Pochhammer symbol

5. Hypergeometric Functions

1 1 0x x y c a b x y ab y Hypergeometric equation( Gauss’ ODE & functions )

Regular singularities at 0,1,x

Solution : 2 1

0

, ; ;!

nn n

n n

a b xy F a b c x

c n

Hypergeometric function(series)

For a, b, c real, range of convergence are :

1 ,1

1 ,1 1

x c a b

x a b c a b

Series diverges for 1a b c

21 11

1 2!

a a b bab xx

c c c

1 1n

a a a a n

01a

Chap 7:

0, 1, 2,c

2 F1 includes many elementary functions.

E.g.

Sum terminates if 2 F1 = polynomial

Properties

2 10

, ; ;!

nn n

n n

a b xF a b c x

c n

0, 1, 2,c

or 0, 1, 2,a b

1

1

ln 1n

n

n

xx

n

0 1

n

n

xx

n

0

1 1

2 !

n

n n

n n

xx

n

2 1ln 1 1,1 ;2 ;x x F x

2nd Solution, Alternative ODE

1 1 0x x y c a b x y ab y

2nd Solution :

2 1

0

, ; ;!

nn n

n n

a b xF a b c x

c n

§ 7.6 :

12 1 1 , 1 ; 2 ;cy x F a c b c c x 2,3,4,c

c = integer y not independent of 2 F1 ( a, b ; c ; x)

additional logarithm term required

Alternative ODE :

2 1 1 11 1 1 2 0

2 2 2

z z zz y a b z a b c y ab y

2

2 2 2 22

1 21 2 2 1 4 0

d c dz y z a b z y z ab y z

d z z d z

11

2x z

11

2y x z y z

0, 1, 2,c

Contiguous Function Relations

2 1 2 1 2 11 , ; ; 1, 1 ; ; 1, 1 ; ;x F a b c x F a b c x F a b c x

2 21 1a b a b c a b

21a b a b

1c a a b b

1c b a b a

2 1 2 1 2 12 , ; ; 1 1, ; ; 1, ; ;a c b a x F a b c x a x F a b c x c a F a b c x

Hypergeometric Representations

2 1

2 ! 1, 2 1;1 ; 1

2 ! 1 2n

nC x F n n x

n

2 1

1, 1 ;1 ; 1

2nP x F n n x

/22

2 1

1! 1, 1 ; 1 ; 1

! 2 ! 2

m

mn m

xn mP x F m n m n m x

n m m

22 2 12

2 ! 1 1, ; ;

2 ! ! 2 2n

n n

nP x F n n x

n n

22 1

2 1 !! 1 1, ; ;

2 !! 2 2n n

F n n xn

22 1 2 12

2 1 ! 3 3, ; ;

2 ! ! 2 2n

n n

nP x F n n x

n n

22 1

2 1 !! 3 3, ; ;

2 !! 2 2n n

F n n xn

2 1

1 1, ; ; 1

2 2nT x F n n x 2 1

3 11 , 2 ; ; 1

2 2nU x n F n n x

22 1

3 11 1, 1 ; ; 1

2 2nV x n x F n n x

2 1

0

, ; ;!

nn n

n n

a b xF a b c x

c n

6. Confluent Hypergeometric Functions

0x y c x y a y Confluent Hypergeometric eq.

Singularities : regular at irregular at0x

Solution : 1 1

0

; ;!

nn

n n

a xF a c x

c n

For a, c real, series converge for all finite x .

211

1 2!

a aa xx

c c c

0, 1, 2,c

x

, ,y M a c x

Sum terminates if 1 F1 = polynomial0, 1, 2,a

E.g. 2

0

2x

terf x d t e

1

0

,

x

t aa x d t e t Re 0a

22 1 3, ,

2 2x M x

1, 1 ,ax M a a x

a

2nd solution :

0x y c x y a y

1 1 , 2 ,cy x M a c c x 2,3,4,c

Standard form :

1, , 1 , 2 ,, ,

sin 1cM a c x M a c c x

y U a c x xc a c c a c

Alternate ODE :

2

2 2 22

2 12 4 0

d c dy x x y z a y z

d x x d z

Integral Representations

1

11

0

, , 1c ax t ac

M a c x d t e t ta c a

0c a

11

0

1, , 1

c ax t aU a c x d t e t ta

Re 0 , 0x a

Techniques for verifying integral representations :

1. g(x,t) or Rodrigues relations.

2. Expand integrand into series & integrate.

3. (a) As solution to ODE. (b) Check normalization.

Confluent Hypergeometric Representations

1

, 2 1, 21 2 2

i xe xJ x M ix

1

, 2 1, 21 2 2

xe xI x M x

22

2 ! 1, ,

! 2n

n

nH x M n x

n

22 1

2 2 1 ! 3, ,

! 2n

n

nH x x M n x

n

, 1,nL x M n x

m

mmn n mm

dL x L x

d x !

, 1,! !

n mM n m x

n m

Further Observations

Advantages for using the (confluent) hypergeometric representations :

1.Asymptotic behavior or normalization easier to evaluate via the integral

representation of M & U.

2.Inter-relationship between special functions becomes clearer.

Self-adjoint version :

/2 1/2 1, 2 1,

2x

kM x e x M k x

Whittaker function

Self-adjoint ODE :

2

2

11 4 04k k

kM M

x x

2nd solution : /2 1/2 1, 2 1,

2x

kW x e x U k x

7. Dilogarithm

2

0

ln 1z

tLi z d t

t

Dilogarithm

Usage :

1.Matrix elements in few-body problems in atomic physics.

