Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
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Hydrogenic Wave Functions 9.3 The Coulomb Green’s Function 161
over the generating function (9.199) for the Laguerre
polynomial [9.23]. Other methods of calculating matrix
elements are discussed in Swainson and Drake [9.22].
The Green’s function
˜
G
(S)
in momentum space is re-
lated to the coordinate space Green’s function G
(S)
via
the Fourier transforms
G
(S)
r, r
; E
=
(
2π
)
−3
exp
i
p· r − p
· r
×
˜
G
(S)
p, p
; E
d
3
p d
3
p
,
(9.110)
˜
G
(S)
p, p
; E
=
(
2π
)
−3
exp
−i
p· r − p
· r
× G
(S)
r, r
; E
d
3
r d
3
r
.
(9.111)
The Green’s function
˜
G
(S)
is a solution of
1
2
p
2
− E
˜
G
(S)
p, p
; E
−
Z
2π
2
1
(
p− p
)
2
×
˜
G
(S)
p
, p
; E
d
3
p
= δ
p− p
.
(9.112)
An explicit closed form expression for
˜
G
(S)
is
˜
G
(S)
p, p
; E
=
δ
p− p
1
2
p
2
−E
+
Z
2π
2
|p− p
|
2
1
2
p
2
−E
1
2
(
p
)
2
−E
×
1 +
νq
1 −ν
×
1 −q
1 +q
2
F
1
1, 1 −ν;2 −ν;
1 −q
1 +q
−
1 +q
1 −q
2
F
1
1, 1 −ν;2 −ν;
1 +q
1 −q
,
(9.113)
where
q =
2E |p− p
|
2
4E
2
−4E p· p
+
(
pp
)
2
. (9.114)
9.3.2 The Green’s Function for the Dirac
Equation
The Green’s function G
D
for the Dirac equation (9.60)
is a 4 × 4 matrix valued solution of
−iα · ∇+β −
Zα
r
−E
G
D
r, r
; E
= δ
r −r
I
4
, (9.115)
where I
4
is the 4 × 4 identity matrix. The partial wave
expansion of G
D
is
G
D
r, r
; E
=
κ,m
G
κ,m
11
−iG
κ,m
12
iG
κ,m
21
G
κ,m
22
,
(9.116)
where
G
κ,m
11
= χ
m
κ
(
θ, φ
)
χ
m†
κ
θ
,φ
g
11
r, r
; E
,
(9.117)
G
κ,m
12
= χ
m
κ
(
θ, φ
)
χ
m†
−κ
θ
,φ
g
12
r, r
; E
,
(9.118)
G
κ,m
21
= χ
m
−κ
(
θ, φ
)
χ
m†
κ
θ
,φ
g
21
r, r
; E
,
(9.119)
G
κ,m
22
= χ
m
−κ
(
θ, φ
)
χ
m†
−κ
θ
,φ
g
22
r, r
; E
.
(9.120)
The identity
δ
r −r
I
4
=
δ
r −r
rr
×
κ,m
χ
m
κ
(
θ, φ
)
χ
m†
κ
θ
,φ
0
0 χ
m
−κ
(
θ, φ
)
χ
m†
−κ
θ
,φ
(9.121)
can be used to show that the radial functions
g
jk
r, r
; E
satisfy the equation
1 − E −
Zα
r
−
d
dr
+
1 −κ
r
d
dr
+
1 +κ
r
−
1 + E +
Zα
r
×
g
11
r, r
; E
g
12
r, r
; E
g
21
r, r
; E
g
22
r, r
; E
=
δ
r −r
rr
10
01
.
(9.122)
The solution to (9.122)is
g
11
r, r
; E
g
12
r, r
; E
g
21
r, r
; E
g
22
r, r
; E
=
(
2λ
)
1+2γ
Γ
(
a
)
Γ
(
3 +2γ
)
×
#
Θ
r
−r
G
<
(
r
)
F
<
(
r
)
G
>
r
F
>
r
+Θ
r −r
G
>
(
r
)
F
>
(
r
)
G
<
r
F
<
r
$
,
(9.123)
Part A 9.3
162 Part A Mathematical Methods
where a is defined by (9.79), Θ is the Heaviside unit
function, defined by
Θ
(
x
)
=
1, x > 0 ,
1
2
, x = 0 ,
0, x < 0,
(9.124)
and the functions G
<
, F
<
, G
>
,andF
>
are special cases
of the homogeneous solutions (9.71 – 9.80):
G
<
(
r
)
=r
γ
exp
(
−λr
)(
1 +E
)
1/2
[
f
2
(
r
)
+ f
1
(
r
)
]
,
(9.125)
F
<
(
r
)
=r
γ
exp
(
−λr
)(
1 −E
)
1/2
[
f
2
(
r
)
− f
1
(
r
)
]
,
(9.126)
G
>
(
r
)
=r
γ
exp
(
−λr
)(
1 +E
)
1/2
[
f
4
(
r
)
+ f
3
(
r
)
]
,
(9.127)
F
>
(
r
)
=r
γ
exp
(
−λr
)(
1 −E
)
1/2
[
f
4
(
r
)
− f
3
(
r
)
]
.
(9.128)
The functions G
<
(
r
)
and F
<
(
r
)
obey the boundary con-
ditions at r = 0. The functions G
>
(
r
)
and F
>
(
r
)
obey
the boundary conditions at r =∞. Integral representa-
tions and expansions for the Dirac Green’s function can
be found in [9.22]and[9.25]. Matrix element evaluation
is discussed in [9.22].
