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Gamma Function mathematics and history. Please send comments and suggestions for improvements to [email protected]. Thanks. More presentations on different subjects can be found on my website at http://www.solohermelin.com.

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Page 1: Gamma function

1

Gamma Function

SOLO HERMELIN

Updated 28.10.12

Page 2: Gamma function

2

SOLO

TABLE OF CONTENT Gamma Function

Gamma Function HistoryGamma Function: Euler’s Second IntegralProperties 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

Page 3: Gamma function

3

SOLO

Gamma Function History

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)

0ln1

0

1

xtdtzt

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

11

!k

n

knk

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

Page 4: Gamma function

4

SOLO

Gamma Function History

Leonhard Euler( 1707– 1783)

0ln1

0

1

xtdtzt

t

x

During the years 1729 and 1730, Euler introduced the following analytic function,

By changing of variables we can obtain more known forms

0ln0

101

1

0

1

xtd

e

ttdetuduz

t

tt

xt

t

txeu

dtedu

u

u

xt

t

022ln0

120

12

2

1

0

1 22

2

2

xtdettdettuduz

t

t

txt

t

txeu

dtetdu

u

u

xt

t

The notation Γ (x) is due to Legendre in 1809, while Gauss used Π (x) = Γ (x+1)

Carl Friedrich Gauss

)1777 – 1855(

Adrien-Marie Legendre (1752 – 1833 )

Gamma Function

Page 5: Gamma function

5

SOLO

t

tt

z

tde

tz

0

1

Proof:

Gamma Function

0& xyixz

t

tt

zt

tt

zt

tt

z

tde

ttd

e

ttd

e

t

1

11

0

1

0

1

For the first part:

xt

xxtx

tdttde

ttd

e

t x

t

t

t

xt

t

xet

tt

yixt

tt

z t 1lim

1110

1

0

1

0

111

0

11

0

1

The first integral converges for any x ≥ δ > 0.

For the second integral, using integration by parts:

t

tt

x

e

t

t

txedv

tu

t

tt

x

e

t

t

txedv

tu

t

tt

xt

tt

yixt

tt

z

tde

txxetx

e

tde

txettd

e

ttd

e

ttd

e

t

t

x

t

x

1

3

/1

1

2

1

2

/1

1

1

1

1

1

1

1

1

2111

1

2

1

Euler’s Second IntegralGamma integral is defined, and converges uniformly for x > 0.

Gamma Function

Page 6: Gamma function

6

SOLO

t

tt

z

tde

tz

0

1

Proof (continue):

Gamma Function

0& xyixz

For the second integral, using integration by parts:

t

tt

x

e

t

t

txedv

tu

t

tt

x

e

t

t

txedv

tu

t

tt

xt

tt

yixt

tt

z

tde

txxetx

e

tde

txettd

e

ttd

e

ttd

e

t

t

x

t

x

1

3

/1

1

2

1

2

/1

1

1

1

1

1

1

1

1

2111

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

tt

t

ttxx

tde

xxxxtdet

xxxx11

1

121

121

Therefore the Gamma integral is defined, and converges uniformly for x > 0.

Gamma integral is defined, and converges uniformly for x > 0.

q.e.d.

Gamma Function

Return to Table of Content

Page 7: Gamma function

7

SOLO

t

tt

z

tde

tz

0

1

Proof :

Gamma Function

0& xyixz

zzz 1

zztdetztdtzeettdetzt

t

tzt

t ud

z

v

t

v

t

u

zdtedvtu

partsby

t

t

tztz

0

1

0

1

0

,

nintegratio0

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 121

121

nxnxxx

nxx

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 nnn

Page 8: Gamma function

8

SOLO

t

tt

z

tde

tz

0

1

Proof :

Gamma Function

!1

1Residue

1

1

nz

n

nzResidues of Gamma Function at x = 0,-1, -2,---,-n,..:

121

nxnxxx

nxx

q.e.d.

