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 ( − 1) , (2 )!! = 2 ⋅ 4 ⋅ 6 (2 − 2)(2 ) = 2 !, (2 + 1)!! = 1 ⋅ 3 ⋅ 5
!! = *
= 2, 3,
!"!"!
!"!"!
,
õ
!"!"!
(2 − 1)(2 + 1) =
!"!"!
(2 + )!! if (2 + + 1)!! if
= 2+ , = 2 + + 1,
2 %
õ
+1
&
' (
+
3 , 2)
+
= 1, 2,
0!! = 1.
A.1.2. Binomial Coefficients Definition: " , .
General case:
! , where + !( − + )! (− / ) / ( / − 1) = (−1) = + !,
= -
" ,
,
. " 0
=
+ !"!"!
+
( / + 1) , ( 1 + 1) ' ( / − 1 + 1) '
'
= 1,
!"!"!
, ,
( / − + + 1) , !
where '
where
!"!"!
(2 ) is the gamma function.
Properties: .0
= 0 for + = −1, −2, or + > 3 , / − 1 . +1 = . −1 = . , . + . +1 = . +1 +1 , 1 +1 1 +1 0 " 0 " 0 " 0 " 0 " 0 - (23 − 1)!! (−1) , - 5 2 - = (−1) −1 4 2 = 22 (23 )!! - −1 - −1 - −1 (−1) (−1) (23 − 3)!! , 5 2 - −2 = 14 2 = 3 22 −1 3 (23 − 2)!! -2 +1 - −4 - −1 −2 2- +1 4 2 = (−1) 2 5 2- , 5 2 - +1 4 2 = 2 5 4 - +1 , "
= 1,
-
!"!"!
/ " ,
" " 5 5
14 2 5 -
= &
- +1 5 2-
22
,
- 4 2 5 -
=
22 &
-
( - −1) 4 2 . 5 -
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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES
A.1.3. Pochhammer Symbol Definition and some properties ( + = 1, 2,
):
!"!"!
' (1 − / ) (/ + 3 ) = (−1) , ' (/ ) ' (1 − / − 3 ) (3 + + − 1)! , ( / )0 = 1, ( / ) - + = ( / ) - ( / + 3 ) , (3 ) = (3 − 1)! , , ' ( / − 3 ), (−1) = ( / )− - = , where / ≠ 1, , 3 ; ' (/ ) (1 − / ) - (23 )! - (23 + 1)! (1) - = 3 !, (1 6 2) - = 2−2 , (3 6 2) - = 2−2 , 3 ! 3 ! (/ )8 +( / )2 (/ ) (/ + + ), (/ + 3 )- = , (/ + 3 ) = . (/ + 7 + )- = (/ )8 , , (/ ), (/ )
( / ) - = / ( / + 1)
( / + 3 − 1) =
!"!"!
'
!"!"!
,
,
,
A.1.4. Bernoulli Numbers Definition: 9 :
2
=
−1
0
exp(−? 2 ) @ ? ,
erfc 2 = 1 − erf 2 =
Expansion of erf 2 into series in powers of 2 as 2 erf 2 = %
2 &
:
exp(−? 2 ) @ ? . ;
0: A
2 2 +1
(−1) ;
2
% &
2
=
% &
exp B −2
2C
< ;
=0
2 2 2 +1 . , 2 + , + 1)!!
,
:
(−1)
8
8 =0
2
C B 12 8 2 8 +1
+G
B |2
|−2 F
−1 CIH
, J
= 1, 2,
!"!"!
A.2.2. Exponential Integral Definition: Ei(2 ) =
:
9 K
>
@ ? ?
−
;
Ei(2 ) = L lim M
+0 N
−L 9 K
> ?
−
@ ?
+> L
:
9 K ?
@ ?PO
for 2
< 0,
for 2
> 0.
;
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A.3. SINE INTEGRAL
AND
657
COSINE INTEGRAL. FRESNEL INTEGRALS
Other integral representations: −: >
Ei(−2 ) = − 9
9 −: >
Ei(−2 ) =
Ei(−2 ) = −2
0
2 ;
0
>
2 ;
9 −: K ;
1
sin ? + ? cos ? 2 2+? 2 sin ? − ? cos ? @ 2 2+? 2
Expansion into series in powers of 2 as 2
ln ?
RR U
Ei(2 ) =
RR
+ ln 2 +
0,
for 2
< 0, > 0.
2
if 2
< 0,
if 2
> 0,
,
!"!"!
Ei(−2 ) =
⋅,+ ! +
=1
For small 2 , li(2 ) ≈
if 0 < 2 < 1, @ ?
2
ln(1 6
> 1. 2
.
) 2
if O
ln ?
1+ L
1: A
li(2 ) =
U
+ ln |ln 2 | +
< ;
+
=1
ln 2 . ⋅, + !
,
A.3. Sine Integral and Cosine Integral. Fresnel Integrals A.3.1. Sine Integral Definition: Si(2 ) =
: >
0
sin ? ?
@ ?
si(2 ) = − >
,
:
sin ? ;
?
= Si(2 ) −
@ ?
W
2
.
Specific values: Si(0) = 0, Properties:
Si(−2 ) = − Si(2 ),
Si( D ) = W
2
si( D ) = 0.
,
si(2 ) + si(−2 ) = − , W
lim : M −
si(2 ) = − . W
;
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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES 0:
Expansion into series in powers of 2 as 2 A
cos ? ;
:
?
U
=
@ ?
:
+ ln 2 + >
0
A
+ ln 2 + 5
< ;
=1
Asymptotic expansion as 2 E F
Ci(2 ) = cos 2 where J
,Y
< −1
D
8
8 2 2
U
,
= 0.5772
!"!"!
(−1) 2 2 . 2 + (2, + )!,
,
− 1)!
(−1) (27
8 =1
= 1, 2,
:
A
@ ?
?
0:
Expansion into series in powers of 2 as 2 Ci(2 ) =
cos ? − 1
+G
C H
|−2 F
B |2
E X
+ sin 2
< −1
8
(−1) (27 )! 8 2 2 +1
8 =0
+G
B |2
|−2 X
−1 C H
,
!"!"!
A.3.3. Fresnel Integrals Definitions: Z
5
(_ ) =
:
1 2
(2 ) = [
1 2 [
sin ?
>
[
?
0
W > W
cos ?
0
^
Expansion into series in powers of _ as _
[
Z
acb
2
(_ ) = \
?
0: A _
` =0
W
(_ ) = 5
2 \ W
Asymptotic expansion as _ Z
e
f
_
acb
=
@ ?
` =0
2 \ W
@ ?
= \
>
] ^
sin ?
] ^
cos ?
0
2 Wb
>
0
2
@ ?
2
,
@ ?
.
b
(−1) _ 2 +1 , b (4 d + 3) (2b d + 1)! (−1) _ 2 . (4 d + 1) (2 d )!