9.4 Special Functions
This section contains a brief list of formulae for the spe-
cial functions which appear in the solutions discussed
above. Derivations, and many additional formulae, can
be found in the standard reference works listed in the bib-
liography. For numerically useful approximations and
available software packages, see Olver et al. [9.12], and
Lozier and Olver [9.26].
9.4.1 Confluent Hypergeometric Functions
The confluent hypergeometric differential equation is
z
d
2
dz
2
+
(
c−z
)
d
dz
−a
w
(
z
)
= 0 .
(9.129)
Equation (9.129) has a regular singular point at r = 0
with indices 0 and 1 −c and an irregular singular point
at ∞. The regular solution to (9.129) is the confluent
hypergeometric function, denoted by
1
F
1
in generalized
hypergeometric series notation. It can be defined by the
series
1
F
1
(
a;c;z
)
=
Γ
(
c
)
Γ
(
a
)
∞
n=0
Γ
(
a+n
)
Γ
(
c+n
)
z
n
n!
.
(9.130)
The series (9.130)for
1
F
1
converges for all finite z
if c is not a negative integer or zero. It reduces to
a polynomial of degree n in z if a =−n where n is
a positive integer and c is not a negative integer or
zero. The function
1
F
1
(
a;c;z
)
is denoted by the symbol
M
(
a, c, z
)
in Abramowitz and Stegun [9.1], in Jahnke
and Emde [9.5], and in Olver [9.9], by
1
F
1
(
a;c;z
)
in
both of Luke’s books [9.6, 7]andinMagnus et al. [9.8],
and by Φ
(
a, c;z
)
in the Bateman project [9.2, 3]and
Gradshteyn and Ryzhik [9.4]. The irregular solution to
(9.129)is
U
(
a, c, z
)
=
Γ
(
1 −c
)
Γ
(
1 +a−c
)
1
F
1
(
a;c;z
)
+
Γ
(
c−1
)
Γ
(
a
)
z
1−c
×
1
F
1
(
1 +a−c;2 −c;z
)
.
(9.131)
The function U
(
a, c, z
)
is multiple-valued, with princi-
pal branch −π<arg z ≤π. It is denoted by the symbol
U
(
a, c, z
)
in Abramowitz and Stegun [9.1], in Magnus
et al. [9.8], and in Olver [9.9], by ψ
(
a;c;z
)
in the first
of Luke’s books [9.6], by U
(
a;c;z
)
in the second of
Luke’s books [9.7], and by Ψ
(
a, c;z
)
in the Bateman
project [9.2, 3]andGradshteyn and Ryzhik [9.4].
The Whittaker functions M
κ,µ
and W
κ,µ
, which are
related to
1
F
1
and U via
M
κ,µ
(
z
)
= exp
−
1
2
z
z
µ+
1
2
×
1
F
1
µ +
1
2
−κ;2µ +1;z
,
(9.132)
W
κ,µ
(
z
)
= exp
−
1
2
z
z
µ+
1
2
× U
µ +
1
2
−κ, 2µ +1, z
,
(9.133)
are sometimes used instead of
1
F
1
and U. For numerical
evaluation and a program, see [9.27, 28].
Part A 9.4
Hydrogenic Wave Functions 9.4 Special Functions 163
The regular solution can be written as a linear com-
bination of irregular solutions via
1
F
1
(
a;c;z
)
=
Γ
(
c
)
Γ
(
c−a
)
e
iπa
U
(
a, c, z
)
+
Γ
(
c
)
Γ
(
a
)
×e
z+iπ
(
a−c
)
U
(
c−a, c, −z
)
,
(9.134)
=
+1 , Im z > 0 ,
−1 , Im z < 0 .
(9.135)
1
F
1
can also be obtained from U as the discontinuity
across a branch cut:
z
c−1
exp
(
−z
)
1
F
1
(
a;c;z
)
=
Γ
(
1 −a
)
Γ
(
c
)
2πi
ze
−iπ
c−1
U
c−a, c, ze
−iπ
−
ze
iπ
c−1
U
c−a, c, ze
iπ
.
(9.136)
The Wronskian of the two solutions is
1
F
1
(
a;c;z
)
d
dz
U
(
a, c, z
)
−U
(
a, c, z
)
d
dz
1
F
1
(
a;c;z
)
=−Γ
(
c
)
z
−c
exp
(
z
)
/Γ
(
a
)
. (9.137)
AformulaforU
(
a, c, z
)
when c is the integer n +1
can be obtained by taking the c → n +1 limit of the
right-hand side of (9.131) to obtain
U
(
a, n +1, z
)
=
(
−1
)
n+1
Γ
(
a−n
)
×
1
n!
ln
(
z
)
1
F
1
(
a;n +1;z
)
+
∞
k=−n
Γ
(
a+k
)
a
k
z
k
Γ
(
a
)(
k+n
)
!
$
,
(9.138)
where
a
k
=
(
−1
)
k+1
(
−k−1
)
! , −n ≤ k ≤−1 ,
[
Ψ
(
k+a
)
−Ψ
(
k+1
)
−Ψ
(
k+n +1
)
]
/k! , k ≥ 0 .
