε
π
ε
y
C
y
zd
z
z
0
1
2
1
1 ∞→−−
→
−
≤
+∫
Ry
C
y
R
R
zd
z
z
R
π
Therefore the integrals on the circular paths are zero for ε→0 and R →∞
( ) ( )
( )z
zz
π
π
sin
1 =−ΓΓ Euler
Reflection Formula
Gamma Function

32
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
yi
y
iy
y
eiud
u
u
eud
u
u ππ
π −
∞ −
−
∞ −
=
+
−
+ ∫∫ 2
11 0
2
0
We obtain
Multiply both sides by yi
e π+
( ) iud
u
u
ee
y
iyiy
πππ
2
10
=
+
− ∫
∞ −
−
( ) ( )yee
i
ud
u
u
iyiy
y
π
π
π ππ
sin
2
10
=
−
=
+ −
∞ −
∫Rearranging we obtain
Since both sides of this equation are meromorphic (analytic) in x (0,1) we canϵ
extend the result for all analytic parts of z C (complex plane).ϵ
( ) ( )
( )[ ] ( )
( )1,0
sin1sin11
1
0
1
0
1
∈=
−
=
+
=
+
=−ΓΓ ∫∫
∞=
=
−−=∞=
=
−
x
xx
ud
u
u
ud
u
u
xx
u
u
yxyu
u
x
π
π
π
π
Substituting y = 1 – x we obtain
( ) ( )
( )z
zz
π
π
sin
1 =−ΓΓ Euler
Reflection Formula
Gamma Function

33
SOLO
Onother Proof
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Start with Weierstrass Gamma Formula
( ) ( )
( )z
zz
π
π
sin
1 =−ΓΓ Euler
Reflection Formula
( ) ∏
∞
=
−
+
=Γ
1 1k
k
x
x
k
x
e
x
e
x
γ
( ) ( ) ∏∏
∞
=
∞
= −
−






−−=
−+
−=
−ΓΓ 1
2
2
2
1
2
1
11
1
kk k
x
k
x
xx
k
x
x
e
k
x
e
k
x
eex
xx
γγ
Use the fact that Γ (-x)=- Γ (1-x)/x to obtain
( ) ( ) ∏
∞
=






−=
−ΓΓ 1
2
2
1
1
1
k k
x
x
xx
Now use the well-known infinite product
( ) ∏
∞
=






−=
1
2
2
1sin
k k
x
xx ππ
q.e.d.
Gamma Function

34
SOLO
Proof
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Other Gamma Function Properties: ( )z
zz
π
π
cos2
1
2
1
=





−Γ





+Γ
Start from
Substitute ½ +z instead of z
( ) ( )
( )z
zz
π
π
sin
1 =−ΓΓ
( )z
z
zz
π
π
π
π
cos
2
1
sin
2
1
2
1
=












+
=





−Γ





+Γ
q.e.d.
Gamma Function

35
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Stirling Approximation Formula:
( ) 121 >>≈+Γ −
xexxx xx
π
( )
( )
( )
( ) ( )
( )( )
∫
∫∫∫
∞=
−=
++−−+
∞=
−=
−−+
∞=
−=
+−
+=
=
∞=
−=
−
=
+=+==+Γ
u
u
uuxxx
u
u
xuxxx
u
u
xxux
uxt
udxtd
t
t
xt
udeex
udueexudxuxetdtex
1
1ln1
1
1
1
1
1
1
111
Proof:
The function f(u) = -u + ln (1 + u) equals zero for u = 0. For other values of u we have
f(u) < 0. This implies that the integrand of the last integral equals 1 at u = 0 and that this
integrand becomes very small for large values of x at other values of u. So for large values of
x we only have to deal with the integrand near u = 0. Note that we have
( ) ( ) ( ) ( ) 0
2
1
2
1
1ln 2222
→Ο+−=Ο+−+−=++−= uforuuuuuuuuuf
This implies that
( )( )
∞→≈ ∫∫
∞=
−∞=
−
∞=
−=
++−
xforduedue
u
u
ux
u
u
uux 2/
1
1ln 2
James Stirling, a contemporary of Euler, also attempted to find a continuous expression for the factorial and came up with what is now known as
Stirling's formula. Although Stirling's formula gives a good estimate of , also for non-integers, it does not provide the exact value. Extensions of his
formula that correct the error were given by Stirling himself and by Jacques Philippe Marie Binet‫ן‬
Gamma Function

