Special functions

CHAPTER

Special functions

4

CHAPTER OUTLINE 4.1 Introduction ............................................................................................................................... 152 4.2 Legendre Functions .................................................................................................................... 152 4.2.1 Legendre polynomials and associated Legendre polynomials........................................152 4.2.2 Recurrence relations for Legendre polynomials...........................................................154 4.2.3 Series of Legendre polynomials.................................................................................154 4.2.4 Legendre functions of first and second kind ...............................................................155 4.2.5 Neumann’s formula for the Legendre functions ..........................................................157 4.3 Laguerre Functions..................................................................................................................... 158 4.3.1 Laguerre polynomials and Laguerre functions .............................................................158 4.3.2 Associated Laguerre polynomials...............................................................................159 4.3.3 Basic integrals over associated Laguerre functions......................................................161 4.4 Hermite Functions ...................................................................................................................... 163 4.4.1 Hermite polynomials ................................................................................................163 4.4.2 Hermite functions....................................................................................................164 4.4.3 Integrals over Hermite functions ...............................................................................165 4.5 Hypergeometric Functions........................................................................................................... 166 4.5.1 Hypergeometric series and differential equation .........................................................166 4.5.2 Confluent hypergeometric functions ..........................................................................168 4.6 Bessel Functions ........................................................................................................................ 170 4.6.1 Bessel functions of integral order ..............................................................................170 4.6.2 Bessel functions of half-integral order .......................................................................171 4.6.3 Spherical Bessel functions .......................................................................................172 4.6.4 Modified Bessel functions ........................................................................................176 4.7 Functions Defined by Integrals .................................................................................................... 178 4.7.1 The gamma function ................................................................................................178 4.7.2 The incomplete gamma function ...............................................................................178 4.7.3 From the gamma function to the exponential integral function.....................................179 4.7.4 The exponential integral function ..............................................................................180 4.7.5 The generalized exponential integral function.............................................................181 4.7.6 Further functions.....................................................................................................181 4.8 The Dirac d-Function .................................................................................................................. 184 4.9 The Fourier Transform................................................................................................................. 185 4.10 The Laplace Transform ............................................................................................................... 188 4.11 Spherical Tensors ...................................................................................................................... 190 Elementary Molecular Quantum Mechanics 2E. http://dx.doi.org/10.1016/B978-0-444-62647-9.00004-X
2013, 2007 Elsevier B.V. All rights reserved

151

152

4.12 4.13 4.14 4.15 4.16

CHAPTER 4 Special functions

4.11.1 Spherical tensors in complex form ..........................................................................190 4.11.2 Spherical tensors in real form .................................................................................191 4.11.3 Generalized spherical tensors .................................................................................195 Orthogonal Polynomials .............................................................................................................. 195 Pade´ Approximants..................................................................................................................... 197 Green’s Functions....................................................................................................................... 199 Problems 4 ................................................................................................................................ 202 Solved Problems ........................................................................................................................ 203

4.1 INTRODUCTION The special functions of mathematical physics and chemistry are mostly obtained in the solution of differential equations like those we studied in the previous chapter (Sneddon, 1956; Abramowitz and Stegun, 1965). Some of these solutions were already met in elementary analysis, such as circular and exponential functions, but others are new, as we saw in the series solution for the harmonic oscillator (Hermite functions) or the atomic one-electron problem (Laguerre and Legendre functions). Other special functions are defined by integrals, like the gamma function or the exponential integral function, while the Dirac d-function of Section 4.8, strictly speaking, is not a function but a distribution.

4.2 LEGENDRE FUNCTIONS 4.2.1 Legendre polynomials and associated Legendre polynomials These polynomials were discussed in Section 3.7.3 of the previous chapter as regular solutions of the second-order differential equation:


1
x2

 d2 Qlm ðxÞ dx2


2x

  dQlm ðxÞ m2 Qlm ðxÞ ¼ 0 þ lðl þ 1Þ
1
x2 dx

(1)

where x ¼ cos q, and l, m are integers with m ¼ jmj  0; l
m  0. Equation (1) is known in potential theory as Legendre’s associated differential equation, which for m = 0 reduces to the so-called Legendre’s differential equation. We saw in Chapter 3 that, under the regularity conditions, the infinite series occurring in the solution of Eqn (1) reduces to a polynomial of degree (l
m) equivalent to the conventional associated Legendre polynomials Pm l ðxÞ (MacRobert, 1947; Hobson, 1965; Abramowitz and Stegun, 1965) to within a constant factor, irrelevant from the standpoint of the differential equation, which was chosen as mþ½ Pm l ðxÞ ¼ ð
1Þ

where ½/ stands for ‘integer part of’.

lþm  2

Qlm ðxÞ

(2)

4.2 Legendre functions

153

In mathematics, the Legendre polynomials of degree l in x are defined through the so-called Rodrigues’ formula (Sneddon, 1956; Hobson, 1965): Pl ðxÞ ¼

l 1 dl
2 x
1 l l 2 l! d x

and the associated Legendre polynomials of degree (l
m) and order m in x by m 
2 m=2 d Pm Pl ðxÞ l ðxÞ ¼ 1
x d xm

(3)

(4)

The associated Legendre polynomials are orthogonal in the interval
1  x  1 and are normalized to (Sneddon, 1956) Z1 2 ðl þ mÞ! m0 (5) d x Pm l ðxÞPl0 ðxÞ ¼ dll0 dmm0 2l þ 1 ðl
mÞ!
1

as can be verified by repeated integration by parts. The explicit form of the first few associated Legendre polynomials up to l ¼ m ¼ 5 is given below.

m¼0

m¼1

m¼2

m¼3

m¼4 m¼5

8 > > > P0 ¼ 1 <

P 1 ¼ x P2 ¼

3x2
1 2

P3 ¼

5x3
3x 2

> > 4 2 5 3 > : P ¼ 35x
30x þ 3 P ¼ 63x
70x þ 15x 4 5 8 8 8   1=2 3
2


 1=2 1=2 > P11 ¼ 1
x2 5x
1 P12 ¼ 1
x2 3x P13 ¼ 1
x2 > < 2 > > :



 1=2 5
3 1=2 15
4 P14 ¼ 1
x2 P15 ¼ 1
x2 7x
3x 21x
14x2 þ 1 2 8 8   
2


 2 2 2 2 2 2 15 > > < P2 ¼ 1
x 3 P 3 ¼ 1
x 15x P4 ¼ 1
x 2 7x
1 > > :

 105
3
 P25 ¼ 1
x2 3x
x 2
  8 3
2 3=2 15 P 3 ¼ 1
x2 3=2 105x > 4 < P3 ¼ 1
x > :


 3=2 105
2 9x
1 P35 ¼ 1
x2 2 n  2

2 P 44 ¼ 1
x2 105 P 45 ¼ 1
x2 945x n
5=2 945 P55 ¼ 1
x2

(6)

154

CHAPTER 4 Special functions

In general l   l
k l! X ðx þ 1Þk l ðx
1Þ Pl ðxÞ ¼ l ðl
kÞ! k! 2 k¼0 k

Pm l ðxÞ where




¼ 1
x

 2 m=2

 l  l
k l! X ðx þ 1Þk
m l þ m ðx
1Þ k ðl
kÞ! ðk
mÞ! 2l k¼m

(7)

(8)

  n! n ¼ is the binomial coefficient (Abramowitz and Stegun, 1965) with n  k. k k!ðn
kÞ!

4.2.2 Recurrence relations for Legendre polynomials We give here without proof some recurrence relations occurring for the Legendre polynomials and their first derivatives in the primed notation (Sneddon, 1956): ðl þ 1ÞPlþ1 ðxÞ ¼ ð2l þ 1ÞxPl ðxÞ
lPl
1 ðxÞ

(9)

P0lþ1 ðxÞ
P0l
1 ðxÞ ¼ ð2l þ 1ÞPl ðxÞ

(10)

P0lþ1 ðxÞ
xP0l ðxÞ ¼ ðl þ 1ÞPl ðxÞ

(11)

4.2.3 Series of Legendre polynomials It may sometimes be useful to express a given function as a series of Legendre polynomials. Because of the linear independence of Pms, any polynomial P(x) of degree l in x can be expressed in terms of Legendre polynomials as l X PðxÞ ¼ Pk ðxÞck (12) k¼0

Now, if we want to expand any given function f(x), defined in the interval jxj  1, into a series of Legendre polynomials in the form N X Pk ðxÞck (13) f ðxÞ ¼ k¼0

assuming the series to be convergent in the given interval, we see that the expansion coefficients are given by Z1 Z1 N N X X 2 2 dlk ¼ cl (14) d x f ðxÞPl ðxÞ ¼ ck d x Pk ðxÞPl ðxÞ ¼ ck 2k þ 1 2l þ1 k¼0 k¼0
1


1

2l þ 1 cl ¼ 2

Z1 d x f ðxÞPl ðxÞ
1

(15)

4.2 Legendre functions

155

so that f(x) is given by the series N X

2k þ 1 Pk ðxÞ f ðxÞ ¼ 2 k¼0

Z1

dx0 f ðx0 ÞPk ðx0 Þ

(16)