2.Perturbation terms in electrodynamics.

Expansion 2

0

ln 1z

tLi z d t

t

2

0

ln 1z

tLi z d t

t

1

10

zn

n

td t

n

21

n

n

z

n

1

0

z

pp

Li tLi z d t

t Poly-logarithm

1

n

pn

z

n

12

21

0 0

z zn

n

Li t td t d t

t n

31

n

n

z

n

3Li z

11

n

n

zLi z

n

01

n

n

Li z z

ln 1 z

11

1 z

0

1

1

z

d tt

1

z

z

For series converges & is real.

Analytic Properties 2

0

ln 1z

tLi z d t

t

2

1

n

n

z

n

Branch point at z = 1.

Conventional choice : Branch cut from z = 1 to z = .

with principal value : 2 21

n

n

zLi z

n

& 1z x x

For series diverges

but integral is finite & complex ( analytic continued ).

& 1z x x

For series diverges

but integral is finite & real ( analytic continued ).

& 1z x x

Mathematica

Since the only pole is at z = 0, the integral is

independent of path as long as it does not

cross the branch cut.

For the path colored blue in figure,

2

0

ln 1z

tLi z d t

t

1

20

1

ln 1ln 1 lnlim

1i

z

ii

e

t ix iLi z d x i d e d t

x e t

1 1 iz z e

branch cut

2

1

ln 11 ln

zt

Li d t i zt

On small circle, set

1 it e id t i e d

1t

On slanted line, set

1 i it r e ln 1 ln 1t t i

1r t

RHS of fig.18.8 & eq.18.159 are not allowed since the path crosses the branch cut.

Properties & Special Values 2

0

ln 1z

tLi z d t

t

2

1

n

n

z

n

2 0 0Li 2 21

11

n

Lin

2 2

2 16

Li

2 2

1

1n

n

Lin

2 1

1

n

sn

sn

1

1s

n

sn

2 ln 1d Li z z

d z z

2

2 2 1 ln ln 16

Li z Li z z z

2

22 2

1 1ln

6 2Li z Li z

z

22 2

1ln 1

1 2

zLi z Li z

z

2

2 112

Li

generates Li2 for all x from those in | x | 1/2.

Proof :both sides & find identity.Set z = 0 or 1 to determine const.

e.g.2

22

12 ln 2

2 6Li

Example 18.7.1. Check Usefulness of Formula

1 2 123 3

1 22 2 2 21 2 12

1

8

r r reI d r d r

r r r

12 1 2r r rj jr r

22

2 2

1 1ln

6 2Li Li

Question: Are the individual terms real?

I real & converges if , , 0 is real

Li2(x) is real for x < 1 both Li2 terms are real.

2ln

1 1

1 1

8. Elliptic Integrals

Example 18.8.1. Period of Simple Pendulum

2

21. .

2

dK E ml

d t

. . cosP E mgl

2

21cos

2

dE ml mgl

d t

cos Mmgl 0

M

d

d t

2cos cos M

d g

d t l

0

2 cos cos

M

M

l dt

g

2 2cos cos 2 sin sin2 2

MM

2 22sin cos2M

sin2sin

sin2M

cos1 2cos2 sin

2M

d d

2

cos cos cos2

M

dd

2 2

2

1 sin sin2M

d

2 20

4 4

1 sin sin2

M

M

l dT t

g

Period

Definitions

2 2

0

\1 sin sin

dF

Elliptic integral of the 1st kind

2 2

0

|1 1

x

d tF x m

t mt

0 1m 2

sin

sin

sin

t

x

m

/2

2

01 sin

dK m

m

Complete Elliptic integral of the 1st kind

1

2 20

1 |1 1

d tF m

t mt

2 2

0

\ 1 sin sinE d

Elliptic integral of the 2nd kind

2

2

0

1|

1

x

mtE x m dt

t

0 1m 2

sin

sin

sin

t

x

m

/2

2

0

1 sinE m d m

Complete Elliptic integral of the 1st kind

1

2

2

0

11 |

1

mtE m dt

t

Series Expansions /2

2

01 sin

dK m

m

/2

2

0

1 sinE m d m

/2

2

00

2 1 !!sin

2 !!n n

n

nK m d m

n

1/2

0

2 1 !!1

2 !!n

n

nx x

n

2

1

2 1 !!1

2 2 !!n

n

nm

n

Ex.13.3.8

Ex.18.8.2

2

1

2 1 !!1

2 2 !! 2 1

n

n

n mE m

n n

2 1

1 1, ;1 ;

2 2 2K m F m

2 1

1 1, ;1 ;

2 2 2E m F m

2 1

0

, ; ;!

nn n

n n

a b xF a b c x

c n

Fig.18.10. K(m) & E(m)

Mathematica

Limiting Values

2

1

2 1 !!1

2 2 !!n

n

nK m m

n

2

1

2 1 !!1

2 2 !! 2 1

n

n

n mE m

n n

0

2K

0

2E

1

2

2

0

1

1

mtE m dt

t

1

2 20

1 1

d tK m

t mt

1

2

0

11

d tK

t

1

0

1 1ln

2 1

t

t

1

0

1E dtIntegrals of the following form can be expressed in terms of elliptic integrals.

4

00

,

x

kk

k

I d t R t a t

E.Jahnke & F.Emde,

“Table of Higher Functions”

1