:
1 cos _ sin _ − [ (_ ) − [ (_ ), 2 2 _ g 2 _ h W W 1 sin _ cos _ + [ (_ ) − [ (_ ), 5 (_ ) = 2 2 _ g 2 _ h W W 1⋅3 1⋅3⋅5⋅7 1 1⋅3⋅5 (_ ) = 1 − + − ijiji , (_ ) = − + (2_ )2 (2_ )4 2_ (2_ )3 h (_ ) =
g
ijiji
.
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A.4. GAMMA AND BETA FUNCTIONS
A.4. Gamma and Beta Functions A.4.1. Gamma Function A.4.1-1. Definition. Integral representations. The gamma function, ' ( k ), is an analytic function of the complex argument for the points k = 0, −1, −2, For Re k > 0, k
everywhere, except
!"!"!
(k ) = '
l
−1 p − q r
`
0
mon
For −(s + 1) < Re k < −s , where s = 0, 1, 2, t
(k ) = l
p −q `
0
−
u
!"!"!
. m
, w
a
v w =0
(−1) x !
−1 r
y mon
. m
A.4.1-2. Some formulas. Euler formula t
v z
Simplest properties: t
!s s
( k ) = lim
( k + 1) k
n
!"!"!
( k ≠ 0, −1, −2,
(k + s )
!"!"!
).
`
( k + 1) = k
t
t
( k ),
(s + 1) = s !,
t
t
(1) = (2) = 1.
Symmetry formulas: t
t
( k ) (− k ) = − t |
t
{
sin(
k
1 +k 2
}
k {
t |
t
, ( k ) (1 − k ) = { ) sin( { 1 { −k } = . 2 cos( k ) k
)
,
{
Multiple argument formulas: t
(2 k ) = t
t
(3 k ) = (s k
22
−1 t
[ n
33
{
(k )
−1 ~ 2
n
2
t
t |
(k )
k
t |
+
1 } , 2
k
+
{
1 } 3
) = (2 )(1− {
)~ 2
v
−1 ~ 2
s
v n
t |
b −1 t | v
k
k
2 } , 3
+ +
=0
d s
}
.
Fractional values of the argument: t | t |
1 } = , { 2 1 − } = −2 2
t |
{
,
t |
s
1 } = { (2s − 1)!!, 2 2 v 1 2 { − s } = (−1) . v 2 v (2s − 1)!! +
Asymptotic expansion (Stirling formula): t
(k ) =
2
p − { n
k n
−1 ~ 2
1+
1 −1 12 k
+
1 −2 288 k
+ (k
−3
)
(|arg | k < ). {
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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES
A.4.1-3. Logarithmic derivative of the gamma function. Definition:
Functional relations:
t
r
(k ) =
t
ln ( k ) r
t
(k ) . ( n k )
{
), 1 )− ,
=
k
1 ( k ) − (1 + k ) = − ,
k
( k ) − (1 − k ) = − cot(
{
( k ) − (− k ) = − cot(
1 2
(x
+k k
1 2
−k
+ x
−
) = ln x
{
k
k {
k
= w −1 { ab
1
tan( |
+ k
), k
{
d x
=0
. }
Integral representations (Re k > 0):
(k ) =
l
0
p − q `
− (1 + )− m
( k ) = ln k + l
( k ) = − + l
0 1
−1 `
m
m
− (1 − p
m
−1
1− 1 −m n
0
−1 r
n
r m
m
,
− q −1 p − q
)
r n
m
,
,
is the Euler constant. where = − (1) = 0.5772 Values for integer argument: !"!"!
(1) = − ,
ab −1 −1 d v
( s ) = − +
(s = 2, 3,
!"!"!
)
=1
A.4.2. Beta Function Definition:
(_ , ) =
1
l
−1
(1 − )
0 m ^
−1 r
m
m
,
where Re _ > 0 and Re > 0. Relationship with the gamma function:
t
(_ , ) =
t
t
(_ ) ( ) . (_ + )
A.5. Incomplete Gamma and Beta Functions A.5.1. Incomplete Gamma Function Definitions (integral representations):
t
( , _ ) = l
( , _ ) = l
0
^ p −q
^
`
−1 r
m
p −q
m
t
−1 r
m
Re > 0,
,
m
= ( ) − ( , _ ).
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A.6. BESSEL FUNCTIONS
Recurrent formulas:
( + 1, ) =
( , ) − − ,
( + 1, ) =
( , ) + − .
0:
Asymptotic expansions as
( , ) =
Asymptotic expansions as
(−1) + , ! ( + )
=0
( , ) = ( ) −
=0
(−1) + . ! ( + )
:
−1 −
( , ) = ( ) −
−1 (1 − ) (− ) =0
−1 (1 − ) ( , ) = −1 − =0 (− )
+ | |− ,
+ | |−
− 32 < arg < 32 .
Integral functions related to the gamma function: erf =
1
1
2
, 2 ,
1 1 2 , , 2
erfc =
Ei(− ) = − (0, ).
A.5.2. Incomplete Beta Function Definition: ( , ) =
1 0
−1
(1 − )
−1
! ,
where Re > 0 and Re " > 0.
A.6. Bessel Functions A.6.1. Definitions and Basic Formulas A.6.1-1. The Bessel functions of the first and the second kinds. The Bessel function of the first kind, # $ ( ), and the Bessel function of the second kind, % $ ( ) (also called the Neumann function), are solutions of the Bessel equation 2 " &'& + " & + ( 2 − ( 2 )" = 0
and are defined by the formulas # $ ( ) =
*)
) $
)
=0
(−1) ( + 2) +2 , , , ! ( ( + + 1)
% $ ( ) =
# $ ( ) cos ( − # − $ ( ) . sin (
(1)
The formula for % $ ( ) is valid for ( ≠ 0, - 1, - 2, ././. (the cases ( ≠ 0, - 1, - 2, ././. are discussed in what follows). The general solution of the Bessel equation has the form 0 $ ( ) = 1 1 # $ ( ) + 1 2 % $ ( ) and is called the cylinder function.
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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES
A.6.1-2. Some formulas. 2 ( 0 $ ( ) = [ 0 $ −1 ( ) + 0 $ +1 ( )], 1 ( ! 0 $ ( ) = 2 [ 0 $ −1 ( ) − 0 $ +1 ( )] = - 2 0 $ ( ) − 0 $43 1 ( )5 , !
!
!
$ −$ 0 $ ( )] = − − 0 $ +1 ( ), ! [
6 6
1 ! 1 ! $ $
$ $ −
[ # $ ( )] = [ − # $ ( )] = (−1) − − # $ + ( ), # $ − ( ), ! 7 ! 7
# − ( ) = (−1) # ( ), % − ( ) = (−1) % ( ), = 0, 1, 2, ././.
[
!