1

4.2.4 Legendre functions of first and second kind Problems of mathematical physics often involves the Laplace’s equation: 72 VðrÞ ¼

v2 V v2 V v2 V þ 2 þ 2 ¼0 vx2 vy vz

(17)

which can be solved in spherical coordinates by putting Vðr; q; 4Þ ¼ RðrÞQðqÞFð4Þ. Factorization of function V implies the independence of the solutions with respect to the different variables, namely, the R(r) and F(4) terms can be treated as constants in the solution of the q-equation. If we put   1 d 1 d2 F 2 dR r ¼ nðn þ 1Þ; ¼
m2 R dr dr F d42 the solutions of these equations are (Hobson, 1956) RðrÞ ¼ c1 r n þ c2 r
n
1 ;

Fð4Þ ¼ c3 expðim4Þ

(18)

Two solutions of the Laplace’s Eqn (17) must hence have the form Vfrn Unm ðq; 4Þ;

Vfr
n
1 Unm ðq; 4Þ;

Unm ðq; 4Þ ¼ QðqÞexpðim4Þ

(19)

In this way, by multiplying both members by QðqÞ, Eqn (17) gives     1 d dQ m2 sin q þ nðn þ 1Þ
2 Q ¼ 0 sin q dq dq sin q which, by posing Q ¼ w;

cos q ¼ z; sin q ¼ ð1
z2 Þ1=2 , becomes      dw d
m2 w¼0 1
z2 þ nðn þ 1Þ
1
z2 dz dz

(20)

Equation (20) is known as associated Legendre’s differential equation (compare Eqn (188) of Chapter 3 with l ¼ n(n þ 1)). For m ¼ 0 we have the corresponding Legendre’s differential equation:    d
2 dw 1
z þ nðn þ 1Þw ¼ 0 (21) dz dz Even if variable z has been defined as the cosine of a real angle, being therefore real and restricted to the interval (
1,1), the solution of Eqn (21) will now be considered in a more general sense. Assuming the ascending power series expansion wðzÞ ¼ a0 þ a1 z þ a2 z2 þ /

(22)

156

CHAPTER 4 Special functions

we obtain in the usual way the two-term recurrence formula: akþ2 ¼

kðk þ 1Þ
nðn þ 1Þ ðn
kÞðn þ k þ 1Þ ak ¼
ak ðk þ 1Þðk þ 2Þ ðk þ 1Þðk þ 2Þ

k ¼ 0; 1; 2; /

giving the solution wðzÞ as   nðn þ 1Þ 2 nðn
2Þðn þ 1Þðn þ 3Þ 4 z þ z
/ wðzÞ ¼ a0 1
1$ 2 1$ 2$ 3$ 4   ðn
1Þðn þ 2Þ 2 ðn
1Þðn
3Þðn þ 2Þðn þ 4Þ 4 z þ z
/ þ a1 z 1
1$ 2$ 3 1$ 2$ 3$ 4$ 5 which, putting 2 F1 ¼ F, can be rewritten in terms of the hypergeometric functions of Section 4.5.1 as     n nþ1 1 2 n
1 nþ2 3 2 wðzÞ ¼ a0 F
; ; ; z þ a1 zF
; ; ;z 2 2 2 2 2 2 as can be easily seen. When n is a positive integer, one of the series terminates and, if n is even, the solution   n nþ1 1 2 ; ;z (23) w1 ðzÞ ¼ a0 F
; 2 2 2 becomes a polynomial of degree n in even powers of z2, while the remaining solution  n
1 nþ2 3 zF
; ; ; z2 is an infinite series which converges when jzj < 1. 2 2 2 For n ¼ odd, the finite solution is   n
1 nþ2 3 2 ; ; ;z (24) w2 ðzÞ ¼ a1 zF
2 2 2 which is a polynomial of degree n in odd powers of z, while the remaining solution  n nþ1 1 ; ; z2 is an infinite series which converges when jzj < 1. F
; 2 2 2 Now, let us try to obtain a solution of Legendre’s Eqn (21) in the form of a series of descending powers of z: wðzÞ ¼ b0 zq þ b2 zq
2 þ b4 zq
4 þ /

(25)

obtaining for the coefficients the two-term recurrence relation b2kþ2 ¼

ðq
2kÞðq
2k
1Þ b2k ðq
2k
2Þðq
2k
1Þ
nðn þ 1Þ

k ¼ 0; 1; 2; /

Since b
2 ¼ 0, we obtain from the resulting quadratic equation (Problem 4.1) ðq
nÞðq þ n þ 1Þ ¼ 0

4.2 Legendre functions

157

which is satisfied by the two values, q ¼ n and q ¼
n
1, giving rise to the two different solutions   n n
1 2n
1
2 (26) w3 ðzÞ ¼ bzn F
;
;
;z 2 2 2   n þ 1 n þ 2 2n þ 3
2
n
1 (27) w4 ðzÞ ¼ cz F ; ; ;z 2 2 2 By reversing the order of the terms in the series (Hobson, 1965), w3 is seen to be identical to w1 and w2 for n integer even and n integer odd, respectively, whereas w4 is an infinite series converging when jzj > 1. Putting both constants a0 and a1 of w1 and w2 equal to an ¼ ð
1Þ½n=2 n!=f2n ð½n=2!Þ2 g where [t] means ‘integer part of t’, constant b of w3 and constant c of w4 equal to (Sneddon, 1956) b¼

ð2nÞ!

; 2n ðn!Þ2

c¼

2n ðn!Þ2 ð2n þ 1Þ!

we can write the general solution of the Legendre’s differential Eqn (21) as wðzÞ ¼ APn ðzÞ þ BQn ðzÞ where A, B are arbitrary constants, and where   n n
1 2n
1
2 ; ;
;z Pn ðzÞ ¼ bzn F
;
2 2 2

Qn ðzÞ ¼ cz
n
1 F

(28)   n þ 1 n þ 2 2n þ 3
2 (29) ; ; ;z 2 2 2

are called Legendre’s functions of the first and second kind of degree n, respectively. For m s 0 the corresponding solutions of Eqn (20) are the associated Legendre’s functions of the first and second kind, respectively, which in un-normalized form are given as 8 m 
m ðzÞ ¼ 1
z2 m=2 d Pn ðzÞ z ˛ ð
1; 1Þ; > P > > n < dzm


m=2 dm Pn ðzÞ m Pbn ðzÞ ¼ z2
1 z ˛ð1; NÞ dzm

> m > > : Qm ðzÞ ¼
1
z2 m=2 d Qn ðzÞ z ˛ ð
1; 1Þ; n dzm

bm ðzÞ Q n




m=2 dm Pn ðzÞ ¼ z2
1 z ˛ð1; NÞ dzm

(30)

4.2.5 Neumann’s formula for the Legendre functions It is possible to show (Sneddon, 1956) that Qn(x) can be expressed by the integral Z1 1 Pn ðxÞ Qn ðxÞ ¼ dx jxj > 1 real 2 x
x

(31)


1

which is known as Neumann’s formula, and also that, for jxj > 1 Z1 1 xþ1 1 Pn ðxÞ
Pn ðxÞ
Qn ðxÞ ¼ Pn ðxÞln dx 2 x
1 2 x
x
1

(32)

158

CHAPTER 4 Special functions

An expression for Qn(x) with n integer useful for practical calculations is the following: Qn ðxÞ ¼

p 1 x þ 1 X 2n
4k
1 Pn ðxÞln
Pn
2k
1 ðxÞ 2 x
1 k¼0 ð2k þ 1Þðn
kÞ

(33)

where p ¼ ðn
1Þ=2 or p ¼ ðn
2Þ=2 according to n > 0 is odd or even. The first four Qns are then Q0 ðxÞ ¼

1 xþ1 ln ; 2 x
1

3 P1 ðxÞ 2 7 1 Q4 ðxÞ ¼ P4 ðxÞQ0 ðxÞ
P3 ðxÞ
P1 ðxÞ 4 3

Q1 ðxÞ ¼ P1 ðxÞQ0 ðxÞ
1;

5 1 Q3 ðxÞ ¼ P3 ðxÞQ0 ðxÞ
P2 ðxÞ
; 3 6

Q2 ðxÞ ¼ P2 ðxÞQ0 ðxÞ


(34)

4.3 LAGUERRE FUNCTIONS 4.3.1 Laguerre polynomials and Laguerre functions The Laguerre polynomials Ln(x) of degree n in x defined as (Sneddon, 1956) Ln ðxÞ ¼ expðxÞ

dn n ½x expð
xÞ dxn

(35)

are the solution of the Laguerre’s differential equation of the second order: x

d2 Ln ðxÞ dLn ðxÞ þ ð1
xÞ þ nLn ðxÞ ¼ 0 dx2 dx

(36)

where n is a positive integer. The explicit form for the first few Laguerre polynomials up to n ¼ 5 is L0 ðxÞ ¼ 1 L1 ðxÞ ¼ 1
x L2 ðxÞ ¼ 2
4x þ x2

(37)

L3 ðxÞ ¼ 6
18x þ 9x2
x3 L4 ðxÞ ¼ 24
96x þ 72x2
16x3 þ x4 L5 ðxÞ ¼ 120
600x þ 600x2
200x3 þ 25x4
x5 and, in general Ln ðxÞ ¼ n!

n X k¼0

ð
1Þk

  k n x k k!