!1

1

121

1

1211limResidue

11

11

nnn

nxnxxx

nxnxx

n

nxnx

Gamma Function

Page 9: Gamma function

9

SOLO

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

Page 10: Gamma function

10

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Absolute value |Γ )z(|

Real value ReΓ )z(

Imaginary value ImΓ )z(

Gamma Function

Page 11: Gamma function

11

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Absolute value |Γ )z(|

Gamma Function

Page 12: Gamma function

12

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

zzz 1

Let compute

110

0

tt

t

t etde

Therefore for any n positive integer:

!1122112111 nnnnnnnnn

Properties of Gamma Function : 1

2

q.e.d.

Gamma Function

Page 13: Gamma function

13

SOLO Primes

Second definition identical to First

bayxallyfxfyxf ,,1,011

xa by yx 1

yxf 1

yfxf 1

Convex Function :

A Function f (x) is called Convex in an interval (a,b) if for every x,y ϵ (a,b) we have

A Function f (x), defined for x > 0, is called Convex, if the corresponding function

y

xfyxfy

defined for all y > -x, y ≠ 0, is monotonic Increasing throughout the range of definition.

x yx y

yxf

xf

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

xfxfy

xx

xfxfy

2

22

1

11

xxxfxfxxxfxf 1221

1

12

12

12

21 xx

xxxf

xx

xxxfxf

One other equivalent definition:

Page 14: Gamma function

14

SOLO Primes

1,0ln1ln1ln 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 1ln1ln yfxfyxf

Since the logarithm is a momotonic increasing function we obtain

yxyfxfyxf ,1,01 1

Page 15: Gamma function

15

SOLO Primes

t

tt

z

tde

tz

0

1

Proof :

Gamma Function

0& xyixz

1,0ln1ln1ln baba

Properties of Gamma Function :

3Gamma is a Log Convex Function

1

1

0

1

0

1

0

111

0

111

badtetdtet

dtetetdtetba

tbtaInequalityHolder

tbtatba

q.e.d.

Return to Table of Content

Page 16: Gamma function

16

SOLO Primes

t

tt

z

tde

tz

0

1

Proof :

Gamma Function

Other Gamma Function Definitions:

nxxx

nnx

x

n

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

lnln

!1

!ln

!2

!1ln

1

!1ln!ln!1lnln

1

!1ln!2ln

n

n

n

n

nn

x

nnxnn

n

xn

nx

n ln!1

ln1ln

x1 1

yln

0

1

1

ln1ln

x

nn

nn nn

nnx

1

ln1ln1

Carl Friedrich Gauss)1777 – 1855(

Page 17: Gamma function

17

SOLO Primes

t

tt

z

tde

tz

0

1

Proof (continue - 1) :

Gamma Function

Other Gamma Function Definitions:

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

xn

nx

n ln!1

ln1ln

xx nn

nxn ln

!1ln1ln

10 x

!1!11 nnnxnn xx

Use xxxnxnxnx

0

121

xxnxnx

nnx

xxnxnx

nn xx

121

!1

121

!11

nxxx

nnx

x

n

1

!limGauss’ Formula

Euler 1729Gauss 1811

Page 18: Gamma function

18

SOLO Primes

t

tt

z

tde

tz

0

1

Proof (continue - 2) :

Gamma Function

Other Gamma Function Definitions:

xxnxnx

nnx

xxnxnx

nn xx

121

!1

121

!11

xxnxnx

nnx

xxnxnx

nn xx

11

!1

11

!

