13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

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13.4. Sterling’s Series Derivation from Euler-Maclaurin Integration Formula Euler-Maclaurin integration formula : 2 1 2 1 2 0 1 0 1 0 0 2 2 ! n q n p p p q m p B dx f x f m n f n f R p 2 1 f x z x B 2 B 4 B 6 B 8 1/ 6 1/3 0 1/4 2 1/3 0 Let 0 0 1 dx f x z x 1 z 1 2 0 1 1 0 1 2 2 m f m z z 1 1 1 1 ! 1 m m m k z m z k 2 1 2 1 2 ! p p p f x z x 1 3 2 f x z x 3 5 234 f x z x 1 2 2 2 1 1 1 1 1 2 p p p B z z z z

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13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula. Euler-Maclaurin integration formula :. Let. . . . . . . . Stirling’s series. . . Stirling approx. z >> 1 :. . A = Arfken’s two-term approx. using. Mathematica. 13.5.Riemann Zeta Function. - PowerPoint PPT Presentation

Transcript of 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Page 1: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

13.4. Sterling’s SeriesDerivation from Euler-Maclaurin Integration Formula

Euler-Maclaurin integration formula :

2 1 2 12

0 10

1 0 02 2 !

n qnp pp

qm p

Bd x f x f m f f n f n f R

p

2

1f xz x

B2 B4 B6 B81/6 1/30 1/42 1/30

Let 00

1d x f xz x

1z

12

0

1 10 12 2m

f m f f zz

1

11

1!1

mmm

k

z mz k

2 12 1

2 !pp

pf x

z x

13

2f xz x

35

2 3 4f xz x

1 22 2 1

1

1 1 12

pp

p

Bz

z z z

Page 2: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

1 22 2 1

1

1 1 12

pp

p

Bz

z z z

1 2

2 2 11

1 112

pp

p

Bz

z z z

111 1z C d z z 2

1 21

1ln2 2

pp

p

BC z

z p z

2ln 1 1z C d z z

2

2 2 11

1ln ln2 2 2 1

pp

p

BC z z z z

p p z

1lim 1 lnz

z C z

1

1 11k

zz k k

1

1k k

1 0C

21lim ln 1 ln2z

z C z z z

Page 3: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

2

11 2 12 2 zz z z

1 ln 2 ln 22

z

21 ln 22

C

21lim ln 1 ln2z

z C z z z

21 1 1lim ln ln2 2 2z

z C z z z

1 1ln ln2 2

z zz

21lim ln ln2z

z C z z z

21lim ln 2 1 2 2 ln 22z

z C z z z

21 1ln 2 2 2ln 2 2 ln2 2

C z z z

2

1112lim ln ln 2 2ln 2

2 1 2z

z zC z

z

2

2 11

1 1ln 1 ln 2 ln2 2 2 2 1

pp

p

Bz z z z

p p z

Stirling’s series

Page 4: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

2

2 11

1 1ln 1 ln 2 ln2 2 2 2 1

pp

p

Bz z z z

p p z

z >> 1 : 1 1ln 1 ln 2 ln2 2

z z z z

1/2ln 2 z zz e 1/21 2 z zz z e

Stirling approx

A = Arfken’s two-term approx. using1/12 11

12ze

z

Mathematica

Page 5: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

13.5. Riemann Zeta Function

Riemann Zeta Function : 1

1z

n

zn

2 4 6 8 10

" " 2 3 4 5 6 7 8 9 10

" ( )" 1.202 1.037 1.008 1.0026 90 945 9450 93555

z

z

Integral representation : 1

0

11

z

t

tz d tz e

Proof :

1

0

11

tz

t

eRHS dt tz e

1

1 0

1 z mt

m

d t t ez

1

1 0

1 1 z sz

m

d s s ez m

1

1z

m m

s m t

LHS

Mathematica

Page 6: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Definition : Contour Integral

1

111

z

tC

tI d tz e

111

z

A x

xI d xz e

z

1

0

11

z

t

tz d tz e

121

1

zi

B x

x eI d x

z e

2 1z iz e 2 z iz e

1

2

0

11

i

zii

D e

eI i e d

z e

2

21

0

1 i zzi e dz

0 for Re z >1

diverges for Re z <1

1

1

2

111

z

tz i C

tz d tee z

agrees with integral

representation for Re z > 1

C1

Page 7: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Similar to ,

Definition valid for all z (except for z integers).