(9.139)
Here Ψ is the logarithmic derivative of the gamma
function:
Ψ
(
z
)
= Γ
(
z
)
/Γ
(
z
)
.
(9.140)
n is a non-negative integer. When n = 0, the sum from
k =−n to −1 is omitted.
The basic integral representations for
1
F
1
and U are
1
F
1
(
a;c;z
)
=
Γ
(
c
)
Γ
(
a
)
Γ
(
c−a
)
×
1
0
e
zt
t
a−1
(
1 −t
)
c−a−1
dt ,
(9.141)
U
(
a, c, z
)
=
1
Γ
(
a
)
∞
0
e
−zt
t
a−1
(
1 +t
)
c−a−1
dt .
(9.142)
The basic transformation formulae for
1
F
1
and U are
1
F
1
(
a;c;z
)
= e
z
1
F
1
(
c−a;c;−z
)
,
(9.143)
U
(
a, c, z
)
= z
1−c
U
(
a−c+1, 2 −c, z
)
.
(9.144)
The recurrence relations among contiguous functions
are
(
z+2a−c
)
1
F
1
(
a;c;z
)
=
(
a−c
)
1
F
1
(
a−1;c;z
)
+a
1
F
1
(
a+1;c;z
)
,
(9.145)
(
z+a−1
)
1
F
1
(
a;c;z
)
=
(
a−c
)
1
F
1
(
a−1;c;z
)
+
(
c−1
)
1
F
1
(
a;c−1;z
)
,
(9.146)
c
1
F
1
(
a;c;z
)
= c
1
F
1
(
a−1;c;z
)
+z
1
F
1
(
a;c+1;z
)
,
(9.147)
(
a+1 −c
)
1
F
1
(
a;c;z
)
= a
1
F
1
(
a+1;c;z
)
+
(
1 −c
)
1
F
1
(
a;c−1;z
)
,
(9.148)
c
(
z+a
)
1
F
1
(
a;c;z
)
= ac
1
F
1
(
a+1;c;z
)
+
(
c−a
)
z
1
F
1
(
a;c+1;z
)
,
(9.149)
c
(
z+c−1
)
1
F
1
(
a;c;z
)
= c
(
c−1
)
×
1
F
1
(
a;c−1;z
)
+
(
c−a
)
1
F
1
(
a;c+1;z
)
,
(9.150)
(
z+2a−c
)
U
(
a;c;z
)
=U
(
a−1;c;z
)
+a
(
a−c+1
)
U
(
a+1;c;z
)
,
(9.151)
(
z+a−1
)
U
(
a;c;z
)
=U
(
a−1;c;z
)
+
(
c−a−1
)
U
(
a;c−1;z
)
,
(9.152)
(
c−a
)
U
(
a;c;z
)
=−U
(
a−1;c;z
)
+zU
(
a;c+1;z
)
,
(9.153)
Part A 9.4
164 Part A Mathematical Methods
(
a+1 −c
)
U
(
a;c;z
)
= aU
(
a+1;c;z
)
+U
(
a;c−1; z
)
,
(9.154)
(
z+a
)
U
(
a;c;z
)
= a
(
a−c+1
)
U
(
a+1;c;z
)
+zU
(
a;c+1; z
)
,
(9.155)
(
z+c−1
)
U
(
a;c;z
)
=
(
c−a−1
)(
c−1
)
× U
(
a;c−1;z
)
+zU
(
a;c+1;z
)
.
(9.156)
Useful differentiation formulae include
d
dz
1
F
1
(
a;c;z
)
= ac
−1
1
F
1
(
a+1;c+1;z
)
,
(9.157)
d
dz
z
a
1
F
1
(
a;c;z
)
= az
a−1
1
F
1
(
a+1;c;z
)
,
(9.158)
d
dz
e
−z
z
c−a−1
1
F
1
(
a+1;c;z
)
=
(
c−a−1
)
e
−z
z
c−a−2
1
F
1
(
a;c;z
)
,
(9.159)
d
dz
U
(
a, c, z
)
=−aU
(
a+1, c+1, z
)
,
(9.160)
d
dz
z
a
U
(
a, c, z
)
= a
(
a−c+1
)
z
a−1
U
(
a+1, c, z
)
, (9.161)
d
dz
e
−z
z
c−a−1
U
(
a+1, c, z
)
=−e
−z
z
c−a−2
U
(
a, c, z
)
. (9.162)
An important multiplication theorem is
1
F
1
(
a;c;z
1
z
2
)
=
∞
k=0
Γ
(
a+k
)
Γ
(
λ +2k
)
k !Γ
(
a
)
Γ
(
λ +k
)
(
−z
1
)
k
×
2
F
1
(
−k,λ+k;c;z
1
)
×
1
F
1
(
a+k;λ +2k+1;z
2
)
.
(9.163)
The fundamental asymptotic expansion for large z is
U
(
a, c, z
)
∼ z
−a
∞
n=0
Γ
(
a+n
)
Γ
(
1 +a−c+n
)
n!Γ
(
a
)
Γ
(
1 +a−c
)
(
−z
)
−n
,
−
3
2
π<arg z <
3
2
π.
(9.164)
The asymptotic expansion of
1
F
1
for large z is obtained
by using (9.164)and
exp
[
z+iπ
(
a−c
)
]
U
(
c−a, c, −z
)
∼ e
z
z
a−c
∞
n=0
Γ
(
c−a+n
)
Γ
(
1 −a+n
)
n!Γ
(
c−a
)
Γ
(
1 −a
)
z
−n
,
−
5
2
π<arg z <
5
2
π,
(9.165)
which is a consequence of (9.164), on the right-hand
side of (9.134). In the asymptotic expansion (9.165),
and in the asymptotic expansion for
1
F
1
, the change
in the factor exp
[
iπ
(
a−c
)
]
as arg z passes through
zero is compensated by the phase change which comes
from a factor
(
−z
)
a−c
in the asymptotic expansion of
U
(
c−a, c, −z
)
. The change in the factor exp
(
iπa
)
in
the first term of (9.134)asargz passes through zero is
not compensated by any other phase change. However,
this discontinuity occurs in a region in which this first
term is negligible compared to the second term. This
is an example of the Stokes phenomenon [9.29], which
occurs because the single-valued function
1
F
1
is being
approximated by multiple-valued functions. The large z
asymptotic expansion of
1
F
1
is valid for −
3
2
π<arg z <
3
2
π, which is the overlap of the domain of validity of the
expansions (9.164)and(9.165).
Uniform asymptotic expansions for the Whittaker
functions M
κ,µ
and W
κ,µ
introduced in (9.132), (9.133)
have been constructed via Olver’s method. The follow-
ing result [9.9], (p. 412, Ex. 7.3), which holds for x
positive, κ large and positive, and µ unrestricted, gives
the flavor of these approximations:
W
κ,µ
(
4κx
)
=
2
4/3
π
1/2
κ
κ+(1/6)
φ
n
(
κ, µ
)
exp
(
κ
)
x ζ
x −1
1/4
×
Ai
(
4κ
)
2/3
ζ
n
s=0
A
s
(
ζ
)
(
4κ
)
2s
+
Ai
(
4κ
)
2/3
ζ
(
4κ
)
2/3
×
n
s=0
B
s
(
ζ
)
(
4κ
)
2s
+
2n+1,2
(
4κ, ζ
)
.
(9.166)
Here Ai is the Airy function, and
2n+1,2
is an error term
which tends to zero faster than the last term kept when
Part A 9.4
Hydrogenic Wave Functions 9.4 Special Functions 165
κ →∞with n fixed. ζ is related to x by
4
3
ζ
3/2
=
x
2
−x
1/2
−ln
x
1/2
+
(
x −1
)
1/2
,
x ≥ 1 ,
(9.167)
4
3
(
−ζ
)
3/2
= cos
−1
x
1/2
−
x −x
2
1/2
,
0 < x ≤ 1 .
(9.168)
ζ is an analytic function of x in the neighborhood of
x = 1; conversely, x is an analytic function of ζ in
the neighborhood of ζ = 0. The differential form of the
relations (9.167), (9.168)is
2ζ
1/2
dζ = x
−1/2
(
x −1
)
1/2
dxx≥1 ,
(9.169)
2
(
−ζ
)
1/2
dζ = x
−1/2
(
1 −x
)
1/2
dx 0 < x ≤ 1 .
(9.170)
A Taylor series expansion of ζ about x = 1 is most easily
constructed by expanding the x
−1/2
in (9.169)or(9.170)
about x = 1 and integrating term by term. The opening
terms of such an expansion are
ζ =2
−2/3
(
x −1
)
−
1
5
(
x −1
)
2
+
17
175
(
x −1
)
3
+O
(
x −1
)
4
.
(9.171)
The coefficient functions A
s
and B
s
are calculated re-
cursively from
B
s
(
ζ
)
=
1
2ζ
1/2
ζ
0
ψ
(
η
)
A
s
(
η
)
− A
s
(
η
)
dη
η
1/2
,
x ≥ 1 ,
(9.172)
B
s
(
ζ
)
=
1
2
(
−ζ
)
1/2
0
ζ
[
ψ
(
η
)
A
s
(
η
)
−A
s
(
η
)
dη
(
−η
)
1/2
, 0 < x ≤ 1 ,
(9.173)
A
s+1
(
ζ
)
=−
1
2
B
s
(
ζ
)
+
1
2
ψ
(
ζ
)
B
s
(
ζ
)
dζ,
(9.174)
ψ
(
ζ
)
=
4µ
2
−1
ζ
x
(
x −1
)
+
(
3 −8x
)
ζ
4x
(
x −1
)
3
+
5
16ζ
2
.
(9.175)
The functions A
s
(
ζ
)
, B
s
(
ζ
)
,andψ
(
ζ
)
, which appear
to be singular at ζ =0, are actually analytic functions of
ζ at ζ = 0. The first few coefficient functions are:
A
0
(
ζ
)
= 1 ,
(9.176)
B
0
(
ζ
)
=
1
4
ζ
−1/2
x
x −1
3/2
8µ
2
−
1
2
×
x
x −1
2
−
x
x −1
+
5
6
$
−
5
48
ζ
−2
,
(9.177)
A
1
(
ζ
)
=
1
4
ζ
−1
[
B
0
(
ζ
)
−ψ
(
ζ
)
]
+ζ
[
B
0
(
ζ
)
]
2
.
(9.178)
The function φ
n
is calculated from
φ
n
(
κ, µ
)
=
n
s=0
A
s
(
∞
)
(
4κ
)
2s
−
n−1
s=0
lim
ζ→∞
ζ
1/2
B
s
(
ζ
)
(
4κ
)
2s+1
.
(9.179)
The first two φ
n
are
φ
0
(
κ, µ
)
= 1 ,
(9.180)
φ
1
(
κ, µ
)
= 1 −
12µ
2
−1
24κ
+
12µ
2
−1
24κ
2
.
(9.181)
A bound for the error term
2n+1,2
has been given by
Olver ([9.9] p. 410, Eq. 7.13). The extension to the com-
plex case can be found in Skovgarrd [9.30]. Skovgarrd’s
expansions are in powers of (4κ)
−1
instead of (4κ)
−2
,
because the factor 1/φ
n
(
κ, µ
)
has been expanded out
in inverse powers of 4κ in his results; as a conse-
quence, his coefficient functions A
s
and B
s
differ from
Olver’s. The corresponding asymptotic expansions for
U
µ +
1
2
−κ, 2µ +1, 4κz
and for the Laguerre poly-
nomial L
(2µ)
κ−µ−1/2
(
4κz
)
can be constructed with the aid
of (9.133)and(9.188).
Formulae for matrix element integrals can be ob-
tained by inserting the integral representation (9.141)
and interchanging the orders of integration. An example
is
∞
0
z
µ
1
F
1
(
a
1
;c
1
;λ
1
z
)
1
F
1
(
a
2
;c
2
;λ
2
z
)
dz
= F
2
(
µ +1, a
1
, a
2
, c
1
, c
2
;−λ
1
, −λ
2
)
.
(9.182)
Here F
2
is one of the hypergeometric functions of two
variables introduced by Appell, which can be defined
Part A 9.4
166 Part A Mathematical Methods
either by the integral representation ([9.2], Vol. 1, p. 230,
Eq. 2)
F
2
(α,β,β
,γ,γ
;x, y)
=
Γ(γ)Γ(γ
)
Γ(β)Γ(β
)Γ(γ −β)Γ(γ
−β
)
×
1
0
du
1
0
dv u
β−1
v
β
−1
(1 −u)
γ −β−1
× (1 −v)
γ
−β
−1
(1 −ux−vy)
−α
, (9.183)
or by the series expansion ([9.2], Vol. 1, p. 224, Eq. 7)
F
2
(α,β,β
,γ,γ
;x, y)
=
∞
m=0
∞
n=0
Γ
(
α +m +n
)
m!n!Γ
(
α
)
×
Γ
(
β +m
)
Γ
β
+n
Γ
(
γ
)
Γ
γ
Γ
(
β
)
Γ
(
β
)
Γ
(
γ +m
)
Γ
(
γ
+n
)
x
m
y
n
,
(9.184)
which converges for |x|+|y| < 1. Numerical evalua-
tion of F
2
is particularly convenient in the special cases
where it can be expressed in terms of ordinary hyperge-
ometric functions
2
F
1
([9.2], Vol. 1, Chap. 2), which are
easy to calculate, or in terms of elementary functions.
The key is the formula ([9.2], Vol. 1, p. 238, Eq. 3)
F
2
(α,β,β
,α,α; x, y)
= (1 −x)
−β
(1 −y)
−β
×
2
F
1
β, β
;α;
xy
(1 −x)(1 −y)
,
(9.185)
which shows that it is necessary to get Appell func-
tions F
2
in which the first, fourth, and fifth parameters
are equal. This can be achieved by exploiting whatever
freedom in the choice of a
1
, a
2
, c
1
,andc
2
is available,
and by using identities such as
F
2
(α +2,β,β
,α+1,α+1; x, y)
= (α +1)
−1
β
y
× F
2
(α +1,β,β
+1,α+1,α+1;x, y)
+
F
2
(α +1,β,β
,α+1,α+1; x, y)
+2(α +1)
−2
ββ
xy
× F
2
(α +2,β+1,β
+1,α+2,α+2;x, y)
+(α +1)
−1
βx
× F
2
(α +1,β+1,β
,α+1,α+1; x, y),(9.186)
to obtain Appell functions F
2
for which the reduction
(9.185) can be used.
9.4.2 Laguerre Polynomials
The Laguerre polynomials L
(α)
n
are the polynomial so-
lutions of the differential equation
z
d
2
dz
2
+
(
α +1 −z
)
d
dz
+n
L
(α)
n
(
z
)
= 0 .
(9.187)
They are a special case of the confluent hypergeometric
function:
L
(α)
n
(
z
)
=
Γ
(
n +α +1
)
n!Γ
(
α +1
)
1
F
1
(
−n;α +1;z
)
=
(
−1
)
n
n!
U
(
−n,α+1, z
)
,
(9.188)
and are given explicitly by
L
(α)
n
(
z
)
=
n
k=0
Γ
(
α +n+1
)
k !
(
n −k
)
!Γ
(
α +k+1
)
(
−z
)
k
.
(9.189)
The L
(α)
n
are sometimes called generalized Laguerre
polynomials, because L
(0)
n
, which is often denoted by
L
n
, is the polynomial introduced by Laguerre. This La-
guerre polynomial differs from the “associated Laguerre
function” for which the symbol L
p
q
(with pand q both in-
tegers) is often used in the physics literature. The relation
between the two is
L
p
q
(
z
)
physics
=
(
−1
)
p+q
q!L
( p)
q−p
(
z
)
.
(9.190)
The first three L
(α)
n
are
L
(α)
0
(
z
)
= 1 ,
(9.191)
L
(α)
1
(
z
)
= α +1 −z ,
(9.192)
L
(α)
2
(
z
)
=
1
2
(
α +1
)(
α +2
)
−
(
α +2
)
z+
1
2
z
2
.
(9.193)
Equation (9.188) shows that the irregular solution U
does not supply a linearly independent second solution.
A second solution which remains linearly independent
and finite when α is a positive integer is
M
(α)
n
(
z
)
=−Γ
(
α
)
z
−α
1
F
1
(
−n−α;1 −α;z
)
−cos
(
πα
)
Γ
(
1 −α
)
L
(α)
n
(
z
)
.
(9.194)
The Wronskian of the two solutions is
L
(α)
n
(
z
)
d
dz
M
(α)
n
(
z
)
−M
(α)
n
(
z
)
d
dz
L
(α)
n
(
z
)
= Γ
(
n +α +1
)
z
−α−1
exp
(
z
)
/n! . (9.195)
Part A 9.4
Hydrogenic Wave Functions 9.4 Special Functions 167
AformulaforM
(α)
n
(
z
)
when α is a positive integer m can
be obtained by taking the α → m limit of the right-hand
side of (9.194) to obtain
M
(m)
n
(
z
)
= ln
(
z
)
L
(m)
n
(
z
)
+
∞
k=−m
(
n +m
)
!
(
k+m
)
!
b
k
z
k
,
(9.196)
where
b
k
=
−
(
−k−1
)
!/
(
n −k
)
!, −m ≤ k ≤−1 ,
(
−1
)
k
[
Ψ
(
n +1 −k
)
−Ψ
(
k+1
)
−Ψ
(
k+m +1
)
]
/
[
k !
(
n −k
)
!
]
,
0 ≤ k ≤ n ,
(
−1
)
n
(
k−n −1
)
!/k! , k ≥ n+1 .
(9.197)
The Rodrigues formula is
L
(α)
n
(
z
)
=
1
n!
z
−α
exp
(
z
)
d
n
dz
n
z
n+α
exp
(
−z
)
.
(9.198)
The generating function is
(
1 −w
)
−α−1
exp
−
wz
1 −w
=
∞
n=0
L
(α)
n
(
z
)
w
n
.
(9.199)
The Christoffel–Darboux formula is
n
k=0
k !
Γ
(
k+α +1
)
L
(α)
k
(
w
)
L
(α)
k
(
z
)
=
(
n +1
)
!
L
(α)
n
(
w
)
L
(α)
n+1
(
z
)
−L
(α)
n+1
(
w
)
L
(α)
n
(
z
)
Γ
(
n +α +1
)(
w −z
)
.
(9.200)
The orthonormality relation is
∞
0
x
α
exp
(
−x
)
L
(α)
n
(
x
)
L
(α)
n
(
x
)
dx
= (n!)
−1
Γ
(
n +α +1
)
δ
n,n
. (9.201)
Other useful integration formulae include
∞
0
x
α+1
exp
(
−x
)
L
(α)
n
(
x
)
L
(α)
n
(
x
)
dx
=
Γ
(
n +α +1
)
n!
−nδ
n,n
+1
+
(
2n+α +1
)
δ
n,n
−
(
n +α +1
)
δ
n,n
−1
,
(9.202)
∞
0
x
α−1
exp
(
−x
)
L
(α)
n
(
x
)
L
(α)
n
(
x
)
dx
=
Γ
(
n
<
+α +1
)
n
<
!α
, n
<
= min
n, n
,
(9.203)
∞
0
x
α−2
exp
(
−x
)
L
(α)
n
(
x
)
L
(α)
n
(
x
)
dx
=
Γ
(
n
<
+α +1
)
n
<
!α
(
α +2
)
)
α
(
α +1
)
n +n
+4α +5
×
(
1 −n
<
)
+2
(
n +1
)
n
+1
+2α
n +n
*
,
n
<
= min
n, n
.
(9.204)
The differentiation and recursion relations are
z
d
dz
L
(α)
n
(
z
)
= nL
(α)
n
(
z
)
−
(
n +α
)
L
(α)
n−1
(
z
)
,
(9.205)
L
(α)
n+1
(
z
)
= (n +1)
−1
(
2n+α +1 −z
)
L
(α)
n
(
z
)
−
(
n +α
)
L
(α)
n−1
(
z
)
.
(9.206)
Additional relations can be obtained as special cases of
the relations listed above for the confluent hypergeomet-
ric function by using the relation (9.188).
The coefficients c
k
in a Laguerre polynomial expan-
sion such as
F
(
x
)
=
∞
k=0
k !
Γ
(
k+α +1
)
c
k
×
(
βx
)
α/2
exp
−
1
2
βx
L
(α)
k
(
βx
)
(9.207)
are given by the integral
c
k
= β
∞
0
F
(
x
)(
βx
)
α/2
exp
−
1
2
βx
L
(α)
k
(
βx
)
dx .
(9.208)
Part A 9.4
168 Part A Mathematical Methods
The rate of convergence of the expansion (9.207)is
determined by the asymptotic behavior of the integral
(9.208)forlargek. A convenient way of extracting this
asymptotic behavior will be described with the special
case
α = 0 ,
(9.209)
F
(
x
)
=
(
x +c
)
ν
exp
(
−γx
)
, (9.210)
as an example. Since the discussion is intended to be il-
lustrative rather than exhaustive, only the case γ<
1
2
β
will be considered. The method is based on the generat-
ing function (9.199). The generating function g
(
z
)
for
the coefficients c
k
is given in general by
g
(
z
)
=
∞
k=0
c
k
z
n
= β
(
2+α
)
/2
(
1 −z
)
−α−1
G
β
(
1 +z
)
2
(
1 −z
)
,
(9.211)
where G is the Laplace transform
G
(
λ
)
=
∞
0
x
α/2
F
(
x
)
exp
(
−λx
)
dx . (9.212)
The asymptotic analysis of the expansion coefficient c
k
begins with the Cauchy integral for c
k
,whichis
c
k
=
1
2πi
C
z
−k−1
g
(
z
)
dz , (9.213)
where the contour C is a small circle which runs
counterclockwise around the origin. The contour C
is deformed to give integrals which can be evaluated
via standard methods for the asymptotic evaluation of
integrals [9.31,32]. For the example (9.209), (9.210),
g
(
z
)
= βc
ν+1
(
1 −z
)
−1
U
[
1,ν+2, t
(
z
)
]
, (9.214)
t
(
z
)
=
(
β +2γ
)
c+
(
β −2γ
)
cz
2
(
1 −z
)
.
(9.215)
This g
(
z
)
has a branch point on the negative real
axis at z =−
(
β +2γ
)
/
(
β −2γ
)
, and a combination of
a branch point and an essential singularity on the positive
real axis at z = 1. It is convenient to take the associated
branch cuts to run from −∞ to −
(
β +2γ
)
/
(
β −2γ
)
on the negative real axis, and from +1to+∞ on the
positive real axis. The contour C can be deformed into
the sum of two contours which run clockwise around the
branch cuts. Then
c
k
= c
(1)
k
+c
(2)
k
, (9.216)
where c
(1)
k
is the contribution from a contour which
runs clockwise around the branch cut from −∞ to
−
(
β +2γ
)
/
(
β −2γ
)
on the negative real axis, and c
(2)
k
is the contribution from a contour which runs clock-
wise around the branch cut from +1to+∞ on the
positive real axis. The asymptotic behavior of c
(1)
k
can
be extracted in straightforward fashion via the method
of Darboux ([9.9], pp. 309–315, 321) ([9.32], pp. 116–
122). The result is
c
(1)
k
=
2β
β +2γ
4β
(
β +2γ
)(
β −2γ
)
ν
(
−1
)
k
× k
ν
β −2γ
β +2γ
k
1 +O
k
−1
.
(9.217)
The contribution c
(2)
k
requires a somewhat different strat-
egy. The first step writes it as an integral along the real
axis from +1to+∞ of the jump across the branch cut.
Evaluating the jump yields
c
(2)
k
=
β
Γ
(
−ν
)
∞
0
dxx
ν
β +
1
2
β −γ
x
−ν−1
×exp
−
1
2
β −γ
c−
(
k+1
)
ln
(
1 +x
)
−βcx
−1
.
(9.218)
The integrand in (9.218) has a saddle point at
x ≈
[
βc/
(
k+1
)
]
1/2
. The asymptotics for
(
k+1
)
c
large can be extracted via the method of steepest
descent (also known as the saddle point method)
([9.9], pp. 136–138), ([9.32], pp. 85–103), [9.31], (see
Chapt. 8). However, if k+1 is large with
(
k+1
)
c small
or moderate, the steepest descent approximation breaks
down because the saddle point is too close to the branch
point of the integrand at x = 0. The breakdown can be
curedbyusingthesmallx approximations [β +(1/2β −
γ)x]
−ν−1
≈ β
−ν−1
and ln(1 +x) ≈ x −1/2(k +1)
−1
βc
to obtain
c
(2)
k
=
2β
Γ
(
−ν
)
c
(
k+1
)
β
(
ν+1
)
/2
exp
(
βc
)
× K
ν+1
2
(
k+1
)
βc
1 +O
k
−1/2
.
(9.219)
The function K
ν+1
which appears in (9.219)is
a modified Bessel function of the third kind
in standard notation. The large z approximation
K
ν+1
(
z
)
≈ π
1/2
(
2z
)
−1/2
exp
(
−z
)
can be used to re-
cover the result of a steepest descent approximation to
Part A 9.4
Hydrogenic Wave Functions 9.4 Special Functions 169
(9.218), which is
c
(2)
k
=
π
1/2
βc
ν+1
Γ
(
−ν
)
βc
k+
1
2
−
(
2ν+3
)
/4
×exp
γc−2
βc
k+
1
2
1/2
×
1 +O
k
−1
.
(9.220)
The approximation (9.219) remains valid as c →0. The
approximation (9.220) does not.
The generating function method outlined above has
one very nice feature: if the singularity in the complex
z-plane which dominates the asymptotics is known, the
analysis can be inverted to obtain a “convergence accel-
eration function” which builds in this singularity, and has
no other singularities in the finite complex z plane. The
difference between the original F
(
x
)
and this conver-
gence acceleration function will have an expansion of the
form (9.207) which converges faster than the expansion
of F
(
x
)
. Examples can be found in [9.33–36].