Take the limit n → ∞

xxnxnx

nn

nx

xxnxnx

nn x

n

x

n

x

n 11

!lim

11lim

11

!lim

1

1,011

!lim

x

xxnxnx

nnx

x

n

Substitute n+1 for n

nxxx

nnx

x

n

1

!limGauss’ Formula

Page 19: Gamma function

19

SOLO Primes

t

tt

z

tde

tz

0

1

Let substitute x + 1 for x

Gamma Function

Other Gamma Function Definitions:

1,011

!lim

x

xxnxnx

nnx

x

x

n

n

q.e.d

nxxx

nnx

x

n

1

!limGauss’ Formula

Proof (continue - 3) :

1,011

!lim

1lim

11

!lim1

1

1

xxxxxnxnx

nn

nx

nx

xnxnx

nnx

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.

Return to Table of Content

Page 20: Gamma function

20

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Other Gamma Function Definitios:

Start from Gauss Formula xx nn

lim

q.e.d

constantMascheroni-Euler57721566.0ln1

2

11lim

11

nn

kx

e

x

ex

n

k

k

xx

Weierstrass’ Formula

Proof :

nx

nxx

x

eeee

xx

nx

nx

n

xxnxnx

nnx

n

xxx

nnx

xx

n

11

11

1

11

111

11

!:

211

2

11ln

11

1

2

11ln

11

1limlim

k

k

xxn

k

k

x

nnx

nn

n

kx

e

x

e

kx

e

xexx

Karl Theodor Wilhelm Weierstrass

(1815 – 11897)

Gamma Function

Return to Table of Content

Page 21: Gamma function

21

SOLO Primes

t

tt

z

tde

tz

0

1

Gamma Function

Some Special Values of Gamma Function:

q.e.d

2222/1

02

0

22 t

t

uut

duudt

t

t

t

udetdt

e

n

nnnnnn

2

125312/12/112/32/12/12/12/1

12531

21

2/12/32/1

2/1

2/1

2/32/1

nnnn

nn

nn

2/1

n

nn

2

125312/1

12531

212/1

n

nnn

Proof:

Return to Table of Content

Page 22: Gamma function

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 nn

xnfn ln

!1ln1ln

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

t

t

tz tdetz0

1 nxxx

nnz

x

n

1

!lim

Page 23: Gamma function

23

SOLO

Bohr-Mollerup-Artin Theorem:

Harald August Bohr ) 1887 – 1951(

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

Proof (continue-1):

By the second property xfxxxnxnxnf 121

xx nn

xfxxxnxnn ln

!1

121ln1ln

We found

Since lan is a monotonic increasing function, we have

121

!1

121

!11

xnxnxx

nnxf

xnxnxx

nn xx

x

xxx

n

n

xnxnxx

nnxf

xnxnxx

nn 1

11

!

11

!

n

n

1

t

t

tz tdetz0

1 nxxx

nnx

x

n

1

!limor Gauss Formula

Emil Artin(1898 – 1962)

Hamburg University

Johannes Mollerup(1872 – 1937)

Gamma Function

Page 24: Gamma function

24

SOLO

Bohr-Mollerup-Artin Theorem:

q.e.d.

Harald August Bohr ) 1887 – 1951(

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 Johannes Mollerup

(1872 – 1937)

Proof (continue - 2):

xxx

nxnxnxx

nnxf

xnxnxx

nn

1

111

!

11

!

t

t

tz tdetz0

1 nxxx

nnx

x

n

1

!limor Gauss Formula

By taking n → ∞ we obtain

1

11lim

11

!lim

11

!lim

x

n

x

x

n

x

x

n nxnxnxx

nnxf

xnxnxx

nn

But this is possible only if xxf

Emil Artin(1898 – 1962)

Hamburg University

Gamma Function

Page 25: Gamma function

25

SOLO

t

tt

z

tde

tz

0

1

Gamma Function Gamma integral is defined, and converges uniformly for x > 0.