(38)

We now verify in detail that L5(x) does verify the differential equation with n ¼ 5: xL005 ðxÞ þ ð1
xÞL05 ðxÞ þ 5L5 ðxÞ ¼ 0

(39)

4.3 Laguerre functions

159

In fact, using the primed notation for derivatives, we have from Eqn (37) L05 ðxÞ ¼
600 þ 1200x
600x2 þ 100x3
5x4 L005 ðxÞ

¼ 1200
1200x þ 300x
20x 2

3

(40) (41)

and substituting into Eqn (39) and adding all terms 1200x
1200x2 þ 300x3
20x4
600 þ 1200x
600x2 þ 100x3
5x4 þ 600x
1200x2 þ 600x3
100x4 þ 5x5

(42)

þ 600
3000x þ 3000x2
1000x3 þ 125x4
5x5 ¼ 0 as it must be. The Laguerre polynomials are not orthogonal in themselves, but the Laguerre functions 1 expð
x=2ÞLn ðxÞ n!

(43)

form an orthonormal set in the interval 0  x  N: ZN dx expð
xÞLn ðxÞLn0 ðxÞ ¼ ðn!Þ2 dnn0

(44)

0

as can be proved by integration by parts. Recurrence relations for the Laguerre polynomials are (Sneddon, 1956) Lnþ1 ðxÞ ¼ ð2n þ 1
xÞLn ðxÞ
n2 Ln
1 ðxÞ

(45)

L0n ðxÞ ¼ nL0n
1 ðxÞ
nLn
1 ðxÞ

 L00nþ1 ðxÞ ¼ ðn þ 1Þ xL00n ðxÞ
L0n ðxÞ

(46)

and for the derivatives

xL00n ðxÞ

¼ ðx


1ÞL0n ðxÞ
nLn ðxÞ

(47) (48)

the last equation being Eqn (36) defining Ln(x).

4.3.2 Associated Laguerre polynomials The m-th derivative (m  n) of the Laguerre polynomial Ln(x) of order n in x

dm Ln ðxÞ dm dn n m ¼ m expðxÞ n ½x expð
xÞ Ln ðxÞ ¼ dxm dx dx

(49)

160

CHAPTER 4 Special functions

is called the associated Laguerre polynomial Lm n ðxÞ of degree (n
m) and order m in x, and is the solution of the associated Laguerre’s differential equation of the second order: x

d2 Lm dLm ðxÞ n ðxÞ þ ðm þ 1
xÞ n þ ðn
mÞLm n ðxÞ ¼ 0 2 dx dx

(50)

where n, m (m  n) are positive integers. The explicit form of the first few associated Laguerre polynomials up to m ¼ 5 is L11 ðxÞ ¼
1 L12 ðxÞ ¼
4 þ 2x;

L22 ðxÞ ¼ 2

L13 ðxÞ ¼
18 þ 18x
3x2 ;

L23 ðxÞ ¼ 18
6x;

L14 ðxÞ ¼
96 þ 144x
48x2 þ 4x3 ;
 L34 ðxÞ ¼
96 þ 24x; L44 x ¼ 24

L33 ðxÞ ¼
6

L24 ðxÞ ¼ 144
96x þ 12x2 ;

L15 ðxÞ ¼
600 þ 1200x
600x2 þ 100x3
5x4 ; L35 ðxÞ ¼
1200 þ 600x
60x2 ;

L25 ðxÞ ¼ 1200
1200x þ 300x2
20x3 ;

L45 ðxÞ ¼ 600
120x;

and, in general m Lm n ðxÞ ¼ ð
1Þ n!

(51)

nX
m

ð
1Þk k! k¼0



L55 ðxÞ ¼
120  n xk mþk

(52)

Once the explicit form of a Laguerre polynomial is known, the corresponding associated Laguerre polynomial is readily evaluated through the appropriate derivative. We now verify in detail that L25 ðxÞ does verify the differential Eqn (50) with n ¼ 5, m ¼ 2, n
m ¼ 3: 00

0

xL25 ðxÞ þ ð3
xÞL25 ðxÞ þ 3L25 ðxÞ ¼ 0 Using the primed notation for the derivatives, we have from Eqn (51) 0

L25 ðxÞ ¼
1200 þ 600x
60x2 00 L25 ðxÞ

¼ 600
120x

and, substituting into the differential equation and adding all terms, we find

 600x
120x2 þ
3600 þ 1800x
180x2

 þ 1200x
600x2 þ 60x3 þ 3600
3600x þ 900x2
60x3 ¼ 0

(53) (54)

(55)

as it must be. In the previous chapter, we discussed two such associated Laguerre differential equations which we solved by the series expansion technique after solution of the asymptotic differential equations resulting at the singular points, (1) the radial equation for the atomic one-electron system and (2) the x-equation for the free hydrogen atom in parabolic coordinates.

4.3 Laguerre functions

161

Recurrence relations for the associated Laguerre polynomials are (Sneddon, 1956) m m
1 Lm ðxÞ
n2 Lm nþ1 ðxÞ ¼ ð2n þ 1
xÞLn ðxÞ
mLn n
1 ðxÞ

(56)

m m
1 Lm n ðxÞ ¼ nLn
1 ðxÞ
nLn
1 ðxÞ

(57)

4.3.3 Basic integrals over associated Laguerre functions The general integral over the product of two orthogonal Laguerre functions involving the associated Laguerre polynomials of degree k and k0 was originally given by Schroedinger (1926a) in terms of generalized binomial coefficients as ZN

0

m dx x p expð
xÞLm mþk ðxÞLm0 þk0 ðxÞ 0 0

0

¼ ð
1Þmþkþm þk p!ðm þ kÞ!ðm0 þ k0 Þ!

0 minðk;k X Þ

k¼0

 ð
1Þk

p
m k
k



p0
m0



k0
k


p
1 k

  i a generalized binomial coefficient defined by j 8   < 1 ðj ¼ 0Þ i ¼ iði
1Þ/ði
j þ 1Þ ðj > 0Þ j : j!



(58)

where p is a non-negative integer, and

(59)

with j a non-negative integer and i any integer. The generalized binomial coefficient becomes identical to an ordinary binomial coefficient whenever i  j. This formula was put into a more convenient form by Figari (2010) in terms of ordinary binomial coefficients as ZN

0

0

mþm m dx x p expð
xÞLm ðm þ kÞ!ðm0 þ k0 Þ! mþk ðxÞLm0 þk0 ðxÞ ¼ ð
1Þ

0



kþk X0 k¼0

ð
1Þk ðp þ kÞ!

0 minðk X;kÞ



mþk

1 ðk
sÞ!s! m þ k
s s¼maxð0;k
kÞ



m0 þ k 0 m0 þ s



(60)

For the calculation of the first-order Stark effect, we introduced in Chapter 3 the auxiliary integral Il(m,k) as ZN Il ðm; kÞ ¼ 0

2 dx x mþl expð
xÞ Lm mþk ðxÞ

(61)

162

CHAPTER 4 Special functions

From the formulae above, for m0 ¼ m; k0 ¼ k; p ¼ m þ l, we obtain ZN

2 dx xmþl expð
xÞ Lm Il ðm; kÞ ¼ mþk ðxÞ 0

¼ ðm þ lÞ!½ðm þ kÞ!

2

k X

" ð
1Þ

l

k¼maxð0;k
lÞ

k
k

since "

l k
k

"

# ¼

!2

mþlþk

# ¼

(62)

!

k

ðm þ l þ kÞ!

k¼maxð0;k
lÞ ½ðl


k þ kÞ!2 ½ðk
kÞ!2 k!

ðk ¼ kÞ

1

lðl
1Þ/½l
ðk
k
1Þ > : ðk
kÞ! 8  l! l > < ¼ k
k ðk
kÞ!ðl
k þ kÞ! ¼ > : 0


m
l
1 k

8 > <

k X

#

k

k X

¼ ðm þ lÞ!½ðm þ kÞ!2


m
l
1

k
k

k¼0

¼ ðl!Þ2 ½ðm þ kÞ!2

#2 "

l

k

ðk < kÞ ðk
l  k  kÞ ðk < k
lÞ (63)

8 <

ðk ¼ 0Þ

1

: ð
m
l
1Þð
m
l
2Þ/ð
m
l
kÞ k! ! mþlþk ðm þ l þ kÞ! ¼ ð
1Þk ¼ ð
1Þk k!ðm þ lÞ! k

ðk  0Þ

(64)

ðk  0Þ

We give below the explicit formulae for the first three auxiliary integrals with l ¼ 0, 1, 2: ZN

2 I0 ðm; kÞ ¼ dx xm expð
xÞ Lm mþk ðxÞ 0

¼ ½ðm þ kÞ!2

¼

½ðm þ kÞ!3 k!

k X k¼maxð0;kÞ ½ð
k

ðm þ kÞ! 2

(65) 2

þ kÞ! ½ðk
kÞ! k!

4.4 Hermite functions

ZN I1 ðm; kÞ ¼

163

2 dx xmþ1 expð
xÞ Lm mþk ðxÞ

0

¼ ½ðm þ kÞ!2

k¼maxð0;k
1Þ ½ð1


ðm þ 1 þ kÞ! k þ kÞ!2 ½ðk
kÞ!2 k!