36
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Stirling Approximation Formula:
( ) 121 >>≈+Γ −
xexxx xx
π
Proof (continue):
( ) ( )( )
∞→≈=+Γ ∫∫
∞=
−∞=
−−+
∞=
−=
++−−+
xfordueexudeexx
u
u
uxxx
u
u
uuxxx 2/1
1
1ln1 2
1
∞→== −
∞=
−∞=
−−
=
=
∞=
−∞=
−
∫∫ xforxdtexdue
t
t
t
xtu
xtdud
u
u
ux
π
π
22 2/12/1
/2
/2
2/ 22

If we set we have by using the normal integralxtu /2=
therefore:
( ) ∞→≈+Γ −
xexxx xx
π21
q.e.d.
Gamma Function

37
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
Duplication and Multiplication Formula:
( ) ( ) 0Re2
22
1
12
>Γ=





+ΓΓ −
zzzz z
π Legendre Duplication Formula
1809
)1752–1833(
Proof:
( ) ( ) ( ) ( )
( ) ( ) ( )2/1,2sin22sin2
2sin22sincos2,
21
2/
0
1221
0
1221
2/
0
1221
2/
0
1212
zBdd
ddzzB
zzzzz
zzzz
⋅=⋅⋅==
⋅==
−−−−−
−−−−
∫∫
∫∫
ππ
ππ
ττττ
ϕϕϕϕϕ
( ) ( )
( )
( ) ( ) ( ) ( )
( )
0Re
2/1
2/1
22/1,2,
2
2121
>
+Γ
Γ⋅Γ
⋅=⋅==
Γ
Γ⋅Γ −−
z
z
z
zBzzB
z
zz zz
We have
therefore
q.e.d( )

( ) 0Re2
22
1
12
2
1
>Γ=





+ΓΓ −






Γ
zzzz z
π
Gamma Function

38
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
( ) ( )( )
( )znn
n
n
z
n
z
n
zz znn
Γ=




 −
+Γ





+Γ





+ΓΓ −− 2/12/1
2
121
π
Gauss
Multiplication
Formula
Proof:
n
z
1
=
(1777 – 1855)( )( )
nn
n
nn
n 2/1
2121
−
=




 −
Γ





Γ





Γ
π
 Euler
Multiplication
Formula
Gamma Function
Define the function: ( ) 




 −+
Γ




 +
Γ





Γ=
n
nx
n
x
n
x
nxf x 11
: 
This function has the following properties:
1 ( )
( )xfx
n
x
n
x
n
nx
n
x
n
x
nn
n
nx
n
nx
n
x
n
x
nxf
x
x
⋅=





Γ⋅⋅




 −+
Γ




 +
Γ




 +
Γ⋅=





 +
Γ




 −+
Γ




 +
Γ




 +
Γ=+
↓
+
121
121
1 1




39
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
( ) ( )( )
( )znn
n
n
z
n
z
n
zz znn
Γ=




 −
+Γ





+Γ





+ΓΓ −− 2/12/1
2
121
π
Gauss
Multiplication
Formula
Proof (continue – 1):
(1777 – 1855)
Gamma Function
Since (ln nx
)”=(x ln n)”=(ln n)’=0, and each Γ ((x+k)/k) is log convex.
f (x) is log convex.
( ) ( ) 





Γ





Γ⋅





Γ==Γ=
n
n
nn
naaf nn 
21
11
So using Bohr-Mollerup-Artin Theorem we can write: f (x) = an Γ(x)
where an is a constant, to be found, and Γ (1)=1 (the third condition of the Theorem).
2
Therefore
Use Gauss’ Formula for Gamma Function with x=k/n
( ) ( )pnknkk
npp
p
n
k
n
k
n
k
pp
n
k pn
k
p
n
k
p ++
=






+





+
=





Γ
+
∞→∞→ 

1
!
lim
1
!
lim

40
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
( ) ( )( )
( )znn
n
n
z
n
z
n
zz znn
Γ=




 −
+Γ





+Γ





+ΓΓ −− 2/12/1
2
121
π
Gauss
Multiplication
Formula
(1777 – 1855)
Gamma Function
( ) ( )pnknkk
npp
n
k pn
k
p ++
=





Γ
+
∞→ 
1
!
lim
Since k = 1,2,…,p
( ) ( ) [ ] ( ) ( )[ ] ( ) ( )[ ]
( ) ( )( )!1!
11211
nppnn
pnnpnnnnnpnknkk
p
k
⋅+=⋅+=
⋅+⋅+++⋅⋅=⋅++∏=