$
0 $ ( )] =
$
0 $ −1 ( ),
A.6.1-3. The Bessel functions for ( = - 2
# 1 8 2 ( ) = 9
# 3 8 2 ( ) = 9
2
# +1 8 2 ( ) = 9
where = 0, 1, 2, ././. : # −1 8 2 ( ) = 9
sin , 6
1
sin − cos
# −3 8 2 ( ) = 9 )
, 7
2
2
cos , 6
−
1
cos − sin
) , [ 8 2] * ) 2 (−1) ( + 2 )! , , sin − ) )2 2 (2 )! ( − 2 )! (2 =0
+ cos − # − −1 8 2 ( ) = 9
1 2,
[( −1) 8 2]
*)
2
=0
, (−1) ( + 2 + 1)! , , (2 ) + 1)! ( − 2 − 1)! (2 )2
) , [ 8 2] * ) 2 (−1) ( + 2 )! , , cos + 2 (2 )! ( − 2 )! ) (2 )2 =0
− sin +
[( −1) 8 2]
2
=0
2
% 1 8 2 ( ) = − 9
, (−1) ( + 2 + 1)! , , (2 + 1)! ( − 2 − 1)! (2 )2
*)
% −1 8 2 ( ) = 9
cos ,
% +1 8 2 ( ) = (−1) +1 # − −1 8 2 ( ),
2
, 7
)
+1
,
)
+1
,
sin ,
% − −1 8 2 ( ) = (−1) # +1 8 2 ( ).
A.6.1-4. The Bessel functions for ( = - , where = 0, 1, 2, ././. Let ( = be an arbitrary integer. The relations
# − ( ) = (−1) # ( ),
% − ( ) = (−1) % ( )
are valid. The function # ( ) is given by the first formula in (1) with ( = , and % ( ) can be obtained from the second formula in (1) by proceeding to the limit ( . For nonnegative , % ( ) ) ) ; ; can be represented in the form ) 2
% ( ) = # ( ) ln − 2 ; ;
where (1) = −< ,
−1
1 :)
=0
,
( − − 1)! 2 , !
−2
−
1 *)
=0
(−1) 2
) −1 = , −1 ( ) = −< + , < = 0.5772 ././. is the Euler constant, =1
,
+2 ;
,
( + 1) + ( + + 1) , , , ! ( + )!
( ) = [ln ( )] & is the
logarithmic derivative of the gamma function.
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A.6. BESSEL FUNCTIONS
A.6.1-5. Wronskians and similar formulas: >
( # $ , # −$ ) = −
2
>
sin( ( ),
2 sin( ( ) # $ ( ) # − $ +1 ( ) + # − $ ( ) # $ −1 ( ) = , > Here, the notation ( ? , @ ) = ? @ & − ? & @ is used.
(# $ , % $ ) =
2
,
# $ ( ) % $ +1 ( ) − # $ +1 ( ) % $ ( ) = −
2
.
A.6.2. Integral Representations and Asymptotic Expansions A.6.2-1. Integral representations. The functions # $ and % $ can be represented in the form of definite integrals (for > 0): # $ ( ) = % $ ( ) =
0
0
A cos( sin B − ( B ) ! B − sin ( A sin( sin B − ( B ) ! B −
For | ( | < 12 , > 0,
$
0
$DC
(
0
exp(− sinh − ( ) ! ,
+
− $DC
cos ( )
− sinh C
! .
$
21+ − sin( ) ! , 18 2 ( 1 − ( ) ( 2 − 1) $ +1 8 2 1 2 $ $ 21+ − cos( ) ! % $ ( ) = − . 18 2 ( 1 − ( ) ( 2 − 1) $ +1 8 2 1 2 # $ ( ) =
For ( > − 21 , # $ ( ) =
For ( = 0, > 0, # 0 ( ) =
$ 8 2 2( + 2) $ A cos( cos ) sin2 ! 18 2 ( 1 + ( ) 0 2
2
0
sin( cosh ) ! ,
% 0 ( ) = −
For integer ( = = 0, 1, 2, ././. , 1
# ( ) =
2
A cos(
0
2
# 2 ( ) = # 2 +1 ( ) =
8 2
0
A
0
A
8 2
− sin ) !
(Poisson’s formula).
2
cos( cosh ) ! .
0
(Bessel’s formula),
cos( sin ) cos(2 ) ! ,
sin( sin ) sin[(2 + 1) ] ! .
A.6.2-2. Integrals with Bessel functions:
0
where
H
$ E + +1 $ 2 ( F + ( + 1) G ( ( + 1) H
F +( +1
F +( +3
2
, Re( F + ( ) > −1, 2 2 4 ( J , K , L ; ) is the hypergeometric series (see Section 10.9 of this supplement),
E # $ ( ) ! =
,
, ( +1; − I
I
2 cos( ( ) G (− ( ) + $ +1 F + ( + 1 F +( +3 =− $ , ( + 1, , −I E % $ ( )! E I I 2 ( F + ( + 1) I H 2 2 4 0 I $ 2 2 G (( ) F −( +3 $ F −( +1 − , 1−( , , −I , Re F > |Re ( | − 1. E − +1 F − ( +1I H 2 2 4
M
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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES
A.6.2-3. Asymptotic expansions as | | N 2
# $ ( )= 9 I
I
−1
W 4 − 2( R − R I 4 S TVU X
cos Q P
: O
I
X
X
(−1) ( Y , 2Z )(2 )−2 + [ (| |−2 )\ I
=0 −1
W 4 − 2Y R − R − sin Q I 4 S T UX ^ _
2
( )= ` I
W 4a − 2 Y R − R 4 S T UX
sin Q
R a P
−1
1 (4 Y X 22 Z !
2
− 1)(4 Y
2
X
I
=0
X
e
X
=0
− 32 ) b/b/b [4 Y
U
−1
X
+ 1)(2a )−2
(−1) ( Y , 2Z
f
X −1 + [ (|a |−2 −1 )\ ] ,
=0 2
U
− 1)2 ] =
− (2Z
c
( 12
Z ! c
+Y +Z ) . +Y −Z )
( 21
R a f 2 g (a ) = (−1) g (cos a + sin a ) + [ (a −2 ), R a f 2 g +1 (a ) = (−1) g +1 (cos a − sin a ) + [ (a −2 ).
A.6.2-4. Asymptotic for large Y ( Y N _
X −1 + [ (| |−2 −1 )\ ] , I U
(−1) ( Y , 2Z )(2a )−2 + [ (|a |−2 )\
For nonnegative integer d and large a , e
U
+ 1)(2 )−2
(−1) ( Y , 2Z
W 4a − 2 Y R − R + cos Q 4 S T UX
where ( Y , Z ) =
I
O
):
1 a e Q i 2 Y S 2R Y
(a ) h
_
^ _
,
(a ) h − `
2
Q i
a
2Y S
R Y
−
_
,
where a is fixed, _ f
21 j 3 1 , (2 k 3) Y 1 j 3
(Y ) h
32 j 3
^ _
(Y ) h −
c
21 j 3 1 . (2 k 3) Y 1 j 3
31 j 6
c
A.6.3. Zeros and Orthogonality Properties of Bessel Functions A.6.3-1. Zeros of Bessel functions. _
^ _
Each of the functions f (a ) and (a ) has infinitely many real zeros (for real Y ). All zeros are simple, except possibly for the point a = 0. The zeros l X of f 0 (a ), i.e., the roots of the equation f 0 (l X ) = 0, are approximately given by l X
= 2.4 + 3.13 (Z
− 1)
= 1, 2, b/b/b ),
(Z
with maximum error 0.2%. A.6.3-2. Orthogonality properties of Bessel functions. _
1 m . Let n = n X be positive roots of the Bessel function f (n ), where Y > −1 and Z = 1, 2, 3, b/b/b _ Then the set of functions f (n X o k p ) is orthogonal on the interval 0 ≤ o ≤ p with weight o : q
r 0
f
_ s n X o p
t
f
_ s n u o p
t
o v o = w
0
1 2 2p
y f _ z ( n X ){ 2 = 1 p 2 f _ 2 ( n X ) +1 2
if Z if Z
≠x , =x .
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A.6. BESSEL FUNCTIONS
_
2 m . Let n = n X be positive zeros of the Bessel function derivative f z (n ), where Y > −1 and _ Z = 1, 2, 3, / b b/b Then the set of functions f (n X o k p ) is orthogonal on the interval 0 ≤ o ≤ p with weight o : q r if Z ≠ x , }'~ 0 _ s n X o _ s n u o 2 s o v o Y _ f = | 1 2 f 1 − 2 f 2 (n X ) if Z = x . p t p t 2p 0 n X
t
_
_
3 m . Let n = n X be positive roots of the transcendental equation n f z (n ) + f (n ) = 0, where Y > −1 _ and Z = 1, 2, 3, b/b/b Then the set of functions f (n X o k p ) is orthogonal on the interval 0 ≤ o ≤ p with weight o : q r if Z ≠ x , }~ 0 _ s n X o _ s n u o o v o = | 1 2s 2−Y 2 _2 X f f 1+ f (n ) if Z = x . p t p t 2p 2 0 n X
t
4 m . Let n = n X be positive roots of the transcendental equation _ ^ _ _ ^ _ f ( X ) ( X p ) − f ( X p ) ( X ) = 0 ( Y > −1, Z Then the set of functions _
_
( X o ) = f ( X o )
satisfying the conditions weight o : q
_
r
_
( X o )
^ _
_
( X p ) − f ( X p )
( X p ) =
_
_
^ _
= 1, 2, 3, b/b/b ). = 1, 2, 3, b/b/b ,
( X o ), Z
( X ) = 0 is orthogonal on the interval p ≤ o ≤ with }'~ 0
( u o )o v o = |
_ _ f 2( X p ) − f 2( X ) _ 2 2X f 2( X )
2
if Z
≠x ,
if Z
=x .
5 m . Let n = n X be positive roots of the transcendental equation ^ _ ^ _ _ _ f z ( X ) z ( X p ) − f z ( X p ) z ( X ) = 0
( Y > −1, Z
= 1, 2, 3, b/b/b )
Then the set of functions _
^ _ ^ _ _ _ ( X o ) = f ( X o ) z ( X p ) − f z ( X p ) ( X o ),
satisfying the conditions weight o : q r
_
( X o )
_
_
z ( X p ) = }~ 0
( u o )o v o = |
_
Z
= 1, 2, 3, b/b/b ,
z ( X ) = 0 is orthogonal on the interval p
2
2 2X
s
y f _ z ( p ){ 2 s Y 2 − 1 − y f _ z ( ){ 2 p 2 2 t
Y 2 1 − 2 2 t
≤ o ≤ with if Z
≠x ,
if
=x .
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) _ ^ _ ( ) = f ( ) + ( ),
Asymptotics for N
_ (1) _ (2)
2 = −1.
0: 2
(1) 0 (
)h
(2) 0 (
2 ) h − ln ,
Asymptotics for | | N
_ (2) _ ^ _ ( ) = f ( ) − ( ),
ln ,
_ (1) ( ) h _ (2)
( ) h
( Y )_ c ( k 2) ( Y )_ ( c k 2)
−
(Re Y > 0), (Re Y > 0).
: O
( ) h `
2 1 1 D exp − 2 Y − 4
(− < arg < 2 ),
( ) h `
2 1 1 D exp − − 2 Y − 4
(−2 < arg < ).
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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES
A.7. Modified Bessel Functions A.7.1. Definitions. Basic Formulas A.7.1-1. The modified Bessel functions of the first and the second kinds. _
_
The modified Bessel functions of the first kind, (a ), and the second kind, (a ) (also called the Macdonald function), of order Y are solutions of the modified Bessel equation + a z − (a
a 2 'z z
_
M
M M
and are defined by the formulas (a ) =
u =0
+ Y 2 ) = 0
2
u (a k 2)2 + , x ! ( Y + x + 1)
(a ) =
c
− − , 2 sin Y
(see below for (a ) with Y = 0, 1, 2, b/b/b ). A.7.1-2. Some formulas. The modified Bessel functions possess the properties − (a ) =
2 Y (a ) = a [ 1 (a ) = [ v a 2 v
− g (a ) = (−1) g g (a ),
(a );
−1 ( a
)−
+1 ( a
)],
−1 ( a
)+
+1 ( a
)],
2 Y (a ) = −a [ −1 (a ) − +1 (a )], v 1 (a ) = − [ −1 (a ) + +1 (a )]. v a 2 1 2,
A.7.1-3. Modified Bessel functions for Y = d 2
1 j 2 (a ) = ` 3 j 2 (a ) = `
2 a
−
+1 2 ( ) =
1 2
− −1 2 ( ) =
1 2
© ª ©
1
a
d = 0, 1, 2, b/b/b
where d = 0, 1, 2, b/b/b : 2
−1 j 2 (a ) = `
sinh a ,
sinh a + cosh ¥ a ,
a
cosh a ,
2
1
−3 2 ( ) = − cosh + sinh , a ¥ ¥ ¨ ¤*¥ (−1) (¦ + § )! (¦ + § )! − ¤:¥ − (−1) , ¥ ¢ ¡ £ =0 § ! (¦ − § )! (2 ) ¡ £ =0 § ! (¦ − § )! (2 ) ¥ ¥ ¨ *¤ ¥ (−1) (¦ + § )! (¦ + § )! − ¤:¥ + (−1) , ¢¡ £ =0 § ! (¦ − § )! (2 ) ¡ £ =0 § ! (¦ − § )! (2 )
1 2 ( ) =
+1 2 ( ) =
−
2 ¡ £ ©
© ª
,
− −1 2 ( ) =
3 2 ( ) = −
2
¤*¥
2 ¡ £
«
1+
1
−
¬ ¡ £ ¥
(¦ + § )!