¼

½ðm þ kÞ!2 ½kðm þ kÞ! þ ðm þ k þ 1Þ! k!

¼

½ðm þ kÞ!3 ðm þ 2k þ 1Þ k!

ZN I2 ðm; kÞ ¼

k X

(66)

2 dx xmþ2 expð
xÞ Lm mþk ðxÞ

0

¼ ð2!Þ2 ½ðm þ kÞ!2

k X k¼maxð0;k
2Þ ½ð2
k

ðm þ 2 þ kÞ! þ kÞ!2 ½ðk
kÞ!2 k!

¼

½ðm þ kÞ!2 ½kðk
1Þðm þ kÞ! þ 4kðm þ k þ 1Þ! þ ðm þ k þ 2Þ! k!

¼

 ½ðm þ kÞ!3 2 m þ 3mð2k þ 1Þ þ 6kðk þ 1Þ þ 2 k!

(67)

These integrals were used in Chapter 3 for the calculation of the normalization and z-integrals for the free hydrogen atom in parabolic coordinates.

4.4 HERMITE FUNCTIONS 4.4.1 Hermite polynomials The Hermite polynomials Hn(x) of degree n in x defined as (Sneddon, 1956)
 dn
 Hn ðxÞ ¼ ð
1Þn exp x2 exp
x2 n dx are the solution of the Hermite differential equation of the second order: d2 Hn ðxÞ dHn ðxÞ þ 2nHn ðxÞ ¼ 0
2x 2 dx dx

(68)

(69)

where n is a positive integer. These polynomials were found in Section 3.6 of the previous chapter in the study of the onedimensional linear harmonic oscillator.

164

CHAPTER 4 Special functions

The explicit form for the first few Hermite polynomials up to n ¼ 5 is H0 ðxÞ ¼ 1 H1 ðxÞ ¼ 2x H2 ðxÞ ¼ 4x2
2 H3 ðxÞ ¼ 8x3
12x

(70)

H4 ðxÞ ¼ 16x4
48x2 þ 12 H5 ðxÞ ¼ 32x5
160x3 þ 120x and, in general nðn
1Þ nðn
1Þðn
2Þðn
3Þ ð2xÞn
2 þ ð2xÞn
4 1! 2! nðn
1Þðn
2Þðn
3Þðn
4Þ ð2xÞn
6 þ /
3!

Hn ðxÞ ¼ ð2xÞn


(71)

As an example, we now verify that H5(x) does verify the differential equation with n ¼ 5: H500 ðxÞ
2xH50 ðxÞ þ 10H5 ðxÞ ¼ 0

(72)

In fact, we have from Eqn (70) H50 ðxÞ ¼ 120
480x2 þ 160x4

(73)

H500 ðxÞ ¼
960x þ 640x3

(74)

Substituting in Eqn (72) and adding all terms, we have
960x þ 640x3
240x þ 960x3
320x5 þ 1200x
1600x3 þ 320x5 ¼ 0

(75)

as it must be. Recursion relations are Hnþ1 ðxÞ ¼ 2xHn ðxÞ
2nHn
1 ðxÞ

(76)

Hn0 ðxÞ ¼ 2xHn ðxÞ
Hnþ1 ðxÞ

(77)

Hn00 ðxÞ ¼ 2xHn0 ðxÞ
2nHn ðxÞ

(78)

and for the derivatives

the latter equation being no more than the differential Eqn (69) determining Hn(x).

4.4.2 Hermite functions A differential equation related to the Hermite’s differential equation is the following:
 j00 ðxÞ þ l
x2 jðxÞ ¼ 0

(79)

4.4 Hermite functions

165

which for l ¼ 2n þ 1 we recognize as Eqn (113) of Section 3.6 of Chapter 3. We have shown there that the general regular solutions of Eqn (79) are given in this case by:
 (80) jn ðxÞ ¼ Nn exp
x2 =2 Hn ðxÞ where Nn is the normalization factor given by Eqn (136) of Chapter 3. Functions jn ðxÞ are called Hermite functions, and are orthonormal in the interval (
N,N) as shown by Eqn (87). The recursion relations for jn ðxÞ follow from those of the Hermite polynomials: jnþ1 ðxÞ ¼ 2xjn ðxÞ
2njn
1 ðxÞ

(81)

j0n ðxÞ ¼ xjn ðxÞ
jnþ1 ðxÞ

(82)


 j00n ðxÞ ¼ x2
2n
1 jn ðxÞ

(83)

and for the derivatives

which for l ¼ 2n þ 1 is Eqn (79) defining jn ðxÞ. We verify the recursion relations Eqn (81–83) by the direct calculation of the second derivative of j4 . We first recall that Eqn (76–78) give: H4 ¼ 2xH3
6H2 ;

H40 ¼ 2xH4
H5 ;

H400 ¼ 2xH40
8H4

Direct calculation then gives

 
j04 ¼ exp
x2 =2 H40
xH4 ¼ exp
x2 =2 ½ð2xH4
H5 Þ
xH4  
¼ exp
x2 =2 ðxH4
H5 Þ which is Eqn (82), and



  j004 ¼ exp
x2 =2 H400
2xH40 þ x2
1 H4


   ¼ exp
x2 =2 2xH40
8H4
2xH40 þ x2
1 H4

 ¼ exp
x2 =2 x2
9 H4

(84)

(85)

(86)

which is Eqn (83), or Eqn (79), as it must be.

4.4.3 Integrals over Hermite functions In the quantum mechanical treatment of the harmonic oscillator and its spectroscopy are of great importance integrals involving products of pairs of the Hermite functions jn ðxÞ and jm ðxÞ. From the definition Eqn (83), the following relations are easily established (Sneddon, 1956): ZN pffiffiffiffi d x jm ðxÞjn ðxÞ ¼ 2n n! pdnm (87) Imn ¼
N

expressing the orthogonality of functions jm and jn for m s n, and the normalization integral for m ¼ n: ZN pffiffiffiffi d x jn ðxÞjn ðxÞ ¼ 2n n! p (88) In ¼
N

166

CHAPTER 4 Special functions

Furthermore, we give below some integrals occurring in the selection rules for the spectroscopy of the harmonic oscillator: ZN 1 d x xjm ðxÞjn ðxÞ ¼ nIm;n
1 þ Im;nþ1 (89) 2
N

so that ZN d x xjm ðxÞjn ðxÞ ¼ 0

if msn  1

(90)

pffiffiffiffi d x xjn ðxÞjnþ1 ðxÞ ¼ 2n ðn þ 1Þ! p

(91)


N

and ZN
N

Similarly, we have for integrals involving the first derivative ZN dx

jm ðxÞj0n ðxÞ


N

8 <0

if msn
1 pffiffiffiffi ¼ p if m ¼ n
1 : n pffiffiffiffi
2 ðn þ 1Þ! p if m ¼ n þ 1 2n
1 n!

(92)

4.5 HYPERGEOMETRIC FUNCTIONS 4.5.1 Hypergeometric series and differential equation If a, b, c are parameters and z the complex variable, the series 1þ

a$b aða þ 1Þ $ bðb þ 1Þ 2 zþ z þ/ 1$c 1 $ 2 $ cðc þ 1Þ

(93)

is a generalization of the well-known geometric series 1 þ z þ z2 þ /

(94)

and is called hypergeometric series. Series (93) converges absolutely if jzj < 1 provided c is not zero or a non-negative integer, diverges if jzj > 1, while if jzj ¼ 1 the series converges absolutely if c > a þ b. Abramowitz and Stegun (1965) define series (93) as the hypergeometric function 2 F1 ða; b; c; zÞ: 2 F1 ða; b; c; zÞ

¼

N GðcÞ X Gða þ kÞGðb þ kÞ zk k! GðaÞGðbÞ k¼0 Gðc þ kÞ

(95)

where the suffixes 2 and 1 denote that there are two parameters of the type a and one parameter of the type c.

4.5 Hypergeometric functions

167

Using the properties of the G functions given in Section 4.7.1, if we introduce the Pochhammer’ symbol Gða þ kÞ ða þ k
1Þ! ¼ ðaÞk ¼ GðaÞ ða
1Þ! ¼

ða þ k
1Þða þ k
2Þ/½a þ k
ðk
2Þ½a þ k
ðk
1Þða þ k
kÞða
1Þ! ða
1Þ!