Differentiation of Gamma Function:

q.e.d

0,2!11'

ln

01'''

ln

constantMascheroni-Euler57721566.0111'

ln

11

1

122

2

2

2

1

xnkx

n

x

x

xd

dx

xd

d

kxx

xxxx

xd

d

kxkxx

xx

xd

d

kn

n

n

n

n

n

k

k

Proof :

Start from Weierstrass Formula

1 1k

k

xx

kx

e

x

ex

11

1lnlnlnkk k

x

k

xxxx

11 1

111

lnkk

kxk

kxx

xd

d

0111111

ln0

21

221

2

2

kkk kxkxxkxkxxd

dx

xd

d

0

1

1 !11'ln

kn

n

n

n

n

n

kx

n

x

x

xd

dx

xd

d

Gamma Function

We can see that

1

11

1 1

11lim

1

1

1

1'1ln

n

n

kn kk

xxd

d

Return to Table of Content

Page 26: Gamma function

26

SOLO

1

0

11 1,s

s

zy sdsszyBBeta Function

Beta Function is related to Gamma Function:

u

u

uy

duudt

utt

t

ty udeutdety0

12

20

1 22

2

zy

zyzyB

,

Proof:

In the same way:

v

v

vz vdevz0

12 2

2

u

u

v

v

vuuzy vdudevuzy0 0

1212 22

4

Use polar coordinates:

drdrdrdr

rdrd

vrv

uruvdud

rv

ru

cossin

sincos

//

//

sin

cos

2/

0

1212

0

12

0

2/

0

121212

sincos22

sincos4

2

2

drder

drderzy

zy

zy

r

r

rzy

r

r

rzyzy

Euler’s First Integral

Gamma Function

Page 27: 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

zyzyB

,

Proof (continue):

2/

0

1212 sincos2

dzyzy zy

Change variables in the integral using dsds cossin2sin 2

zyBsdssds

s

yzzy ,1sincos21

0

112/

0

1212

zyBzyzy ,Therefore q.e.d.

Use z→y and y → 1 - z

u

u

zu

u

z

z

zu

us

u

udsd

s

s

zz

udu

u

u

ud

u

u

u

u

dssszzBzz

0

1

021

11

1

1

0

1

1111

1

11,11

2 q.e.d.

Gamma Function

Return to Table of Content

Page 28: Gamma function

28

SOLO

Proof

yzBzyzyBzyyzyzBzyB

,,,,

Use y → 1 - z

u

u

zu

u

z

z

zu

us

u

udsd

s

s

zz

udu

u

u

ud

u

u

u

u

dssszzBzz

0

1

021

11

1

1

0

1

1111

1

11,11

2

t

tt

z

tde

tz

0

1

Gamma Function

Other Gamma Function Properties: zzz

sin

1 Euler Reflection Formula

Gamma Function

Page 29: Gamma function

29

SOLO

Proof (continue - 1)

u

u

x

udu

uxx

0

1

11

t

tt

z

tde

tz

0

1

Gamma Function

Other Gamma Function Properties:

R

RC

C

1

planeu

uRe

uIm

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.

C

yR yyi

C

yR y

udu

uud

u

ueud

u

uud

u

u

R1111

2

Define y = 1 – x, and assume x,y ϵ (0,1)

zzz

sin

1 Euler Reflection Formula

Gamma Function

Page 30: Gamma function

30

SOLO

Proof (continue - 1)

u

u

x

udu

uxx

0

1

11

t

tt

z

tde

tz

0

1

Gamma Function

Other Gamma Function Properties:

R

RC

C

1

planeu

uRe

uIm

This path encloses the pole u=-1 of that has the residue1

u

u y

yi

eu

yy

euu

ui

11

Residue

By the Residue Theorem

For z ≠ 0 we have

yzyzyzyy zeeez lnlnReln

zzz

sin

1 Euler Reflection Formula

yiy

eu

y

C

yR yiy

C

yR y

eiu

uui

u

uizd

z

zud

u

uezd

z

zud

u

u

i

R

21

1lim2

1Residue2

1111

1

2

Gamma Function

Page 31: Gamma function

31

SOLO

Proof (continue - 2)

t

tt

z

tde

tz

0

1

Gamma Function

Other Gamma Function Properties:

R

RC

C

1

planeu

uRe

uIm

yi

C

yR yiy

C

yR y

eizdz

zud

u

uezd

z

zud

u

u

R

2

11112

For the second and forth integral we have

0lnlnReln zzeeezyzyzyzyy

z

z

z

z

z

zyyy

111

Hence for small ε we have:

and for large R we have:

01

21

01

y

C

y

zdz

z

01

21

1

Ry

C

y

R

Rzd

z

z

R

Therefore the integrals on the circular paths are zero for ε→0 and R →∞

zzz

sin

1 Euler Reflection Formula

Gamma Function

Page 32: Gamma function

32

SOLO

Proof (continue - 3)

t

tt

z

tde

tz

0

1

Gamma Function

Other Gamma Function Properties:

R

RC

C

1

planeu

uRe

uIm

yiy

iyy

eiudu

ueud

u

u

2

11 0

2

0

We obtain

Multiply both sides by yie

iudu

uee

yiyiy 2

10

yee

iud

u

uiyiy

y

sin

2

10

Rearranging we obtain

Since both sides of this equation are meromorphic (analytic) in x ϵ (0,1) we can extend the result for all analytic parts of z ϵ C (complex plane).

1,0sin1sin11

10

1

0

1

xxx

udu

uud

u

uxx

u

u

yxyu

u

x

Substituting y = 1 – x we obtain

zzz

sin

1 Euler Reflection Formula

Gamma Function

Page 33: Gamma function

33

SOLO

Onother Proof

t

tt

z

tde

tz

0

1

Gamma Function

Other Gamma Function Properties:

Start with Weierstrass Gamma Formula

zzz

sin

1 Euler Reflection Formula

1 1k

k

xx

kx

e

x

ex

12

22

1

2 1111

kk k

x

k

xxx

k

xx

e

kx

e

kx

eexxx

Use the fact that Γ (-x)=- Γ (1-x)/x to obtain

12

2

11

1

k k

xx

xx

Now use the well-known infinite product

12

2

1sink k

xxx

q.e.d.

Gamma Function

Page 34: Gamma function

34

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Proof

t

tt

z

tde

tz

0

1

Gamma Function

Other Gamma Function Properties: zzz

cos2

1

2

1

Start from

Substitute ½ +z instead of z

zzz

sin

1

zz

zz

cos

21

sin2

1

2

1

q.e.d.

Gamma Function

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Page 35: Gamma function

35

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Stirling Approximation Formula:

121 xexxx xx

u

u

uuxxx

u

u

xuxxxu

u

xxuxuxt

udxtd

t

t

xt

udeex

udueexudxuxetdtex

1

1ln1

1

1

1

11

1

111Proof:

The function f(u) = -u + ln (1 + u) equals zero for u = 0. For other values of u we havef(u) < 0. This implies that the integrand of the last integral equals 1 at u = 0 and that thisintegrand 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

02

1

2

11ln 2222 uforuuuuuuuuuf

This implies that

xforduedueu

u

uxu

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

Page 36: Gamma function

36

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Stirling Approximation Formula:

121 xexxx xxProof (continue):

xfordueexudeexxu

u

uxxxu

u

uuxxx 2/1

1

1ln1 2

1

xforxdtexduet

t

txtu

xtdud

u

u

ux

22 2/12/1/2

/2

2/ 22

If we set we have by using the normal integralxtu /2

therefore:

xexxx xx21

q.e.d.