§ ! (¦ − § )! (2 )
=0
, .
A.7.1-4. Modified Bessel functions for Y = ¦ , where ¦ = 0, 1, 2, // If Y = ¦ is a nonnegative integer, then ©
( ) = (−1) +1 ( ) ln
1¤ + 2 2®
−1
(−1)
®
=0
1 ¤° + (−1) 2 ® ²
2® −
(¦ − ¯
« 2¬ +2 ® =0
« ±2 ¬
− 1)!
¯ ! ²
(¦ + ¯
²
+ 1) + (¯ + 1) ; ¯ ! (¦ + ¯ )!
¦ = 0, 1, 2, // ,
where ( ³ ) is the logarithmic derivative of the gamma function; for ¦ = 0, the first sum is dropped.
© 2002 by Chapman & Hall/CRC Page 666
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A.7. MODIFIED BESSEL FUNCTIONS
A.7.1-5. Wronskians and similar formulas: ´
2
( /µ , − µ ) = −
2 sin( Y )
/µ ( ) − µ +1 ( ) − − µ ( ) ·µ −1 ( ) = − ± ± ± ± ± ´ where ( ¸ , ¹ ) = ¸ ¹ º − ¸ º ¹ . £
´
sin( Y ),
,
©
( ¶µ ,
/µ ( )
©
1
,
© µ +1 ( ) + ± ·µ +1 ( ) ± ±
± ±
µ )=−
1
µ ( )= ±
,
±
£
A.7.2. Integral Representations and Asymptotic Expansions A.7.2-1. Integral representations. The functions µ ( ) and
©
µ ( ) can be represented in terms of definite integrals:
±
µ ±
1
µ /µ ( ) = exp(− )(1 − 2 ) −1 » 2 ¿ 1» 22µ ± ¼ (Y + 1 ) ½ −1 2 ± ± ¾ ¾ ¾ © ° ¿ µ ( )= exp(− cosh ) cosh( Y ) ½ 0 ± ± ¾ ¾ ¾ © 1 ° µ ( )= cos( sinh ) cosh( Y ) ¿ cos À 12 Y Á ½ 0 ± ± ¾ ¾ ¾ © 1 ° ¿ µ ( )= sin( sinh ) sinh( Y ) sin À 12 Y Á ½ 0 ¾ ± ± ¾ ¾ For integer Y = ¦ ,
©
( )= ±
)=
0(
1 °
±
¾
¾
cos( sinh ) ¿
½ 0
±
exp( cos ) cos(¦ ) ¿
½ 0Â
±
=
¾
¾
( > 0, Y > − 12 ), ±
( > 0), ±
( > 0, −1 < Y < 1), ±
( > 0, −1 < Y < 1). ±
(¦ = 0, 1, 2, // ), ¾
°
½ 0
cos( ) ¿ 2 ± +¾ 1 ¾
( > 0). ¾
±
A.7.2-2. Integrals with modified Bessel functions: £
½ 0
± Ã
where ½ 0
µ
+ +1 2 Ä +Y +1 Ä +Y +3 = µ , Re( Ä + Y ) > −1, , , Y + 1; ¼ 2 ( Ä + Y ± Ã + 1) ( Y + 1) Å « 2 2 ±4 ¬ ± ± ( Æ , Ç , È ; ) is the hypergeometric series (see Section 10.9 of this supplement),
/µ ( ) ¿ Å © £ ± Ã
±
µ ( )¿ ±
±
=
2
µ −1 ¼ (Y )
Ä −Y +1
Ä −Y +1 ± Ã µ 2− −1 ¼ (− Y ) + Ä +Y +1
− µ +1
Å
2 «
Ä +Y +1
+ µ +1
Å
± Ã
2 «
A.7.2-3. Asymptotic expansions as Ì /µ ( ) = ¡ £ 2Ë ± ©
µ ( )= ±
Ì ±
Ë
−
±
¤
® =1
−
±
É
(−1)
1+ Í
2 ¡ £
The terms of the order of Ð (
® =1
Í
−1
Ä −Y +3
2
, 1+Y ,
,
Ä +Y +3
2
2
±4 ¬ 2
,
±4 ¬
,
Re Ä > |Re Y | − 1.
: ±
¤
1+ Í
, 1−Y ,
®
Ê
(4 Î
(4 Î
2
2
− 1)(4 Î
− 1)(4 Î
2
2
− 32 ) // [4 Î ® ¯ ! (8 ) ±
− 32 ) // [4 Î ® ¯ ! (8 )
) are omitted in the braces.
2
2
− (2¯
− (2¯
− 1)2 ] − 1)2 ]
Ï ,
Ï .
±
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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES
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 Ñ ± Ñ ± º'º − =0 and are defined by the formulas 1
Ai( ) = Ë
±
1
Bi( ) = ´
Wronskian:
Ô
Ë
±
±
£ £ °
1 3 3
cos À
½ 0
¾
° ½ 0 Ò
exp À
− 13 3
± ¾
¾
ÁDÓ ¿ .
Á + sin À 13 3 +
+
¾
Ai( ), Bi( ) Õ = 1 Ö Ë . ±
Á ¿ ,
+
± ¾
¾
± ¾
¾
±
A.8.1-2. Connection with the Bessel functions and the modified Bessel functions: Ai( ) = Ai(−± ) = ± Bi( ) = Ù ±
Bi(− ) = Ù ±
1 3 1 3
ÒØ×
±
ÒÛÚ
1± 3 1 3
Ò× ±
ÒÛÚ ±
−1 » 3 ( ³
)−
−1 » 3 ( ³
)+
−1 » 3 ( ³
)+
−1 » 3 ( ³
)−
)Ó = Ë
1 » 3 (³
× Ú ×
1» 3 (³
)Ó ,
1» 3 (³
)Ó ,
1 » 3 (³
Ú
−1 Ù
1 3
©
1» 3 (³
³ = 23 3 » 2 , ±
),
±
)Ó .
A.8.2. Power Series and Asymptotic Expansions 0:
A.8.2-1. Power series expansions as ±
É
Ai( ) = È 1 ¸ ( ) − È 2 ¹ ( ), ±
Bi( ) = 1 ¸ ( )=1+ 3! ±
¹ ( )= ±
where È
1
=3
−2 » 3
2 + 4!
3
± 4
1 ⋅ 4± + 6! 2⋅5 + 7!