(96)

¼ aða þ 1Þða þ 2Þ/ða þ k
1Þ we can rewrite Eqn (95) in the contracted form: 2 F1 ða; b; c; zÞ ¼

N X aða þ 1Þ/ða þ k
1Þbðb þ 1Þ/ðb þ k
1Þ zk cðc þ 1Þ/ðc þ k
1Þ k! k¼0

N X ðaÞk ðbÞk zk a$b aða þ 1Þ $ bðb þ 1Þ 2 ¼1þ zþ z þ/ ¼ ðcÞk k! 1$c 1 $ 2 $ cðc þ 1Þ k¼0

(97)

Even if the two suffixes are important in the further generalization of the hypergeometric functions (Sneddon, 1956), it will suffice for our purposes to omit them if there is no risk of confusion, so simplifying our notation, henceforth denoting 2 F1 simply by F. The hypergeometric function (97) is the solution of the second-order hypergeometric linear differential equation d2 y dy
aby ¼ 0 (98) zð1
zÞ 2 þ ½c
ða þ b þ 1Þy dz dz It can be shown (Sneddon, 1956) that any ordinary linear differential equation of the second order whose singular points are regular points can be transformed by a simple change of variable to the form of Eqn (98). In the series expansion of Eqn (98), the roots of the indicial equations (see Section 3.5 of Chapter 3) are 1. Near z ¼ 0 a ¼ 0;

a¼1
c

(99)

a¼c
a
b

(100)

2. Near z ¼ 1 a ¼ 0; 3. Near z ¼ N a ¼ a;

a¼b

(101)

It can be shown (Sneddon, 1956) that the corresponding solutions of the differential Eqn (98) are 1. Near the origin (z ¼ 0), provided (1
c) is not zero or a positive integer yðzÞ ¼ AFða; b; c; zÞ þ Bz1
c Fða
c þ 1; b
c þ 1; 2
c; zÞ where A and B are arbitrary constants. If c ¼ 1, the two F solutions become identical.

(102)

168

CHAPTER 4 Special functions

2. Near z ¼ 1, the solution is yðzÞ ¼ AFða; b; a þ b
c þ 1; 1
zÞ þ Bð1
zÞc
a
b Fðc
a; c
b; c
a
b þ 1; 1
zÞ (103) 3. Near z ¼ N, the solution is

  yðzÞ ¼ Az
a F a; a
c þ 1; a
b þ 1; z
1 þ Bz
b F b; b
c þ 1; b
a þ 1; z
1

(104)

4.5.2 Confluent hypergeometric functions The confluent hypergeometric functions are solutions of the differential equation called Kummer’s or confluent hypergeometric equation (Abramowitz and Stegun, 1965): x

d2 y dy
ay ¼ 0 þ ðb
xÞ 2 dx dx

(105)

We see that, for b ¼ m þ 1, a ¼ m
n, Eqn (105) coincides with the associated Laguerre differential Eqn (50), so that, in this case, yðxÞ ¼ Lm n ðxÞ. Independent solutions are the Kummer’s function: Mða; b; xÞ ¼

N X ðaÞk xk ðbÞk k! k¼0

(106)

where (a)k and (b)k are the Pochhammer’ symbols (96), and Uða; b; xÞ ¼

  p Mða; b; xÞ Mð1 þ a
b; 2
b; xÞ
x1
b sin pb Gð1 þ a
bÞGðbÞ GðaÞGð2
bÞ

(107)

Properties of these functions are dn ðaÞn Mða; b; xÞ ¼ Mða þ n; b þ n; xÞ d xn ðbÞn 1 Uða; b; xÞ ¼ GðaÞ

(108)

ZN dt expð
xtÞta
1 ð1 þ tÞb
a
1

(109)

0

These functions were discussed in the study of the damping coefficients and the non-expanded multipole analysis of the exact second-order induction energy in Hþ 2 (Magnasco and Figari, 1987a), where use was made of the confluent hypergeometric functions M(a,b, x) and U(a,b, x). For integer positive values of the parameters a, b and (b
a)  1, polynomial expressions useful in the applications can be found for the functions U and M.

4.5 Hypergeometric functions

169

Using the binomial expansion in Eqn (109), we obtain ZN dt expð
xtÞta
1 ð1 þ tÞb
a
1

ða
1Þ!Uða; b; xÞ ¼ 0

¼

¼

b
a
1 X

b
a
1

k¼0

k

x1
b

!

ðk þ a
1Þ! xkþa

b
a
1 X

b
a
1

k¼0

k

(110)

! ðb
2
kÞ!xk

so that Uða; b; xÞ ¼

ðb
2Þ! 1
b x Pb
a
1 ðxÞ ða
1Þ!

(111)

where Pb
a
1(x) is a polynomial of degree (b
a
1) in x with P0 ¼ 1. A convenient polynomial form for the Kummer’s function M(a,b,x) is derived starting from the series expansion (106) for M(1,b,x): N N X X ð1Þk xk xk ¼ ðb
1Þ! ðbÞk k! ðb
1 þ kÞ! k¼0 k¼0 # " N b
2 k X X xk x 1
b 1
b ¼ ðb
1Þ!x expðxÞ
¼ ðb
1Þ!x k! k! k¼0 k¼b
1

(112)

ðbÞn dn Mða; b; xÞ ðaÞn d xn

(113)

Mð1; b; xÞ ¼

Using Eqn (108) Mða þ n; b þ n; xÞ ¼ We can write Mða; b; xÞ ¼

ðb
1Þ! da
1 Mð1; b
a þ 1; xÞ ða
1Þ!ðb
aÞ! d xa
1

(114)

This expression relates the M function of the general parameter a to the derivative of the M function of parameter a ¼ 1, whose polynomial expression was given in Eqn (112). Calculating the derivatives, we obtain "   a
1 X ðb
1Þ! k a
1 ðb
a
1 þ kÞ! a
b
k x ð
1Þ þ ð
1Þa Mða; b; xÞ ¼ expðxÞ ða
1Þ! ðb
a
1Þ! k k¼0 # (115) b
a
1 X ðb
2
kÞ! 1
bþk  x k!ðb
a
1
kÞ! k¼0

170

CHAPTER 4 Special functions

which can be rearranged to ðb
1Þ!ðb
2Þ! x1
b Mða; b; xÞ ¼ ð
1Þa ða
1Þ!ðb
a
1Þ! "

  
1 k b
a
1 a
1 X  b
a
1  b
2 
1 xk X x b
2 k a
1 þ 
expðxÞ ð
1Þ k k k k k! k! k¼0

¼ ð
1Þa

#

k¼0

ðb
1Þ!ðb
2Þ! x1
b ½
expðxÞPa
1 ðxÞ þ Qb
a
1 ðxÞ ða
1Þ!ðb
a
1Þ!

(116)

where P(x) and Q(x) are polynomials in x of degree (a
1) and (b
a
1), respectively, with P0 ¼ Q0 ¼ 1. We shall see later in Section 4.7.2 that the incomplete gamma function g(a, x) can be put in relation to the Kummer’s function M(1,a þ 1,x) through Eqn (175) which gives a polynomial expression for it.

4.6 BESSEL FUNCTIONS 4.6.1 Bessel functions of integral order The Bessel functions Jn(x) of order n in x are the solution of the Bessel’s differential equation of the second order:  d2 Jn ðxÞ d Jn ðxÞ
2 þ x
n2 Jn ðxÞ ¼ 0 þx (117) x2 2 dx dx where n is a positive integer. In Problem 4.2 it is shown that the function Jn(x) can be defined in terms of hypergeometric functions as (Sneddon, 1956)   xn 1 (118) Jn ðxÞ ¼ n 0 F1 n þ 1;
x2 2 n! 4 where

  X   N 1 2 1 1 2 k ¼
x 0 F1 n þ 1;
x 4 k!ðn þ 1Þk 4 k¼0

k  1 1 ¼
x2 4 k¼0 k!ðn þ 1Þðn þ 2Þ/ðn þ kÞ N P

(119)

and (a)k is the Pochhammer’ symbol (96). Recursion relations for the function Jn(x) are easily found to be xJn0 ðxÞ ¼ xJn
1 ðxÞ
nJn ðxÞ xJn00 ðxÞ þ Jn0 ðxÞ ¼

n2 Jn ðxÞ
xJn ðxÞ x

where use was made of the primed notation for the derivatives.

(120) (121)

4.6 Bessel functions

171

4.6.2 Bessel functions of half-integral order When n is not an integer, the differential equation of the second order x2

 d2 yðxÞ dyðxÞ
2 þ x
n2 yðxÞ ¼ 0 þx dx2 dx

(122)

admits as general solution yðxÞ ¼ AJn ðxÞ þ BJ
n ðxÞ

(123)

  xn 1 2 Jn ðxÞ ¼ n F1 n þ 1;
x 2 Gðn þ 1Þ 0 4

(124)

with A, B arbitrary constants, and

When n ¼ n is an integer, the two solutions Jn(x) and J
n(x) are not linearly independent, since J
n ðxÞ ¼ ð
1Þn Jn ðxÞ

(125)

When n ¼ n þ 1/2 is half an integer (n ¼ integer), the functions Jnþ 1 ðxÞ; J
n
1 ðxÞ 2

(126)

2

are called spherical Bessel functions of order n. These functions are simply related to the circular functions sin x and cos x. Using the duplication formula for the gamma function of Section 4.7.1, it can be shown that (Sneddon, 1956)  1=2 X N 2 x2kþ1 (127) ð
1Þk J1=2 ðxÞ ¼ ð2k þ 1Þ! px k¼0 so that

 2 1=2 sin x px

 J1=2 ðxÞ ¼

Again it can be shown that



J
1=2 ðxÞ ¼

2 px

1=2 X N k¼0

ð
1Þk

x2k ¼ ð2kÞ!

