13.4.sterling’s series derivation from euler-maclaurin integration formula euler-maclaurin...
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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
Mathematica
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
Mathematica
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
Mathematica
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
Mathematica
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
Mathematica
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
Mathematica
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