Analytic Continuation

1

1

2

111

z

tz i C

tz d tez e

Poles at 2 0, 1, 2,t n i n

1

1 1

1 1C

z

t C

z

t

t td t d te e

1

112 2

1

zz

C tn

td t i n ie

Re z > 1

1

1 1 1

2 Res ; 21 1 1C

z z

C

z

t t tn

t t td t d t i n ie e e

1/ 2 1

1

2 1z i zi z

n

e e n

C C1 encloses no pole.C C1 encloses all poles.

means n 0

/ 2 3 / 22 1z i z i ze e z

3 / 2 / 2

2

21

1

z i z i z

z i

e ez z

z e

11

1

2 z zz

n

i n n

Mathematica

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Riemann’s Functional Equation

3 / 2 / 2

2

21

1

z i z i z

z i

e ez z

z e

3 / 2 / 2 / 2 / 2

2 1

i z i z i z i z

z i z i z i

e e e ee e e

sin2

sin

z

z

1/ 2 1 / 2z z

z z

1sin

z zz

12 1 sin 12

z zz z z z

1 sin 12z z z

Riemann’s functional equation

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Zeta-Function Reflection Formula

3 / 2 / 2

2

21

1

z i z i z

z i

e ez z

z e

3 / 2 / 2

2

11 / 2 1 / 2

i z i z

z i

z ze ee z z

2 1

1/ 2 1 / 2

z zz z

z z

2

11 2 12 2 zz z z

11 1

2 2 2 2 z

z z z

1/2 1

2 1/ 2

z z

z zz

zeta-function reflection formula

1 /2/2 1/ 2 12

zz zz z z

Page 10: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

12 1 sin 12

z zz z z z

Riemann’s functional equation :

for trivial zeros 0z 2z n 1, 2, 3,n

1

1z

n

zn

converges for Re z > 1

12 1 sin 12

z zz z z z

(z) is regular for Re z < 0.

(0) diverges (1) diverges while (0) is indeterminate.

Since the integrand in is always positive,

(except for the trivial zeros)

or

i.e., non-trivial zeros of (z) must lie in the critical strip

1

0

11

z

t

tz d tz e

0 Re 1z z

1 0 Re 1z z

0 Re 0z z

0 Re 1z

Page 11: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Critical Strip

1

1

n

zn

zn

Consider the Dirichlet series :

Leibniz criterion series converges if , i.e., 1lim 0zn n Re 0z

1 1

1 12 1 2z z

n n

zn n

1 1

1 1 122z z z

n nn n

11 2 z z

for 11 2 z

zz

Re 0z

11

1Res ;1 lim

1 2 zz

z zz

1 ln 212 zz e

1 1ln 2

1

12 2 ln 2z

zdd z

1 ln 2 1

1

ln 1n

n

xx

n

1

1

1z

n

zn

Page 12: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

(0)

2 1

1/ 2 1 / 2

z zz z

z z

0

10 lim

/ 2z

zz

102

0

Res ;11 1

limRes ; 0

02

z

szs

z

Simple poles :

Res ;112 Res ; 0

ss

Res ;1 1z

1

limz n

nk

n kzz z k

Res ; 0 1z

Page 13: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Euler Prime Product Formula

1 1 1 1 1 11 12 3 5 7 9 11s s s s s ss

1

1z

n

zn

1 1 1 1 1 1 1 1 1 112 3 4 5 6 7 8 9 10 11s s s s s s s s s ss

( no terms ) 1

2 sn

1 1 1 1 11 1 12 3 5 7 11s s s s ss

( no terms )

13 sn

primes

11 1sp

sp

primes

11 s

p

sp

Euler prime product formula

Page 14: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Riemann Hypothesis

Riemann found a formula that gives the number of primes less than a

given number in terms of the non-trivial zeros of (z).

Riemann hypothesis :

All nontrivial zeros of (z) are on the critical line Re z ½.

Millennium Prize problems proposed by the Clay Mathematics Institute.

1. P versus NP

2. The Hodge conjecture

3. The Poincaré conjecture (proved by G.Perelman in 2003)

4. The Riemann hypothesis

5. Yang–Mills existence and mass gap

6. Navier–Stokes existence and smoothness

7. The Birch and Swinnerton-Dyer conjecture

Page 15: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

13.6. Other Related Functions

1. Incomplete Gamma Functions

2. Incomplete Beta Functions

3. Exponential Integral

4. Error Function

, , ,a x a x

Page 16: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Incomplete Gamma Functions

1

0

,x

t aa x d t e t

1, t a

x

a x d t e t

, ,a x a x a

Integral representation:

Re 0a

0,t

x

ex d tt

Ei x 0x Exponential integral

Page 17: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Series Representation for (n, x)

1

1 0

1 !1 !

!

xnx n k t

k

ne x n d t e

n k

1

0

,x

t nn x d t e t 1 2

00

1x

xt n t ne t n d t e t

1 2 3

0

1 2x

x n x n t ne x n e x n d t e t

1

1

11 ! 1!

nx n k x

k

n e x en k

1

11 ! 1!

nx n k

k

n e xn k

1

0

1, 1 ! 1!

nx s

s

n x n e xs

s n k

1, 2, 3,n

Page 18: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Series Representation for (n, x)

1

1

1 !1 !

!

nx n k t

k x

ne x n d t e

n k

1, t n

x

n x d t e t

1 21t n t n

xx

e t n d t e t

1 2 31 2x n x n t n

x

e x n e x n d t e t

1

1

11 !!

nx n k x

k

n e x en k

1

11 !!

nx n k

k

n e xn k

1

0

1, 1 !!

nx s

s

n x n e xs

s n k

1, 2, 3,n

Page 19: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Series Representation for (a, x) & (a, x)

For non-integral a :

0

,!

na n

n

a x x xn a n

0x

See Ex 1.3.3 & Ex.13.6.4

1

0

1, ~ a xn

n

aa x x e

a n x

x

1

0

1a xnn

n

x e a nx

Pochhammer symbol

1 1n

a a a a n

01a

1

0

10, ~ xnn

n

x x e nx

0

!xn

nn

e nx x

1a a

Relation to hypergeometric functions: see § 18.6 .

Page 20: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Incomplete Beta Functions

1

11

0

, 1 qpB p q d t t t

11

0

, 1x

qpxB p q d t t t 0 ,1 & 0x p

, 0p q

0

1!

p nn

n

qx x

n p n

Ex.13.6.5

Relation to hypergeometric functions: see § 18.5.

Page 21: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Exponential Integral Ei(x)

0,t

x

ex d tt

Ei x 1E x

t

x

eEi x P d tt

x teP d t

t

0x

0x

1 0E

P = Cauchy principal value

1

t x

n n

eE x d tt

E1 , Ei analytic continued.

Branch-cut : (x)–axis.

1 0E x i Ei x i

1 11 0 02

Ei x E x i E x i

Mathematica

Page 22: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Series Expansion

1 0lim ,a

E x a a x

0

1

lim!

naa n

an

xa x xa n a n

0

1

lim!

nan

an

a a xx

a n a n

0

,!

na n

n

a x x xn a n

0

0 0

1 1lim lim

a a

a a

a a x a x xa a a

01

z

zz

d z d xd z d z

1 1 ln x

ln lnz

z z x zd xx e x xd z

lndz zd z

ln x

1

1

ln!

nn

n

E x x xn a n

For x << 1 :

For x >> 1 : 10

!0,x

nn

n

e nE x xx x

Page 23: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Sine & Cosine Integrals

0

0

0

sin sin

cos 1 cosln

lnln

x

x

x

x

x

t tSi x d t si x d tt t

t tCi x x d t ci x d tt t

d tli x P Ei xt

0

sin2

tSi x si x d tt

Ci(z) & li(z) are multi-valued.

Branch-cut : (x)–axis. 1li

0

1 cosx tCin x d tt

is an entire function

0

cosx td tt

not defined

Mathematica

Page 24: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

1

cos

sinx

x

t

x

t

x

tCi x d tt

tsi x d tt

eEi x d tt

eE x d tt

t s x

12

Ei i x Ei i xi

11

s xeE x d ss

t

x

eEi x d tt

1

sin sxsi x d ss

1

12

i s x i s xe esi x d si s

1

cos sxCi x ci x d ss

1 112

E ix E ixi

1

s xed ss

1

12

i s x i s xe eci x d ss

1

2Ei ix Ei i x 1 1

12

E ix E ix

Ei i x ci x i si x 1E ix ci x i si x

Series expansions : Ex.13.6.13. Asymptotic expansions : § 12.6.

Page 25: 13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula

Error Function

2

0

2 zterf z d t e

221 t

z

erfc z erf z d t e

1erf

Power expansion :

2

0 0

2!

n xn

n

erf x d t tn

2 1

0

2! 2 1

nn

n

xn n

Asymptotic expansion (see Ex.12.6.3) :

221 t

x

erf x d t e

2 2

2

2 112 2

x t

z

e ed tx t

22

2

tt dee d t

t

2

2 10

2 1 !!1

2

nx

n nn

nex

21 1 ,2z

1

0

,x

t aa x d t e t

21 1 ,2z

Mathematica