(128)



2 px

1=2 cos x

and, in general, when n is half an odd integer  1=2 2 Jn ðxÞ ¼ ½ fn ðxÞsin x
gn ðxÞcos x px  J
n ðxÞ ¼

2 px

(129)

(130)

1=2 ð
1Þn
1=2 ½gn ðxÞsin x þ fn ðxÞcos x

(131)

172

CHAPTER 4 Special functions

The explicit expressions of the first few Jn(x) and J
n(x) are 1=2   1 sin x
cos x x   1=2   2 3 3
1 sin x
cos x J5=2 ðxÞ ¼ px x2 x  1=2      2 15 6 15 J7=2 ðxÞ ¼
sin x
2
1 cos x px x3 x x  1=2      2 105 45 105 10
2 þ 1 sin x

cos x J9=2 ðxÞ ¼ px x4 x x3 x 

J3=2 ðxÞ ¼

2 px



   2 1=2 1 sin x þ cos x px x



     2 1=2 3 3 sin x þ 2
1 cos x px x x

J
3=2 ðxÞ ¼ J
5=2 ðxÞ ¼  J
7=2 ðxÞ ¼  J
9=2 ðxÞ ¼

1=2 

2 px

    15 15 6 cos x
1 sin x þ
x2 x3 x

(132)

(133)

      2 1=2 105 10 105 45 sin x þ

2 þ 1 cos x px x3 x x4 x

4.6.3 Spherical Bessel functions To examine in more detail their properties, it is convenient to redefine the spherical Bessel functions of order n (n is a non-negative integer), denoted by jn(x), as the analytic solutions of the differential equation of the second order (Abramowitz and Stegun, 1965):   d2 jn ðxÞ 2 djn ðxÞ nðn þ 1Þ jn ðxÞ ¼ 0 þ 1
þ dx2 x dx x2

(134)

Since x ¼ 0 is a regular singularity, the solution of Eqn (134) can be found by expanding jn(x) in the power series of the variable x: jn ðxÞ ¼

N X k¼0

ak xaþk

(135)

4.6 Bessel functions

173

The indicial equation gives a ¼ n for a0 s 0. Furthermore it is found a1 ¼ 0, so that only even coefficients survive with the recurrence relation: akþ2 ¼


ak ðk þ 2Þðk þ 2n þ 3Þ

(136)

so that by iteration (see Problem 4.2) we obtain a2k ¼ ð
1Þkþ1

ð2n þ 1Þ!! a0 ð2kÞ!!ð2k þ 2n þ 1Þ!!

(137)

the double factorial functions being defined as:

(

ð2nÞ!! ¼ 2n n! ð2n
1Þ!! ¼

ð2n þ 2Þ!! ¼ ð2n þ 2Þð2nÞ!! ð2nÞ! 2n n!

ð2n þ 1Þ!! ¼

ð2n þ 1Þ! 2n n!

(138) 0!! ¼ ð
1Þ!! ¼ 1

Putting ð2n þ 1Þ!!a0 ¼ 1, we obtain for jn(x) the representation as infinite series: jn ðxÞ ¼

N X k¼0

ð
1Þk xnþ2k ð2kÞ!!ð2k þ 2n þ 1Þ!!

(139)

Equation (139) gives immediately the parity of jn(x): jn ð
xÞ ¼ ð
1Þn jn ðxÞ

(140)

and the recurrence relations jn
1 ðxÞ ¼

nþ1 jn ðxÞ þ j0n ðxÞ; x

jnþ1 ðxÞ ¼

n jn ðxÞ
j0n ðxÞ x

(141)

In fact, we observe that N  d nþ1 d X ð
1Þk x2kþ2nþ1 x jn ðxÞ ¼ dx dx k¼0 ð2kÞ!!ð2k þ 2n þ 1Þ!! N X

ð
1Þk x2kþ2n ¼ xnþ1 jn
1 ðxÞ ¼ ð2kÞ!!ð2k þ 2n
1Þ!! k¼0

(142)

and, deriving the left-hand member ðn þ 1Þxn jn ðxÞ þ xnþ1 j0n ðxÞ ¼ xnþ1 jn
1 ðxÞ which proves the first recurrence relation.

(143)

174

CHAPTER 4 Special functions

Likely, for the second recurrence relation, we have  d
n x jn ðxÞ ¼
nx
n
1 jn ðxÞ þ x
n j0n ðxÞ dx N d X ð
1Þk x2k ¼ dx k¼0 ð2kÞ!!ð2k þ 2n þ 1Þ!! ¼ ¼

N X

ð
1Þk ð2kÞx2k
1 ð2kÞ!!ð2k þ 2n þ 1Þ!! k¼0

N X

(144)

kþ1

ð
1Þ ð2k þ 2Þx2kþ1 ð2k þ 2Þ!!ð2k þ 2n þ 3Þ!! k¼0

¼
x
n

N X

ð
1Þk x2kþnþ1 ð2kÞ!!ð2k þ 2n þ 3Þ!! k¼0

¼
x
n jnþ1 ðxÞ from which follows the second recurrence relation. The recurrence relations give for the first derivative of jn(x) the formulae n j0n ðxÞ ¼ jn ðxÞ
jnþ1 ðxÞ x n þ1 j0n ðxÞ ¼
jn ðxÞ þ jn
1 ðxÞ x

(145) (146)

From Eqn (145) and Eqn (146) follows for the second derivative of jn(x) the expression j00n ðxÞ ¼

i nðn
1Þ dhn 2 jn ðxÞ
jn ðxÞ þ jnþ1 ðxÞ jn ðxÞ
jnþ1 ðxÞ ¼ dx x x2 x

(147)

Using for jnþ1(x) the second of the recurrence formulae Eqn (141), it is easily seen that jn(x) does satisfy the starting differential Eqn (134). A finite form for jn(x) is given by the integral representation1 (Abramowitz and Stegun, 1965) Z1 1 n du expðixuÞPn ðuÞ (148) jn ðxÞ ¼ ð
iÞ 2
1

where i is the imaginary unit (i ¼
1) and Pn(u) a Legendre polynomial of degree n with u ¼ cos q. Integral (148) can be calculated by successive integration by parts taking Pn(u) as a finite factor and du exp(ixu) as a differential factor. The successive derivatives of Pn(u) stop at the order n, so that 2

Z1 du expðixuÞPn ðuÞ ¼
1 1

 n X ð
iÞq q¼0 ðixÞ

qþ1

expðixuÞ

dq Pn ðuÞ duq

1 (149)
1

The equivalence with the infinite series (139) for jn(x) is obtained by expanding the imaginary exponential in power series.

4.6 Bessel functions

175

The derivatives of Pn(u) are calculated from the definition (3) of the Legendre polynomials and the Leibnitz’s rule for the n-th derivative of products of functions (Smirnov, 1993a): n   X dn n ðn
kÞ ðkÞ f ð fgÞ ¼ g (150) k dxn k¼0

where

f ðnÞ

¼d

n

f ðxÞ=dxn

denotes the n-th derivative of f(x). We have

dq 1 dnþq P ðuÞ ¼ ½ðu
1Þn ðu þ 1Þn  n duq 2n n! dunþq

¼

nþq 1 X  n þ q dt dnþq
t ðu
1Þn nþq
t ðu þ 1Þn n t t du du 2 n! t¼0

¼

n  n! 1 X n! nþq ðu
1Þn
t ðu þ 1Þt
q n t 2 n! t¼q ðn
tÞ! ðt
qÞ!

(151)

Substituting into Eqn (149) and noting that ðu
1Þn
t ¼ dnt

for u ¼ 1;

ðu þ 1Þt
q ¼ dqt

for u ¼
1

(152)



du expðixuÞPn ðuÞ ¼ in Wn ðxÞexpðixÞ þ Wn ðxÞexpð
ixÞ

(153)

we obtain integral (149) in the form Z1
1

where Wn ðxÞ ¼ ð
iÞnþ1

n X iq ðn þ qÞ!
q
1 x 2q q!ðn
qÞ! q¼0

Hence, we obtain for jn(x) the finite representation  1

jn ðxÞ ¼ Wn ðxÞexpðixÞ þ Wn ðxÞexpð
ixÞ 2

(154)

(155)

Euler’s formulae for the imaginary exponentials expðixÞ ¼ cos x  i sin x

(156)

allow to transform Eqn (155) into an explicit form involving algebraic and trigonometric functions of the variable x, particularly suited for the graphic representation of the function jn(x): jn ðxÞ ¼ ð
1Þn=2 ½ fn ðxÞcos x þ gn ðxÞsin x n ¼ even

(157)

jn ðxÞ ¼ ð
1Þðnþ1Þ=2 ½gn ðxÞcos x
fn ðxÞsin x n ¼ odd

(158)

176

CHAPTER 4 Special functions

where, if [t] ¼ the largest integer t, and fn(x) ¼ 0 for n ¼ 0 hn
1i fn ðxÞ ¼

2 X q¼0

gn ðxÞ ¼

ð
1Þq ðn þ 2q þ 1Þ! x
2q
2 22qþ1 ð2q þ 1Þ!ðn
2q
1Þ! ½n=2 X

ð
1Þq ðn þ 2qÞ!
2q
1 x 22q ð2qÞ!ðn
2qÞ! q¼0

(159)

(160)

The explicit form of the first few spherical Bessel’s functions (SBFs) is given below: sin x j0 ðxÞ ¼ x 1 1 j1 ðxÞ ¼
cos x þ 2 sin x x x   3 1 3
sin x j2 ðxÞ ¼
2 cos x
x x x3     1 15 6 15 j3 ðxÞ ¼ cos x
2
4 sin x
x x3 x x     10 105 1 15 105 j4 ðxÞ ¼
4 cos x þ
þ 5 sin x x2 x x x3 x     1 105 945 5 105 189 j5 ðxÞ ¼

3 þ 5 cos x þ sin x
þ x x 2x2 2x4 2x6 x

(161)

Formulae (161) are simpler than their parent formulae (132) and (133). It is apparent that the behaviour of the SBFs is the wavelike one of the trigonometric functions damped by the presence of the algebraic factors depending on the negative powers of the variable. Using recurrence relation Eqn (145) in the equivalent operatorial form   n 1 d jn ðxÞ (162) jnþ1 ðxÞ ¼
j0n ðxÞ þ jn ðxÞ ¼ xnþ1
x x dx xn by iteration we find the so-called Rayleigh’s formula for jn(x):     1 d n 1 d n sin x n n j0 ðxÞ ¼ x
jn ðxÞ ¼ x
x dx x dx x

(163)

4.6.4 Modified Bessel functions The second-order differential equation  d2 yðxÞ dyðxÞ
2
x þ n2 yðxÞ ¼ 0 x2 þx 2 dx dx

(164)

4.6 Bessel functions

177

which differs from Bessel’s Eqn (122) only in the coefficient of y(x), frequently occurs in problems of mathematical physics.2 Its two solutions in real form are called modified Bessel’s functions, and are defined as (Watson, 1966)  nþ2k 1 x N X 2 (165) In ðxÞ ¼ k!Gðn þ k þ 1Þ k¼0 the modified Bessel’s function of the first kind, and Kn ðxÞ ¼

p I
n ðxÞ
In ðxÞ 2 sin np

(166)

the modified Bessel’s function of the second kind. The physical importance of Kn(x) lies in the fact that it is a solution of Eqn (164) which tends exponentially to zero as x /N through positive values of the variable. These functions were used by Cha1asinski and Jeziorski (1974) in their exact multipole expansion of the second-order induction interaction of two hydrogen atoms discussed in detail in Section 17.3.3 of Chapter 17.3 For integer n, the SBFs of Section 4.6.3 can be related to the modified Bessel’s functions of the second kind by the relation  1=2 2 x
1=2 Knþ 1 ðxÞ (167) jn ðxÞ ¼ 2 p The function xn Kn ðxÞ

(168)

for n > 0 has no singularities for x / 0. Therefore, the function kn ðxÞ ¼ ð2=pÞ1=2 xn Kn ðxÞ

(169)

was called by Shavitt (1963) reduced Bessel function. For half-integer n k
1=2 ¼

kN
1 ðxÞ ¼ 2

expð
xÞ x

(170)

N expð
xÞ X ð2N
k
1Þ! k
N k 2 x x ðk
1Þ!ðN
kÞ! k¼1

(171)

The following recursion formula, practical for computational purposes, holds for all values of n (Magnus et al., 1966): x2 kn
1 ðxÞ ¼ knþ 3 ðxÞ
ð2n þ 1Þknþ 1 ðxÞ 2

2 3

2

2

Equation (164) is obtained in the solution of the Laplace’s equation in cylindrical coordinates. Incorrectly, Cha1asinski and Jeziorski call Kn(x) modified Bessel’s function of the third kind.

(172)

178

CHAPTER 4 Special functions

The reduced Bessel functions were extensively used by Steinborn and co-workers (Weniger and Steinborn, 1983; and references therein) in their studies of multicentre molecular integrals over Slatertype orbitals (STOs) and, more generally, exponential-type orbitals (ETOs) with their related anisotropic generalizations, the so-called B-functions. The B-functions are linear combinations of STOs, having a sensibly more complicated analytical structure than STOs but more appealing properties in multicentre problems, such as simpler Fourier transforms (FTs) and extremely compact convolution integrals (see Section 4.9).

4.7 FUNCTIONS DEFINED BY INTEGRALS 4.7.1 The gamma function The gamma function G is defined by the integral (Sneddon, 1956; Abramowitz and Stegun, 1965) ZN dx expð
xÞxn
1 (173) GðnÞ ¼ 0

where n > 0. A few properties of the gamma function are 1. Gð1Þ ¼ 1 2. Gðn þ 1Þ ¼ nGðnÞ the recurrence formula 3. Gðn þ 1Þ ¼ n! if n is positive integer   pffiffiffiffi 1 ¼ p 4. G 2     1 1 Gð2nÞ ¼ 22n
1 GðnÞG n þ 5. G 2 2 the duplication formula For n a positive integer the duplication formula becomes     1 1 ð2nÞ! ¼ 22n n!G n þ G 2 2 n!nx n/N ðx þ 1Þðx þ 2Þ/ðx þ nÞ the Euler’s formula

6. Gðx þ 1Þ ¼ lim

x>0

Some of these properties are easily derived in Problem 4.3 using the definition of the gamma function and the rule of integration by parts.

4.7.2 The incomplete gamma function The incomplete gamma function g(a,x) is defined by the integral (Abramowitz and Stegun, 1965) Zx gða; xÞ ¼ dt expð
tÞta
1 (174) 0

4.7 Functions defined by integrals

179

and is related to the Kummer’s confluent hypergeometric function of Section 4.5.2 by the relation gða; xÞ ¼ a
1 expð
xÞxa Mð1; a þ 1; xÞ # " a
1 k (175) X x ¼ ða
1Þ!½1
expð
xÞPa
1 ðxÞ ¼ ða
1Þ! 1
expð
xÞ k! k¼0 The difference between the gamma function GðaÞ and the incomplete gamma function g(a,x) defines the new function Gða; xÞ: ZN dt expð
tÞta
1 (176) Gða; xÞ ¼ GðaÞ
gða; xÞ ¼ x

which for a ¼
n negative integer can be expressed by # " n
1 X 1 n
1 k ðn
1
kÞ! k
n x ð
1Þ Gð
n; xÞ ¼ ð
1Þ Eið
xÞ þ ðn
1Þ!expð
xÞx n! ðn
1Þ! k¼0 " ¼

#

(177)

1 ð
1Þn
1 Eið
xÞ þ ðn
1Þ!expð
xÞx
n Pn
1 ðxÞ n!

where Ei(
x) is the exponential integral function defined by Eqn (185) of the next section.

4.7.3 From the gamma function to the exponential integral function From Eqn (6) of the previous section it is possible to derive an expression for Euler’s constant g, which is defined by the series   1 1 1 (178) g ¼ lim 1 þ þ þ / þ
ln n ¼ 0:577 215 665/ n/N 2 3 n In fact, taking the logarithm of (6) we have ln Gðx þ 1Þ ¼ lim ½ln n! þ x ln n
ln ðx þ 1Þ
ln ðx þ 2Þ
/
ln ðx þ nÞ n/N

and, taking its derivative   d 1 1 1 ln Gðx þ 1Þ ¼ lim ln n


/
n/N dx xþ1 xþ2 xþn Letting x/0 we obtain



d ln Gðx þ 1Þ g ¼
lim x/0 dx and, from the definition of Gðx þ 1Þ it follows ZN g ¼
dt expð
tÞln t 0

(179)

(180)

 (181)

(182)

180

CHAPTER 4 Special functions

Integrating by parts the function of x ZN ZN ZN N dt expð
tÞln t ¼
ln td½expð
tÞ ¼
ln t expð
tÞjx þ dt expð
tÞt
1 x

x

ZN

¼ ln x expð
xÞ þ

x

(183)

dt expð
tÞt
1

x

and, taking the limit for x/0, we find ZN ZN g ¼
dt expð
tÞ ln t ¼
lim dt expð
tÞln t x/0

0

2

¼
lim 4

3

ZN dt

x/0

x

expð
tÞ þ ln x5 ¼ lim ½Eið
xÞ
ln x x/0 t

(184)

x

where

ZN Eið
xÞ ¼


dt

expð
tÞ ¼
E1 ðxÞ t

(185)

x

is the exponential integral function (Sneddon, 1956; Abramowitz and Stegun, 1965). We notice that dEið
xÞ expð
xÞ ¼ dx x

(186)

4.7.4 The exponential integral function For E1(x) we have the series expansion E1 ðxÞ ¼
Eið
xÞ ¼
g
ln x


N X xn ð
1Þn n $ n! n¼1

(187)

where g is Euler’s constant, or Eið
xÞ ¼ g þ ln x þ Hence

N X xn ð
1Þn n $ n! n¼1

g ¼ lim ½Eið
xÞ
ln x x/0

(188)

(189)

This relation, which is nothing but Eqn (184) derived before, occurs in the calculation of the twocentre two-electron exchange integral ð1sA 1sB j1sA 1sB Þ between STO functions centred at nuclei A and B (Section 18.7.2 of Chapter 18 of this book) and its higher homologues. This integral was met by Heitler and London (1927) in their wave mechanical calculations on the ground state of the hydrogen molecule, and was first evaluated exactly by Sugiura (1927).