6
±
±
±
3 È 1 ¸ ( ) + È 2 ¹ ( )Ó ,
Ò ± ¤*¥ 1⋅4⋅7 ± 9 ° + + // = 9! ±
7
2⋅5⋅8 + 10!
10
+ // =
¥ ¥ ¥
3 À =0
¤*¥ °
¥
3 À =0
± ± ± ± ¼ −1 » 3 ¼ Ö (2 Ö 3) ≈ 0.3550 and È 2 = 3 Ö (1 Ö 3) ≈ 0.2588.
A.8.2-2. Asymptotic expansions as
1 3
3
Á
(3± § )!¥ ¥
2 3
Á
,
3 +1
(3± § + 1)!
,
.
± of É asymptotic Ê For large values of , the leading terms expansions of the Airy functions are ±
Ai(−± ) Ü Bi( ± ) Ü ÝßÞ
1 −1 » 2 −1 » 4 2Ë −1 » 2 −1 ± » 4
Ai( ) Ü
exp(− ³ ),
sin À ³ +
Ë Ë
−1 » 2 ± −1 » 4
4
Á ,
³ = 23 3 » 2 , ±
exp( ³ ), Â
Ë −1 » 2 ± −1 » 4 cos À ³ + 4 Á . ± Â Reference: M. Abramowitz and I. Stegun (1964).±
Bi(−± ) Ü
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A.9. DEGENERATE HYPERGEOMETRIC FUNCTIONS
TABLE A1 Special cases of the Kummer function à ( Æ , Ç ; ³ ) Æ
³ Ç
Æ
Æ
1
2
±
Æ +1
2
3 2
−¦
1 2 3 2
−¦
−¦
±
−
2 Î +1
2
± ¦ !
(2¦ )! «
±2
2¦ +2
−
±2
¦ !
( ç −1)
¼ (1+ Î ) ¼
2 ±
«
¦ +
( )
2å
á −1 ¿ ¾
¾
2 å +1 (
å å = (−1)
)
æ
±
µ ( ) ±
− 12
×
¾
¿ å
2
2
¿
Laguerre polynomials (è )
( )
×
¾
− å À¡ £ Á , ¡ £ ¦ = 0, 1, 2,± //
±
¡ £ « ±2 ¬
½ 0
Hermite polynomials ä
± ä
−µ
−å 3 2 ¬ ¡ £ « ±2 ¬
Ë
±
2¬
(Ç )å æ å
2
±
± 1 − ä −å
−ã
£ ½ 0 ¡
Error function 2 2 ¿ erf = £ exp(− )
erf
¦ ! 1 − (2¦ +1)! « 2 ¬
2
±
¦ +1
Ë
2
2
â (Æ , ) =
±
−
Incomplete gamma function
(Æ , )
±
Ç
1 2
±
− á/â
Æ
±
Î +
sinh
¡ £
±
1 2
¡ £
1 ±
Æ
Conventional notation à
å + 12 ( ) ±
å
¿ å
−è
( )= ¡ £ ¦± ! ±
é
¿
− å À¡ £
å +è Á ,
± = Ç −1, ± ( Ç ) å = Ç ( Ç +1) // ( Ç + ¦ −1)
Modified Bessel functions µ ( ) ×
±
A.9. Degenerate Hypergeometric Functions A.9.1. Definitions and Basic Formulas A.9.1-1. The degenerate hypergeometric functions à ( Æ , Ç ; ) and ê ( Æ , Ç ; ). ± The degenerate hypergeometric functions à ( Æ , Ç ; ) and ê ( Æ ± , Ç ; ) are solutions of the degenerate Ñ Ñ Ñ± hypergeometric equation ± ë º'º ë + ( Ç − ) ë º − Æ = 0. í In the case Ç ≠ 0, −1, −2, −3, // ,± the function± à ( Æ , Ç ; )í can be represented as Kummer’s series: à (Æ , Ç ; ) = 1 + í ï
±
± í î
ì*í ° =1
(Æ ) ï , (Ç ) !
where ( Æ ) = Æ ( Æ + 1) ð/ð/ð ( Æ + − 1), ( Æ )0 = 1. Table A1 presents some special cases where à can be expressed in terms of simpler functions. î The function ê ( Æ , Ç ; ) is defined as follows: ¼ (1 − Ç ) ¼ ( Ç − 1) î î î 1− ç î ê (Æ , Ç ; ) = ¼ à (Æ , Ç ; ) + ¼ à ( Æ − Ç + 1, 2 − Ç ; ). ( Æ − Ç + 1) (Æ )
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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES
A.9.1-2. Kummer transformation and linear relations. Kummer transformation: ë î î à ( Æ , Ç ; ) = ñ à ( Ç − ò , Ç ; − ),
î î î ê ( ò , Ç ; ) = 1− ç ê (1 + ò − Ç , 2 − Ç ; ).
Linear relations for à : î
î
î
î
( Ç − ò ) à ( ò − 1, Ç ; ) + (2 ò − Ç + ) à ( ò , Ç ; ) − ò à ( ò + 1, Ç ; ) = 0, î î î î î Ç ( Ç − 1) à ( ò , Ç − 1; ) − Ç ( Ç − 1 + ) à ( ò , Ç ; ) + ( Ç − ò ) à ( ò , Ç + 1; ) = 0, î î î ( ò − Ç + 1) à ( ò , Ç ; ) − ò à ( ò + 1, Ç ; ) + ( Ç − 1) à ( ò , Ç − 1; ) = 0,
î î î î Ç/à ( ò , Ç ; ) − Ç/à ( ò − 1, Ç ; ) − à ( ò , Ç + 1; ) = 0, î î î î î Ç ( ò + ) à ( ò , Ç ; ) − ( Ç − ò ) à ( ò , Ç + 1; ) − ò Ç/à ( ò + 1, Ç ; ) = 0, î î î î ( ò − 1 + ) à ( ò , Ç ; ) + ( Ç − ò ) à ( ò − 1, Ç ; ) − ( Ç − 1) à ( ò , Ç − 1; ) = 0.
Linear relations for ê : î î î î ê ( ò − 1, Ç ; ) − (2 ò − Ç + ) ê ( ò , Ç ; ) + ò ( ò − Ç + 1) ê ( ò + 1, Ç ; ) = 0, î î î î î ( Ç − ò − 1) ê ( ò , Ç − 1; ) − ( Ç − 1 + ) ê ( ò , Ç ; ) + ê ( ò , Ç + 1; ) = 0, î î î ê ( ò , Ç ; ) − ò ê ( ò + 1, Ç ; ) − ê ( ò , Ç − 1; ) = 0, î î î î ( Ç − ò ) ê ( ò , Ç ; ) − ê ( ò , Ç + 1; ) + ê ( ò − 1, Ç ; ) = 0, î î î î î ( ò + ) ê ( ò , Ç ; ) + ò ( Ç − ò − 1) ê ( ò + 1, Ç ; ) − ê ( ò , Ç + 1; ) = 0, î î î î ( ò − 1 + ) ê ( ò , Ç ; ) − ê ( ò − 1, Ç ; ) + ( ò − È + 1) ê ( ò , Ç − 1; ) = 0.
A.9.1-3. Differentiation formulas and Wronskian. Differentiation formulas: ó
ò
î
ó å ó î å à (ò , ô ó å ó î å ê (ò , ô
î
ó î à ( ò , ô ; ) = à ( ò + 1, ô + 1; ), ô ó î î ó î ê ( ò , ô ; ) = − ò ê ( ò + 1, ô + 1; ),
Wronskian: ö
( à , ê ) = à ê ëº − à ëº ê = − ¼
(ò )å î à ( ò + õ , ô + õ ; ), (ô )å
î
; )= î
î
; ) = (−1) å ( ò ) å ê ( ò + õ , ô + õ ; ). ¼ (ô ) î −ç ë ñ . (ò )
A.9.1-4. Degenerate hypergeometric functions for õ = 0, 1, 2, ð/ð/ð : î ê ( ò , õ + 1; ) =
(−1) å −1 î î à ( ò , õ +1; ) ln õ ! ¼ (ò ù − õ ) ÷
ì:ù
+ ø =0
(ò ) (õ + 1)
ù úØû
û
ù û û
where õ = 0, 1, 2, ð/ð/ð (the last sum is dropped for õ = 0), derivative of the gamma function, û
(1) = − , where
î
( ò + o ) − (1 + o ) − (1 + õ + o )ü
û
(õ ) = − +
o ! ý
ù
+
å (õ − 1)! þì ù ¼ (ò )
−1 =0
(ò − õ ) (1 − õ )
ù î
ù −å
o !
,
( ÿ ) = [ln ¼ ( ÿ )] º is the logarithmic
*ì å í −1 ï −1 , =1
= 0.5772 ð/ð/ð is the Euler constant.
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A.9. DEGENERATE HYPERGEOMETRIC FUNCTIONS
If ô < 0, then the formula
î
(ò , ô ; ) =
î 1− î ( ò − ô + 1, 2 − ô ; )
î
is valid for any . For ô ≠ 0, −1, −2, −3, ð/ð/ð , the general solution of the degenerate hypergeometric equation can be represented in the form î î = 1 ( ò , ô ; ) + 2 ( ò , ô ; ), and for ô = 0, −1, −2, −3, ð/ð/ð , in the form
=
ú î 1− î î 1 ( ò − ô + 1, 2 − ô ; ) + 2 ( ò − ô + 1, 2 − ô ; )ü .
A.9.2. Integral Representations and Asymptotic Expansions A.9.2-1. Integral representations: 1 ¼ (ô ) î ñ −1 (1 − ) − (ò , ô ; ) = ¼ (ò ) ¼ (ô − ò ) 0 1 î − −1 − −1 ó (ò , ô ; ) = ¼ ø ñ (1 + ) (ò ) 0
−1
ó
(for ô > ò > 0),
(for ò > 0,
î
> 0),
where ¼ ( ò ) is the gamma function. A.9.2-2. Integrals with degenerate hypergeometric functions:
ô −1 î ó î î = ( ò − 1, ô − 1; ) + , (ò , ô ; ) ò −1 1 î ó î î (ò , ô ; ) = ( ò − í 1, ô − 1; í ) + í , 1−ò î í ì*å í +1 (−1) +1 (1 ï ï î − ô ) å − +1 î å î ó î ï =õ ! (ò , ô ; ) (ò − , ô − ; ) + , í í (1 − ò ) (õ − + 1)! =1
î å î ó î (ò , ô ; ) =õ !
ì å í +1 =1
î
A.9.2-3. Asymptotic expansion as | |
í î ï ï î (−1) +1 å − +1 ï (ò − , ô − ; ) + . (1 − ò ) (õ − + 1)!
:
¼ (ô ) ì ( ô − ò ) å (1 − ò ) å î î î ñ − (ò , ô ; ) = ¼ (ò ) õ ! å =0 î (ò , ô ; ) =
(ô ) î (− )− (ô − ò )
å =0
( , ô ; ) = − å =0
where = ( −
−1
(−1) å
−å
+ ,
î
> 0,
( ) å ( − ô + 1) å (− )− å + , !
( ) å ( − ô + 1) å − å + , !
−
< 0,
0, the hypergeometric function can be expressed in terms of a definite integral: ( , , ; ) =
( ) ( ) ( − )
1 0
(
−1
(1 − )& −(
−1
(1 − , )− '
+
,
where ( ) is the gamma function. See M. Abramowitz and I. Stegun (1964) and H. Bateman and A. Erd e´lyi (1953, Vol. 1) for more detailed information about hypergeometric functions. # A.11. Whittaker # Functions
The Whittaker functions equation:
,.
( ) and /
,.
! + 0 −1 + 4
( ) are linearly independent solutions of the Whittaker 1% 2
+1
1 4
−2
23
−2 4
= 0.
The Whittaker functions# are expressed in terms of degenerate hypergeometric functions as # /
5 2 1 . +1 5 2 6 − 5 2 1 , . ( ) = ,.
( ) = .
+1 5 2 6 −
1 2 1 2
% + 2 − , 1 + 22 , 3 ,
% + 2 − , 1 + 22 , 3 .
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A.10. HYPERGEOMETRIC FUNCTIONS
TABLE A2 Some special cases where the hypergeometric function ( , , ; ÿ ) can be expressed in terms of elementary functions
ÿ
−
7 =0
−
− − 8
#
+
1 2
1 2
2
+
1 2
3 2
2
−
1 2
−
2
1−
1 2
−
2
−
2 − 1
1−
3 2
sin2
2−
3 2
sin2
1−
1 2
sin2
+1
+
1 2
+
1 2
1 2
091,: 1 +
1 : 1+
2
3 2 ' + 1: 1 +
2+ 2' + 3
−1
+ 1 : 1+
2: 1 +
2
−
1
1
2
−
1 2
1
3 2
+= +2
(1 + )1−2 ' − (1 − )1−2 ' 2 (1 − 2 )
−2
1 1+ : 1− 3
sin[(2 − 1) ( − 1) sin(2 sin[(2 − 2) ( − 1) sin(2 cos[(2 − 1) cos
1+ : 1− 2
1 =1
8
B >
−1 (
)B
−
7
>
( ).
( ) can be calculated recursively using the relations 7 7 1 2 + 1 B 0 ( ) = 1, B 1 ( ) = , B 2 ( ) = (3 2 − 1), " " " , B +1 ( ) = B ( ) − B 2 +1 +1 7 7 7 = C ( ) have the form The first three functions C
−1 (
).