The contributions c
(1)
k
and c
(2)
k
exhibit two typical
features. The most rapidly varying part, which is
[
(
β −2γ
)
/
(
β +2γ
)
]
k
(9.221)
for c
(1)
k
,and
exp
−2
βc
k+
1
2
1/2
(9.222)
for the steepest descent approximation to c
(2)
k
,isde-
termined by the location of the associated singularity.
The next most important part, which is k
ν
for c
(1)
k
,
and (k +1/2)
−(2ν+3)/4
for the steepest descent approx-
imation to c
(2)
k
, is determined by the nature of the
singularity (i. e., by the value of ν).
9.4.3 Gegenbauer Polynomials
The Gegenbauer polynomials are a special case of the
Jacobi polynomial, and of the hypergeometric function:
C
λ
n
(
z
)
=
Γ
λ +
1
2
Γ
(
2λ +n
)
Γ
(
2λ
)
Γ
λ +n +
1
2
P
(λ−
1
2
,λ−
1
2
)
n
(
z
)
=
Γ
(
2λ +n
)
n!Γ
(
2λ
)
×
2
F
1
−n, n+2λ;λ +
1
2
;
1
2
−
1
2
z
,
(9.223)
and are given explicitly by
C
λ
n
(
z
)
=
[n/2]
k=0
(
−1
)
k
Γ
(
λ +n −k
)
k !
(
n −2k
)
!Γ
(
λ
)
(
2z
)
n−2k
.
(9.224)
The first three C
λ
n
are
C
λ
0
(
z
)
= 1 ,
(9.225)
C
λ
1
(
z
)
= 2λz ,
(9.226)
C
λ
2
(
z
)
= 2λ
(
λ +1
)
z
2
−λ. (9.227)
Additional C
λ
n
can be obtained with the aid of the
recursion relation
C
λ
n+1
(
z
)
= (n+1)
−1
2
(
n +λ
)
zC
λ
n
(
z
)
−
(
n +2λ −1
)
C
λ
n−1
(
z
)
,
(9.228)
which is valid for n ≥1. A generating function is
1 −2zt+t
2
−λ
=
∞
n=0
C
λ
n
(
z
)
t
n
. (9.229)
An important addition theorem, which can be used to
derive addition theorems for spherical harmonics in any
number of dimensions, is
C
λ
n
[
cos
(
γ
)
]
=
Γ
(
2λ −1
)
[
Γ
(
λ
)
]
2
n
=0
2
2
(
2λ +2 −1
)
Γ
(
n +2λ +
)
× Γ
(
n − +1
)
[
Γ
(
λ +
)
]
2
×sin
(
χ
)
sin
χ
× C
λ+
n−
[
cos
(
χ
)
]
C
λ+
n−
cos
χ
× C
λ−
1
2
cos
γ
,
(9.230)
cos
(
γ
)
= cos
(
χ
)
cos
χ
+sin
(
χ
)
sin
χ
cos
γ
.
(9.231)
9.4.4 Legendre Functions
The Legendre polynomials P
(
z
)
are a special case of
the Gegenbauer polynomial, and of the hypergeometric
function:
P
(
z
)
= C
1/2
n
(
z
)
=
2
F
1
−n, n+1;1;
1
2
−
1
2
z
.
(9.232)
Part A 9.4
170 Part A Mathematical Methods
The second (irregular) solution to the Legendre
equation, which appears in (9.43) above, can be defined
by the integral representation
Q
(
z
)
= 2
−−1
1
−1
1 −t
2
(
z−t
)
−−1
dt .
(9.233)
The first two Q
are
Q
0
(
z
)
=
1
2
ln
z+1
z−1
,
(9.234)
Q
1
(
z
)
=
1
2
z ln
z+1
z−1
−1 ,
(9.235)
Additional Q
can be obtained with the aid of the
recursion relation
Q
+1
(
z
)
=
[
(
2 +1
)
zQ
(
z
)
−Q
−1
(
z
)
/
(
+1
)
,
(9.236)
which is valid for ≥ 1.
References
9.1 M. Abramowitz, I. A. Stegun: Handbook of Math-
ematical Functions (Dover, New York 1965)
9.2 A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tri-
comi: Higher Transcendental Functions, Vol.1,2,3
(McGraw-Hill, New York 1955) p. 1953
9.3 A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tri-
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(McGraw-Hill, New York 1954)
9.4 I. S. Gradshteyn, I. W. Ryzhik: Tables of Integrals,
Series, and Products, 4th edn. (Academic Press, New
York 1965)
9.5 E. Jahnke, F. Emde: Tables of Functions with Formu-
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9.6 Y. L. Luke: The Special Functions and Their Approxi-
mations, Vol. 1, 2 (Academic Press, New York 1969)
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9.8 W. Magnus, F. Oberhettinger, R. P. Soni: Formu-
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Sec. 2.5
9.14 L. Pauling, E. Bright Wilson: Introduction to Quan-
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188
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Runge–Lenz vector to the “hidden” O(4) symmetry
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Spectral Theorem and VIII Unbounded Operators. See
Theorem VIII.6
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(1982)
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9.29 R. E. Meyer: SIAM Rev. 31, 435 (1989)
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Confluent Hypergeometric Functions and Whittaker
Functions (Gjellerups, Copenhagen 1966)
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sions of Integrals (Holt, Rinehart, & Winston, New
York 1975) Reprinted Dover, New York 1986
Part A 9