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Supplement A

Special Functions and Their Properties

Throughout Supplement A it is assumed that#

is a positive integer, unless otherwise specified.

A.1. Some Symbols and Coefficients

A.1.1. Factorials

Definitions and some properties:

0! = 1! = 1,#

! = 1 2 3 33 3

(#

1)#

,#

= 2, 3, 33 3

,

(2 # )!! = 2 4 6 3 3 3 (2 # 2)(2 # ) = 2 # !,

(2#

+ 1)!! = 1 3 5 33 3

(2#

1)(2#

+ 1) =2 +1

7 8 9 @

#+

3

2A

,

#!! =

B

(2C

)!! if #

= 2C

,

(2 C + 1)!! if # = 2 C + 1,0!! = 1.

A.1.2. Binomial Coefficients

Definition:

4 D

E =# !

C!(

#

C)!

, whereC

= 1, 33 3

,#

,

4 D

F = (1)

D

(G

)

D

C !=

G(

G

1) 33 3

(G

C

+ 1)

C !, where

C

= 1, 2, 33 3

General case:

4 H

F =9 (

G+ 1)

9 (I

+ 1) 9 (G

I

+ 1), where 9 (

P) is the gamma function.

Properties:

4

0F = 1,

4 D

E = 0 forC

= 1, 2, 33 3

orC

>Q

,

4 H

+1F =

G

I

+ 14 H

F

1=

G

I

I

+ 14 H

F ,

4 H

F +

4 H

+1F =

4 H

+1F

+1,

4

E

1R2

=(1)

E

22E S

E

2E = (1)

E (2 Q 1)!!

(2Q

)!!,

S

E

1 R 2=

(1)E

1

Q

22E

1S

E

1

2E

2 =(1)

E

1

Q

(2Q

3)!!

(2Q

2)!!,

S

2E

+1E

+1R2

= (1)E

24E

1S

E

2E ,

S

E

2E

+1R2

= 22E

S

2E

4E

+1,

S

1R

2E = 2

2E

+1

8

S

E

2E

,S

E

R

2E = 2

2E

8

S

(E

1) R 2E .

2002 by Chapman & Hall/CRC

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A.1.3. Pochhammer Symbol

Definition and some properties (C

= 1, 2, 33 3

):

(G

) E =G

(G

+ 1) 33 3

(G

+Q

1) =9 (

G

+Q

)

9 ( G )= (1)

E9 (1

G)

9 (1 G Q ),

(G

)0 = 1, ( G ) E +

D

= (G

) E (G

+Q

)

D

, (Q

)

D

=(

Q+

C 1)!

(Q

1)!,

(G

) E =9 (

G

Q)

9

(G

)

=(1)

E

(1 G

)E

, whereG

1, 33 3

,Q

;

(1) E = Q !, (1 T 2) E = 22E (2

Q)!

Q!

, (3 T 2) E = 22E (2

Q

+ 1)!

Q!

,

(G

+U C

) E

D

=( G ) V

D

+E

D

(G

)V

D

, (G

+Q

) E =( G )2 E

(G

) E, (

G+

Q)

D

=( G )

D

( G + C ) E

(G

) E.

A.1.4. Bernoulli Numbers

Definition:

P

W X

1

=Y

`

E

=0 a

E

P

E

Q

!

.

The numbers:

a

0 = 1,

a

1 = 1

2,

a

2 =1

6,

a

4 = 1

30,

a

6 =1

42,

a

8 = 1

30,

a

10 =5

66, 3

3 3,

a

2 V +1 = 0 for U = 1, 2, 3 3 3

A.2. Error Functions and Exponential Integral

A.2.1. Error Function and Complementary Error Function

Definitions:

erfP

=2

78 b

X

0

exp(c

2)d c

, erfcP

= 1 erfP

=2

78 b

Y

X

exp(c

2)d c

.

Expansion of erfP

into series in powers ofP

asP e

0:

erfP

=2

78

Y

`

D

=0

(1)

D

P

2

D

+1

(C

)!(2C

+ 1)=

27

8 expf

P

2g

Y

`

D

=0

2

D

P

2

D

+1

2C

+ 1)!!.