Gamma Function

Page 37: Gamma function

37

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Duplication and Multiplication Formula:

0Re222

112

zzzz

z

Legendre Duplication Formula1809

Adrien-Marie Legendre (1752 – 1833 )

Proof:

2/1,2sin22sin2

2sin22sincos2,

212/

0

1221

0

1221

2/

0

12212/

0

1212

zBdd

ddzzB

zzzzz

zzzz

0Re2/1

2/122/1,2,

22121

zz

zzBzzB

z

zz zzWe have

therefore

q.e.d

0Re222

112

2

1

zzzzz

Gamma Function

Page 38: Gamma function

38

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Duplication and Multiplication Formula:

znnn

nz

nz

nzz znn

2/12/12

121

Gauss Multiplication

Formula

Proof:

nz

1

Carl Friedrich Gauss)1777 – 1855(

nn

n

nn

n 2/12121

Euler

Multiplication Formula

Gamma Function

Define the function:

n

nx

n

x

n

xnxf x 11

:

This function has the following properties:

1

xfxn

x

n

x

n

nx

n

x

n

xnn

n

nx

n

nx

n

x

n

xnxf

x

x

121

1211 1

Page 39: Gamma function

39

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Duplication and Multiplication Formula:

znnn

nz

nz

nzz znn

2/12/12

121 Gauss

Multiplication Formula

Proof (continue – 1):Carl Friedrich Gauss

)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.

n

n

nnnaaf nn

2111

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

pnk

nk

nk

pp

n

k pn

k

p

n

k

p

1!lim

1

!lim

Page 40: Gamma function

40

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Duplication and Multiplication Formula:

znnn

nz

nz

nzz znn

2/12/12

121 Gauss

Multiplication Formula

Proof (continue – 2):Carl Friedrich Gauss

)1777 – 1855(

Gamma Function

pnknkk

npp

n

k pn

k

p

1!lim

Since k = 1,2,…,p

!1!

11211

nppnn

pnnpnnnnnpnknkkp

k

!!

lim!

!lim

21 2

11

11

pnn

pnpn

pnn

pnpn

n

n

nnna

npnn

p

n

n

npnn

pn

Page 41: Gamma function

41

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Duplication and Multiplication Formula:

znnn

nz

nz

nzz znn

2/12/12

121 Gauss

Multiplication Formula

Proof (continue – 3):Carl Friedrich Gauss

)1777 – 1855(

Gamma Function

!!

lim2

11

pnn

pnpna

npnn

pn

Use the identity

npp pnpn

pnn

pn

n

pnpn

1

!

!lim1

21

11lim1

to an to get

2

1

2

112

11

!

!lim

!

!

!

!lim1

!

!lim

n

pnn

pn

npnn

p

npnn

pn

ppn

npn

pnpn

pnn

pnn

pnpn

pnn

pnpna

Page 42: Gamma function

42

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Duplication and Multiplication Formula:

znnn

nz

nz

nzz znn

2/12/12

121 Gauss

Multiplication Formula

Proof (continue – 4):Carl Friedrich Gauss

)1777 – 1855(

Gamma Function

to an to get

2

1

!

!lim

n

pnn

pn

ppn

npna

pepp pp

2

1

2! pepnpn pnpn2

1

2!

2

1

2

1

2

1

2

1

2

1

2

2

2

lim n

pepn

nep

nan

npnpn

pn

n

pp

pn

Use Stirling’s Approximation formula xexxx xx21

Page 43: Gamma function

43

SOLO

t

tt

z

tde

tz

0

1

Gamma Function

Duplication and Multiplication Formula:

znnn

nz

nz

nzz znn

2/12/12

121 Gauss

Multiplication Formula

Proof (continue – 4):Carl Friedrich Gauss

)1777 – 1855(

Gamma Function

2

1

2

1

2 nan

n

xan

nx

n

x

n

xnxf nx

11:

We have

or xnn

nx

n

x

n

x xn

2

1

2

1

211

Define x = n z to obtain

znnn

nz

nzz

znn

2

1

2

1

211 q.e.d

Return to Table of Content

Page 44: Gamma function

44

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References

Internethttp://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.pdfhttp://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 2009J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007D. 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.pdfhttp://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf

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Page 45: Gamma function

April 10, 2023 45

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TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA