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

Euler-Maclaurin integration formula :

2 1 2 12

0 10

10 0

2 2 !

n qnp pp

qm p

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

p

2

1f x

z x

B2 B4 B6 B8

1/6 1/30 1/42 1/30

Let 00

1d x f x

z x

1

z

1

20

1 10 1

2 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 x

z x

3

5

2 3 4f x

z x

1 2

2 2 11

1 11

2p

pp

Bz

z z z

1 2

2 2 11

1 11

2p

pp

Bz

z z z

1 2

2 2 11

1 11

2p

pp

Bz

z z z

111 1z C d z z 2

1 21

1ln

2 2p

pp

BC z

z p z

2ln 1 1z C d z z

2

2 2 11

1ln ln

2 2 2 1p

pp

BC z z z z

p p z

1lim 1 lnz

z C z

1

1 1

1k

zz k k

1

1

k k

1 0C

2

1lim ln 1 ln

2zz C z z z

2

11 2 1

2 2 zz z z

1ln 2 ln 2

2z

2

1ln 2

2C

2

1lim ln 1 ln

2zz C z z z

2

1 1 1lim ln ln

2 2 2zz C z z z

1 1ln ln

2 2z z

z

2

1lim ln ln

2zz C z z z

2

1lim ln 2 1 2 2 ln 2

2zz C z z z

2

1 1ln 2 2 2ln 2 2 ln

2 2C z z z

2

11

12lim ln ln 2 2ln 22 1 2z

z zC z

z

2

2 11

1 1ln 1 ln 2 ln

2 2 2 2 1p

pp

Bz z z z

p p z

Stirling’s series

2

2 11

1 1ln 1 ln 2 ln

2 2 2 2 1p

pp

Bz z z z

p p z

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

2 2z 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 1

112

zez

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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

1

1

z

t

tz d t

z e

Proof :

1

0

1

1

tz

t

eRHS d t t

z 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

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Definition : Contour Integral

1

11

1

z

tC

tI d t

z e

11

1

z

A x

xI d x

z e

z

1

0

1

1

z

t

tz d t

z e

12

1

1

zi

B x

x eI d x

z e

2 1z iz e 2 z iz e

1

2

0

1

1i

zi

iD 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

1

11

z

tz i C

tz d t

ee z

agrees with integral

representation for Re z > 1

C1

Similar to ,

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

Analytic Continuation

1

1

2

1

11

z

tz i C

tz d t

ez 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

11

2 21

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

2z zz

n

i n n

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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 e

e e e

sin2

sin

z

z

1

/ 2 1 / 2

z z

z z

1sin

z zz

12 1 sin 12

z zz z z z

1sin 1

2z z z

Riemann’s functional equation

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

1

1 / 2 1 / 2

i z i z

z i

z ze e

e z z

2 1

1/ 2 1 / 2

zz

z zz z

2

11 2 1

2 2 zz z z

11 1

2 2 2 2 z

z zz

1/2 1

2 1/ 2

z z

z zz

zeta-function reflection formula

1 /2/2 1/ 2 1

2zz z

z z z

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

1

1

z

t

tz d t

z e

0 Re 1z z

1 0 Re 1z z

0 Re 0z z

0 Re 1z

Critical Strip

1

1

n

zn

zn

Consider the Dirichlet series :

Leibniz criterion series converges if , i.e., 1

lim 0zn n Re 0z

1 1

1 1

2 1 2z z

n n

zn n

1 1

1 1 12

2z z zn 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

11

ln 2

1

122 ln 2

zzd

d z

1 ln 2 1

1

ln 1n

n

xx

n

1

1

1z

n

zn

(0)

2 1

1/ 2 1 / 2

zz

z zz z

0

10 lim

/ 2z

z

z

10

2

0

Res ;1

1 1lim

Res ; 0

02

z

s

z

sz

Simple poles :

Res ;11

2 Res ; 0

s

s

Res ;1 1z

1

limz n

nk

n kz

z z k

Res ; 0 1z

Euler Prime Product Formula

1 1 1 1 1 11 1

2 3 5 7 9 11s s s s s ss

1

1z

n

zn

1 1 1 1 1 1 1 1 1 11

2 3 4 5 6 7 8 9 10 11s s s s s s s s s ss

( no terms ) 1

2s

n

1 1 1 1 11 1 1

2 3 5 7 11s s s s ss

( no terms ) 1

3s

n

primes

11 1

sp

sp

primes

1

1 sp

sp

Euler prime product 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

13.6. Other Related Functions

1. Incomplete Gamma Functions

2. Incomplete Beta Functions

3. Exponential Integral

4. Error Function

, , ,a x a x

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 t

t

Ei x 0x Exponential integral

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

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

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

For non-integral a :

0

,!

n

a n

n

a x x xn a n

0x

See Ex 1.3.3 & Ex.13.6.4

1

0

1, ~ a x

nn

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, ~ x

nnn

x x e nx

0

!xn

nn

e n

x x

1a a

Relation to hypergeometric functions: see § 18.6 .

Incomplete Beta Functions

1

11

0

, 1qpB 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.

Exponential Integral Ei(x)

0,t

x

ex d t

t

Ei x 1E x

t

x

eEi x P d t

t

x te

P d tt

0x

0x

1 0E

P = Cauchy principal value

1

t x

n n

eE x d t

t

E1 , Ei analytic continued.

Branch-cut : (x)–axis.

1 0E x i Ei x i

1 1

10 0

2Ei x E x i E x i

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Series Expansion

1 0lim ,a

E x a a x

0

1

lim!

naa n

an

xa x x

a n a n

0

1

lim!

nan

an

a a xx

a n a n

0

,!

n

a n

n

a x x xn a n

0

0 0

1 1lim lim

a a

a a

a a x a x x

a a a

01

z

zz

d z d x

d z d z

1 1 ln x

ln lnz

z z x zd xx e x x

d z

lnd

z zd z

ln x

1

1

ln!

n

n

n

E x x xn a n

For x << 1 :

For x >> 1 : 10

!0,

xn

nn

e nE x x

x x

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 t

t t

t tCi x x d t ci x d t

t t

d tli x P Ei x

t

0

sin

2

tSi x si x d t

t

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

Branch-cut : (x)–axis.

1li

0

1 cosx tCin x d t

t

is an entire function

0

cosx td t

tnot defined

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1

cos

sin

x

x

t

x

t

x

tCi x d t

t

tsi x d t

t

eEi x d t

t

eE x d t

t

t s x

1

2Ei ix Ei i x

i

1

1

s xeE x d s

s

t

x

eEi x d t

t

1

sin sxsi x d s

s

1

1

2

i s x i s xe esi x d s

i s

1

cos sxCi x ci x d s

s

1 1

1

2E ix E ix

i

1

s xed s

s

1

1

2

i s x i s xe eci x d s

s

1

2Ei ix Ei i x 1 1

1

2E ix E ix

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

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

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

n

n

n

xn n

Asymptotic expansion (see Ex.12.6.3) :

221 t

x

erf x d t e

2 2

2

2 11

2 2

x t

z

e ed t

x t

2

2

2

tt de

e d tt

2

2 10

2 1 !!1

2

nx

n nn

ne

x

21 1,

2z

1

0

,x

t aa x d t e t

21 1,

2z

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