Asymptotic expansion of erfcP

asP e h

:

erfcP

=1

7 8 expf

P

2g i p

1`

V =0

(1)V

f

1

2

g

V

P

2 V +1+

q f|

P|2

p

1g r ,

t= 1, 2, 3

3 3

A.2.2. Exponential Integral

Definition:

Ei(P

) = b

X

Y

W u

c

d cfor

P< 0,

Ei(P

) = limv w

+0 x

b

v

Y

W u

c

d c

+ b

X

v

W u

c

d c yfor

P> 0.


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Other integral representations:

Ei( P ) = W X

b

Y

0

P sin c + c cos c

P

2 +c

2d c for P > 0,

Ei(P

) = W X

b

Y

0

Psin

c

ccos

c

P

2 +c

2d c

forP

< 0,

Ei(P

) = P

b

Y

1

W X u lnc d c

forP

> 0.

Expansion into series in powers of P as P e 0:

Ei(P

) =

+ ln(P

) +Y

`

D

=1

P

D

C

C

!if

P< 0,

+ lnP

+Y

`

D

=1

P

D

C

C

!if

P> 0,

where

= 0.5772 33 3

is the Euler constant.

Asymptotic expansion asP e h

:

Ei(P

) = W X

E

`

D

=1

(1)

D

(C

1)!

P

D

+

E ,

E 0,

9

(k

) =l

0m n

1

m

.

For (

+ 1) < Rek

<

, where

= 0, 1, 2, 33 3

,

( k ) = l

0

=0

(1)

! m n

1

m

.

A.4.1-2. Some formulas.

Euler formula

(k

) = lim

z

! n

k(

k+ 1) 3

3 3(

k+

)

(k

0, 1, 2, 33 3

).

Simplest properties:

( k + 1) = k

( k ),

( + 1) = !,

(1) =

(2) = 1.

Symmetry formulas:

( k )

( k ) = {k sin(

{

k ),

( k )

(1 k ) = {sin(

{

k ),

|

12

+k }

|

12

k }

={

cos({

k)

.

Multiple argument formulas:

(2k

) =22

n

1

{

(k

) |

k

+1

2}

,

(3k

) =33

n

1 ~ 2

2{

(k

) |

k+

1

3}

|

k+

2

3}

,

( k

) = (2 { )(1

) ~ 2

n

1~2

1

=0

|

k

+

}

.

Fractional values of the argument:

| 1

2}

=

{

,

|

1

2}

= 2

{

,

|

+1

2}

=

{

2

(2

1)!!,

| 1

2

}= (1)

2

{

(2 1)!!.

Asymptotic expansion (Stirling formula):

(k

) = 2{

n

k

n

1~2

1 + 112

k

1 + 1288

k

2 +

(k

3)

(|arg |k

0):

(k

) =l

0

(1 +m

)

n

m

1

m

,

(k

) = lnk

+l

0

m

1 (1 )1

n

m

,

(k

) =

+l

1

0

1 m

n

1

1 m

m

,

where

=

(1) = 0.5772 33 3


Values for integer argument:

(1) = ,

( ) = +

1

=1

1 ( = 2, 3, 3 3 3 )

A.4.2. Beta Function

Definition:

(

,

) =l

1

0m

1(1 m

)

1

m

,

where Re

> 0 and Re

> 0.

Relationship with the gamma function:

(

,

) =

(

)

(

)

( + ) .

A.5. Incomplete Gamma and Beta Functions

A.5.1. Incomplete Gamma Function

Definitions (integral representations):

( , ) = l 0

m

1

m

, Re > 0,

(

,

) =l

m

1

m

=

(

)

(

,

).


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2m

. Letn

=n

be positive zeros of the Bessel function derivative z

(n

), where

> 1 and

= 1, 2, 3,

Then the set of functions

(n

o

k

) is orthogonal on the interval 0 o

with

weight o :

0

n

o

n

o

o o = |

}~

0 if

,

1

2

2

1

2

n

2

2 ( n ) if = .

3 m . Let n = n be positive roots of the transcendental equationn z

( n ) +

( n ) = 0, where > 1

and = 1, 2, 3,


( n ok

) is orthogonal on the interval 0 o

with weight o :

0

n

o

n

o

o o =|

}~

0 if ,

1

2

2

1 +

2 2

n

2

2 ( n ) if = .

4 m . Let n = n be positive roots of the transcendental equation

(

)

(

)

(

)

(

) = 0 (

> 1,

= 1, 2, 3,

).


(

o ) =

(

o )

(

)

(

)

(

o ),

= 1, 2, 3,

,

satisfying the conditions

( ) =

( ) = 0 is orthogonal on the interval o with

weight o :

( o )

(

o ) o o = |

}~

0 if ,2

2

2

2 ( ) 2

( )

2

( )if = .

5 m . Let n = n be positive roots of the transcendental equation

z

(

)

z

(

)

z

(

)

z

(

) = 0 (

> 1,

= 1, 2, 3,

)


( o ) =

( o )

z

( ) z

( )

( o ), = 1, 2, 3,

,

satisfying the conditions

z

(

) =

z

(

) = 0 is orthogonal on the interval

o

withweight o :

( o )

(

o ) o o = |}

~

0 if ,

2 2

2

1

2

2

2

z

( ) {2

z

(

){

2

1

2

2

2

if = .

A.6.4. Hankel Functions (Bessel Functions of the Third Kind)

The Hankel functions of the first kind and the second kind are related to Bessel functions by (1) ( ) =

( ) +

( ), (2) ( ) =

( )

( ), 2 = 1.

Asymptotics for x

0:

(1)0

( ) 2

ln , (1) ( )

( )

( k

2) (Re > 0),

(2)0

( ) 2

ln , (2) ( )

( )

( k

2) (Re > 0).

Asymptotics for | |x y

:

(1) ( ) 2

exp

12

14

( < arg < 2 ),

(2) (

)

2

exp

12

1

4

(2 < arg

< ).


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A.8. Airy Functions

A.8.1. Definition and Basic Formulas

A.8.1-1. The Airy functions of the first and the second kinds.

The Airy function of the first kind, Ai(

), and the Airy function of the second kind, Bi(

), are

solutions of the Airy equation

= 0

and are defined by the formulas

Ai(

) =1

0

cos 13

3 +

,

Bi(

) =1

0

exp 13

3 +

+ sin 13

3 +

.

Wronskian:

Ai(

), Bi(

) = 1

.

A.8.1-2. Connection with the Bessel functions and the modified Bessel functions:

Ai(

) =1

3

1

3(

)

1

3(

)

=

1 1

3

1

3(

),

=

2

3

3 2

,Ai(

) = 13

1 3( ) +

1 3( ) ,

Bi(

) = 13

1 3( ) +

1 3( ) ,

Bi(

) = 13

1 3( )

1 3( ) .

A.8.2. Power Series and Asymptotic Expansions

A.8.2-1. Power series expansions as

0:

Ai(

) = 1 (

) 2 (

),

Bi(

) =

3

1 (

) + 2 (

) ,

(

) = 1 +1

3!

3 +1 4

6!

6 +1 4 7

9!

9 +-

- -

=

=0

3

1

3

3

(3

)!,

(

) =

+2

4!

4 +2 5

7!

7 +2 5 8

10!

10 +-

- -

=

=0

3

2

3

3

+1

(3

+ 1)!,

where 1 = 32

3

(2 3) 0.3550 and 2 = 31

3

(1 3) 0.2588.

A.8.2-2. Asymptotic expansions as

.

For large values of

, the leading terms of asymptotic expansions of the Airy functions are

Ai(

) 12

1 2

1 4 exp( ), = 23

3 2,

Ai(

)

1 2

1 4 sin

+

4

,

Bi(

)

1 2

1 4 exp(

),

Bi(

)

1 2

1 4

cos

+

4

.