( ) ( )
( )
( ) ( )
( )!
!
lim
!
!
lim
21 2
1
1
1
1
pnn
pnp
n
pnn
pnp
n
n
n
nn
na
n
pnn
p
n
n
npnn
p
n
⋅+
=
⋅+
=





Γ





Γ⋅





Γ=
+
+
∞→
++
+
∞→



41
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
( ) ( )( )
( )znn
n
n
z
n
z
n
zz znn
Γ=




 −
+Γ





+Γ





+ΓΓ −− 2/12/1
2
121
π
Gauss
Multiplication
Formula
(1777 – 1855)
Gamma Function
( ) ( )
( )!
!
lim
2
1
1
pnn
pnp
na
n
pnn
p
n
⋅+
=
+
+
∞→
Use the identity
( )
( ) ( )npp pnpn
pnn
pn
n
pnpn ⋅
⋅
⋅
⋅+
=





⋅
+





⋅
+⋅





⋅
+=
∞→∞→
1
!
!
lim1
2
1
1
1lim1 
to an to get
( ) ( )
( )
( ) ( )
( )
( )
( ) ( )
( )
( ) 2
1
2
1
12
1
1
!
!
lim
!
!
!
!
lim1
!
!
lim −
⋅
∞→
+
+
∞→
+
+
∞→
⋅
=
⋅⋅
⋅+
⋅
⋅+
=⋅
⋅+
= n
pnn
pn
n
pnn
p
n
pnn
p
n
ppn
np
n
pnpn
pnn
pnn
pnp
n
pnn
pnp
na

42
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
( ) ( )( )
( )znn
n
n
z
n
z
n
zz znn
Γ=




 −
+Γ





+Γ





+ΓΓ −− 2/12/1
2
121
π
Gauss
Multiplication
Formula
(1777 – 1855)
Gamma Function
to an to get
( )
( ) 2
1
!
!
lim −
⋅
∞→
⋅
= n
pnn
p
n
ppn
np
na
∞→≈ −
+
pepp p
p
2
1
2! π ( ) ( ) ( )
∞→⋅≈⋅ ⋅−+⋅
pepnpn pnpn
2
1
2! π
( ) ( )
( ) 2
1
2
1
2
1
2
1
2
1
2
2
2
lim n
pepn
nep
na
n
n
pnpn
pn
n
p
p
p
n
−
−
⋅−+⋅
⋅−
+
∞→
=
⋅








= π
π
π
Use Stirling’s Approximation formula ( ) ∞→≈+Γ −
xexxx xx
π21

43
SOLO
( ) ∫
∞=
=
−
=Γ
t
t
t
z
td
e
t
z
0
1
Gamma Function
( ) ( )( )
( )znn
n
n
z
n
z
n
zz znn
Γ=




 −
+Γ





+Γ





+ΓΓ −− 2/12/1
2
121
π
Gauss
Multiplication
Formula
(1777 – 1855)
Gamma Function
( ) 2
1
2
1
2 na
n
n
−
= π
( ) ( )xa
n
nx
n
x
n
x
nxf n
x
Γ=




 −+
Γ




 +
Γ





Γ=
11
: 
We have
or ( ) ( )xn
n
nx
n
x
n
x xn
Γ=




 −+
Γ




 +
Γ





Γ
+−−
2
1
2
1
2
11
π
Define x = n z to obtain
( ) ( ) ( )znn
n
n
z
n
zz
znn
Γ=




 −
+Γ





+ΓΓ
+−−
2
1
2
1
2
11
π q.e.d

44
SOLO
References
Internet
http://en.wikipedia.org/wiki/
G.B. Arfken, H.J. Weber, “Mathematical Methods for Physicists”, Academic Press,
Fifth Ed., 2001
http://www.frm.utn.edu.ar/analisisdsys/material/function_gamma.pdf
http://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf
Gamma Function
M.Abramowitz & I.E. Stegun, ED., “Handbook of Mathematical Functions”,
Dover Publication, 1965,
H.Vic Dannon,”Riemann Zeta Function: The Riemann Hypothesis Origin, the
Factoriztion Error, and the Count of the Primes”, Gauge Institute Journal, Vol.
5, No. 4, November 2009
J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis
University of Copenhagen, June 2007
D. Miličić, “Notes on Riemann Zeta Function”,
http://www.math.utah.edu/~milicic/zeta.pdf
P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010),
http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf
http://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf

January 6, 2015 45
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA

Gamma function

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