7 1 1+ 7 3 2 − 1 1 + 3 1+ , C 1 ( ) = ln − 1, C 2 ( ) = − . ln ln 2 1− 2 1− 4 1− 2 The polynomials B ( ) have the implicit representation C
)=
0 (
5
7
2]
[
( ) = 2− 7 B
7
7
>
(−1) > F > F =0
2 −2
7
7
where [ G ] is the integer part of a number G .
7
>
−2
7
> ,
A.12.2. Zeros of Legendre Polynomials and the Generating Function All zeros of B ( ) are real and lie on the interval −1 < < +1; the functions B ( ) form an orthogonal system on the7 interval −1 ≤ ≤ +1, with 7 H +1 0 if M ≠ N , 2 B I (J ) B (J ) K J = L > if M = N . −1 2M + 1 The generating function is O
S
1 1 − 2 PQJ + P
2
= R I
I (J ) P =0
I
(| P | < 1).
T
A.12.3. Associated Legendre Functions The associated Legendre functions I > (J ) of order N T
I > (J ) = (1 − J
2
)> U
2
KT >
K J > T 0 I (J ) =
I (J ),
are defined by the formulas
M = 1, 2, 3, " " " ,
= 0, 1, 2, " " " N
It is assumed by definition that I (J ). T The functions I > (J ) form Tan orthogonal system on the interval −1 ≤ J ≤ +1, with T
H
+1 −1
#
W!X 0 2 (M + N )! I > (J ) > (J ) K J = V T T 2M + 1 (M − N )!
The functions I > (J ) (with N (1 − J 2 )−1 , that is, T H
+1 −1
%
if M ≠ , %
if M = .
≠ 0) are orthogonal on the interval −1 ≤ J ≤ +1 with weight #
W!X 0 I > (J ) > (J ) K J = V (M + N )! T (1 −T J 2 ) N (M − N )!
%
if M ≠ , %
if M = .
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A.14. MATHIEU FUNCTIONS
A.13. Parabolic Cylinder Functions A.13.1. Definitions. Basic Formulas The Weber parabolic cylinder function Y Z ( [ ) is a solution of the linear differential equation: \ ]^! ] + _ − 1 [ 4
2
+ ` + 12 a \ = 0,
where the parameter ` and the variable [ can assume arbitrary real or complex values. Another linearly independent solution of this equation is the function Y − Z −1 (b,[ ); if ` is noninteger, then Y Z (− [ ) can also be taken as a linearly independent solution. The parabolic cylinder functions can be expressed in terms of degenerate hypergeometric functions as Y Z ( [ ) = 21 U
2
exp _ − 14 [ 2 a c
_d d
1 2
_ 1 2
−
a Z
Z _ − 2 , 12 , 12 [
a e
2
2
−1 a +2 U
d
For nonnegative integer ` = M , we have I Y I ( [ ) = 2− U
2
1
2
exp _ − 41 [ 2 a g I _ 2−1 U 2 [ a ,
I g I ( [ ) = (−1) exp _ [
2
_ −2a _ d _ −Z [ 2a e
1 2
Z
− 2 , 32 , 12 [ 2 Qa f .
M = 0, 1, 2, h h h ;
I
K a K [ I exp _ − [
2
a ,
where g I ( [ ) is the Hermitean polynomial of order M .
A.13.2. Integral Representations and Asymptotic Expansions Integral representations: Y Z ([ ) = i Y Z ([ ) =
2 j k exp _
d
1 2 4[ a
Z
0
I
=0
exp _ − 21 2 a cos _ [
l 0
l
− Z −1
R l
exp _ − [ l
−
l
− 12 k ` a K
1 2 2 a
l
K
l
l
for
Re ` > −1,
for
Re ` < 0.
: n I
S
exp _ − 41 [ 2 a c o
Z R
H
1 exp _ − 41 [ 2 a (− ` )
Asymptotic expansion as | [ | m Y Z ([ ) = [
H
Z
(−2) _ − 2 a I _ M !
1 2
−
Z
2
a I
1
+p [ 2I
_ | [ |−2 o
−2
a f
for | arg [ |
ce2 I
2I
S
+1
=R =0 v
>
2
=0 v
2 I +1 2 +1
>
S
>
se2 I
S
+1
=R >
>
=0 x
=0 x
r
cos(2N +1)J r
v 2I 2
se2 I = R
+1 +1 sin(2 N
r r
+1)J r
x
v
>
v
2I 4
2x I 2 +2
>
x −r
S
=0
v
2I 2
>
)2
I
= ( ô 2 I −4) 22 ; I = ( ô 2 I −4N x 2) 22 > 2I N ≥2 x 2 −2 ,
S
=0
v
R
(
( R =0
>
>
>
x
=0
> S
2 I +1 2 2 +1 ) = 1
( R
>
I 2 I +1 = ( ô 2 I +1 −1− r ) 12 +1; 3 2x I +1 x 2 2 I +1 2 +3 = [ ô 2 I +1 −(2 N +1) ] 2 +1 > > 2 I +1 − r 2 −1 , N ≥ 1 x x
(
>
>
x
S
)2 + R
2 if M = 0 =w 1 if M ≥ 1
>
v
sin 2N J ,
r
2I 0
(
I 2 I +1 = ( q 2 I +1 −1− r ) 12 +1; 3 2v I +1 2 2 I +1 v 2 +3 = [ q 2 I +1 −(2 N +1) ] 2 +1 > > 2 I +1 − r 2 −1 , N ≥ 1 v
se0 = 0 2I 2 >
I
= q 2 I 20 ; I I r = (v q 2 I −4) 22v −2 r 20 ; 2 v 2I v v r 2I ) 2 2 +2 = ( q 2 I −4 N > > 2I v − r 2 −2 , N ≥ 2v r
2I 4
cos 2N J >
2I 2
Normalization conditions
x
2I 2
>
)2 = 1
2 I +1 2 2 +1 )
>
=1
A.14.1-2. Properties of the Mathieu functions. The Mathieu functions possess the following properties: I
ce2 I (J , − r ) = (−1) ce2 I y se2 | (z , − r ) = (−1) |
−1
k
2
se2 | y
−z , r { , k
2
ce2 |
− z , r { , se2 |
+1 ( z +1 ( z
, − r ) = (−1) | se2 | , − r ) = (−1) | ce2 |
k y
+1 +1
k y
2 2
−z , r { , −z , r { .
Selecting sufficiently large number N and omitting the term with the maximum number in the recurrence relations (indicated in Table A3), we can obtain approximate relations for eigenvalues q | (or ô | ) with respect to parameter r . Then, equating the determinant of the corresponding homogeneous linear system of equations for coefficients | (or | ) to zero, we obtain an algebraic > > equation for finding q | ( r ) (or ô | ( r )). v x For fixed real r ≠ 0, eigenvalues q | and ô | are all real and different, while if
r >0
then q
0