Reference: M. Abramowitz and I. Stegun (1964).


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A.9.1-2. Kummer transformation and linear relations.

Kummer transformation:

( , ;

) =

( , ;

), ( , ;

) = 1

(1 + , 2 ;

).

Linear relations for

:

( ) ( 1, ;

) + (2 +

) ( , ;

)

( + 1, ;

) = 0,

( 1) ( , 1;

) ( 1 +

) ( , ;

) + ( )

( , + 1;

) = 0,

(

+ 1)

(

,

;

) (

+ 1,

;

) + (

1)

(

,

1;

) = 0,

(

,

;

)

(

1,

;

)

(

,

+ 1;

) = 0,

(

+

)

(

,

;

) (

)

(

,

+ 1;

)

(

+ 1,

;

) = 0,

( 1 +

) ( , ;

) + ( ) ( 1, ;

) ( 1) ( , 1;

) = 0.

Linear relations for

:

( 1, ;

) (2 +

) ( , ;

) + ( + 1) ( + 1, ;

) = 0,

( 1) ( , 1;

) ( 1 +

) ( , ;

) +

( , + 1;

) = 0,

( , ;

)

( + 1, ;

) ( , 1;

) = 0,

( ) ( , ;

)

( , + 1;

) + ( 1, ;

) = 0,

(

+

)

(

,

;

) +

(

1)

(

+ 1,

;

)

(

,

+ 1;

) = 0,

(

1 +

)

(

,

;

)

(

1,

;

) + (

+ 1)

(

,

1;

) = 0.

A.9.1-3. Differentiation formulas and Wronskian.

Differentiation formulas:

( , ;

) =

( + 1, + 1;

),

( , ;

) =

( + 1, + 1;

),

( , ;

) =( )

( )

( + , + ;

),

( , ;

) = (1) ( )

( + , + ;

).

Wronskian:

( , ) =

=

( ) ( )

.

A.9.1-4. Degenerate hypergeometric functions for

= 0, 1, 2,

:

( , + 1;

) =(1) 1

! ( ) ( , +1;

) ln

+

=0

( )

( + 1)

( + o )

(1 + o )

(1 + + o )

o ! +

( 1)! ( )

1

=0

( )

(1 )

o !,

where

= 0, 1, 2,

(the last sum is dropped for

= 0),

(

) = [ln (

)] is the logarithmic

derivative of the gamma function,

(1) =

,

(

) =

+

1

=1

1,

where

= 0.5772



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If < 0, then the formula

(

,

;

) = 1

(

+ 1, 2

;

)

is valid for any

.

For

0, 1, 2, 3,

, the general solution of the degenerate hypergeometric equation can

be represented in the form =

1

( , ;

) + 2

( , ;

),

and for

= 0, 1, 2, 3,

, in the form

= 1

1

(

+ 1, 2

;

) + 2

(

+ 1, 2

;

)

.

A.9.2. Integral Representations and Asymptotic Expansions

A.9.2-1. Integral representations:

( , ;

) = ( )

(

)

(

)

1

0

1(1 )

1

(for > > 0),

(

,

;

) =1

(

)

0

1(1 +

)

1

(for

> 0,

> 0),

where ( ) is the gamma function.

A.9.2-2. Integrals with degenerate hypergeometric functions:

( , ;

)

= 1

1

( 1, 1;

) +

,

( , ;

)

=1

1

( 1, 1;

) +

,

( , ;

)

= !

+1

=1

(1)

+1(1 )

+1

(1 )

(

+ 1)!

(

,

;

) +

,

( , ;

)

= !

+1

=1

(1)

+1

+1

(1 )

(

+ 1)!

(

,

;

) +

.

A.9.2-3. Asymptotic expansion as |

| :

( , ;

) = (

)

( )

=0

(

)

(1

)

!

+ !

,

> 0,

( , ;

) ="

( )

"

( )(

)

#

=0

($

)

($

+ 1)

% !(

&) +

!,

&< 0,

($

, ;&

) =&

#

=0

(1) (

$)

($

+ 1)

% !&

+ !

,

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