4.7 Functions defined by integrals

181

The complete form for this exchange integral, here given in the so-called charge density notation, is Z

Z ð1sA 1sB j1sA 1sB Þ ¼

Z

dr2

dr1

½1sA ðr2 Þ1sB ðr2 Þ ½1sA ðr1 Þ1sB ðr1 Þ r12

(190)

dr1 KAB ðr1 Þ½1sA ðr1 Þ1sB ðr1 Þ

¼

where KAB(r1) is the two-centre exchange potential at the space point r1 due to the two-centre charge distribution of electron 2 at r2: Z ½1sA ðr2 Þ1sB ðr2 Þ (191) KAB ðr1 Þ ¼ dr2 r12 The calculation of this difficult integral is based on the expansion of the inverse of the interelectron distance r12 in spheroidal coordinates and is explained in detail in Section 18.7.2 of Chapter 18 of this book. The series occurring for two-centre spherical 1s orbitals with equal orbital exponents breaks down after two terms, but an infinite number of terms is needed when the orbital exponents are different. This is the case of the four-centre two-electron integral evaluated by Musso and Magnasco (1971) in terms of a Gauss–Legendre four-dimensional numerical integration using appropriate recursion formulae for the auxiliary functions.

4.7.5 The generalized exponential integral function In the calculation of molecular multicentre exchange integrals over STOs (Magnasco et al., 1998; Magnasco et al., 1999; Magnasco and Rapallo, 2000) occurs the generalized exponential integral of order n En(r) (Abramowitz and Stegun, 1965) defined as ZN En ðrÞ ¼

dx expð
rxÞx
n

(192)

1

with n a non-negative integer and Re(r) > 0. The high accuracy needed in its numerical calculation can be achieved through multiple precision arithmetics using recurrence relations and accurate Gaussian quadrature techniques (Ralston, 1965; Demidovic and Maron, 1981; Bachvalov, 1981). It is of crucial importance to start the recursion with extremely accurate terms.

4.7.6 Further functions The basic indefinite integral occurring in atomic or molecular calculations involving STOs with exponential radial decay is defined by the primitive function Z n X n! xn
k ð
1Þk Fn ðxÞ ¼ dx expð
rxÞxn ¼ expð
rxÞ ðn
kÞ! ð
rÞkþ1 k¼0 # " n! ðrxÞn ðrxÞn
1 ðrxÞ2 þ þ/þ þ rx þ 1 ¼
nþ1 expð
rxÞ (193) n! ðn
1Þ! 2! r ¼


n! rnþ1

expð
rxÞ

n X ðrxÞk k¼0

k!

182

CHAPTER 4 Special functions

where n is a non-negative integer and r a real positive number, a result that can be obtained by repeated integration by parts. From these relations follow some definite integrals needed in one-centre atomic problems: " # Zu n k X n! ðruÞ dx expð
rxÞxn ¼ nþ1 1
expð
ruÞ (194) k! r k¼0 0

ZN dx expð
rxÞxn ¼

n! rnþ1

expð
ruÞ

n X ðruÞk k¼0

u

k!

(195)

giving by addition the well-known integral ZN dx expð
rxÞxn ¼

n! rnþ1

(196)

0

and the following auxiliary functions needed in two-centre molecular problems (Roothaan, 1951b): ZN An ðrÞ ¼

dx expð
rxÞxn ¼

n! rnþ1

expð
rÞ

n X rk k¼0

1

k! (197)

Z1 dx expð
rxÞxn ¼ ð
1Þnþ1 An ð
rÞ
An ðrÞ

Bn ðrÞ ¼
1

In some calculations are needed the integrals Zu n X n! ðruÞk dx expð
rxÞxn ¼ An ðrÞ
nþ1 expð
ruÞ k! r k¼0

(198)

1

ZN dx expð
rxÞxn ¼

n! rnþ1

expð
ruÞ

n X ðruÞk k¼0

u

k!

(199)

whose addition gives the An(r) function. Properties of the Bn(r) functions are Bn ð
rÞ ¼ ð
1Þn Bn ðrÞ;

Bn ð0Þ ¼

2 den nþ1

(200)

where e ¼ even. Recurrence relations between the auxiliary functions are sometimes needed in numerical calculations. For instance, it is easily shown from the definitions that 1 An ðrÞ ¼ ½nAn
1 ðrÞ þ rA0 ðrÞ r

(201)

4.7 Functions defined by integrals

183

In fact An ðrÞ ¼

n! rnþ1

n
1 k X r

expð
rÞ

k¼0

k!

"

¼

þ

n! rnþ1

n
1 k X r

expð
rÞ

rn n!

#

(202)

n ðn
1Þ! expð
rÞ 1 þ expð
rÞ ¼ ½nAn
1 ðrÞ þ rA0 ðrÞ k! r rn r r k¼0

The generalized auxiliary functions are defined as ZN m
dx expð
rxÞxj x2
1 Tjm ðrÞ ¼

(203)

1

Z1


m dx expð
rxÞxj 1
x2

(204)


m=2 m dx expð
rxÞxj 1
x2 Pl ðxÞ

(205)

Gjm ðrÞ ¼
1

Z1 Bljm ðrÞ

¼
1

ZN Hlmpq ðr1 ; r2 Þ ¼

ZN dx

1


m=2 m


Q^l ðx> ÞP^m dy expð
r1 xÞexpð
r2 yÞxp yq x2
1 y2
1 l ðx< Þ (206)

1

^m ^m j, l, m, p, q being non-negative integers, Re(r) > 0, Pm l ; Pl ; Ql normalized associated Legendre functions of first and second kind, x< and x> the lesser and the greater of x and y. Tjm(r) and Gjm(r) are generalizations of the elementary auxiliary functions Aj(r) and Bj(r) defined above, while Bljm ðrÞ and Hlmpq(r1,r2) were introduced by Ruedenberg (1951) as a generalization of the auxiliary integral ZN Hlpq ðrÞ ¼ Hlqp ðrÞ ¼

ZN dx

1

dy exp½
rðx þ yÞxp yq Q^l ðx> ÞP^l ðx< Þ

(207)

1

occurring in Rosen calculation (1931) of the exchange integrals over STOs in his quantum mechanical treatment of the ground state of the hydrogen molecule. It is seen that Tjm(r) and Gjm(r) can be written as finite sums of An and Bn functions:   m X mþk m Ajþ2k ðrÞ ð
1Þ (208) Tjm ðrÞ ¼ k k¼0   m X k m Bjþ2k ðrÞ ð
1Þ (209) Gjm ðrÞ ¼ k k¼0

While evaluation of Tjm(r) by Eqn (208) is straightforward through use of recurrence relations, for some values of indices j, m, Gjm(r) may be affected by strong numerical instabilities due to cancellation of terms of similar magnitude. Stability in the evaluation of Gjm(r) with very high numerical

184

CHAPTER 4 Special functions

accuracy (14–15 significant figures) can be obtained by expanding in series the exponential in Eqn (204). Similar problems are met in the evaluation of Bljm ðrÞ of Eqn (205), and again numerical stability to about 14–15 significant figures is achieved by the series expansion of the exponential. We may point out here that the accurate evaluation of the double integral (206) is the most timeconsuming step in the evaluation of two-centre molecular integrals in quantum chemistry calculations.

4.8 THE DIRAC d-FUNCTION The Dirac d-function is a distribution which may be visualized as an infinitely sharp Gaussian function (spike) selecting a given value of a function f(x), its main property being Z dx dðx
aÞf ðxÞ ¼ f ðaÞ (210) where the integration is over all possible values of definition of the variable x. The operation of multiplying f(x) by d(x
a) and integrating over all values of x is hence simply equivalent to replacing a for x in the argument of the original function. The Dirac d-function is met when expanding any regular function f(x) in the complete set f4k ðxÞg ^ say of the eigenfunctions of a Hermitian operator A, f ðxÞ ¼

X

4k ðxÞCk

(211)

k

the expansion coefficients being given by

Z

dx0 4k ðx0 Þf ðx0 Þ

Ck ¼

(212)

Using Dirac’s notation, Eqn (212) can be written as

and expansion Eqn (211) as

Ck ¼ h4k j f i

(213)

 X  jf ¼ j4k > h4k j f

(214)

k

where

X X 4k ðxÞ4k ðx0 Þ ¼ dðx
x0 Þ j4k >< 4k j ¼ k

(215)

k

if the set is complete (see Section 1.1.7 of Chapter 1). So, Eqn (215) expresses the completeness of the set f4k ðxÞg of regular functions in terms of the Dirac d-function which can hence be said to be the identity operator ^ 1. More precisely, dðx
x0 Þ is recognized as the kernel of the integral operator ^dðxÞ ¼ ^1: Z ^dðxÞf ðxÞ ¼ dx0 dðx
x0 Þf ðx0 Þ ¼ f ðxÞ (216)

4.9 The Fourier transform

185

Roughly speaking, as said before, we say that the Dirac d-function corresponds to the identity operator when working on continuous functions. It can be shown (Sneddon, 1956) that further properties of the Dirac d-function are dð
xÞ ¼ dðxÞ and, for a > 0 dðaxÞ ¼

1 dðxÞ; a


 1 d a2
x2 ¼ ½dðx
aÞ þ dðx þ aÞ 2a

(217)

(218)

It is also often said that the Dirac d-function is the derivative of the Heaviside unit function H(x), which is defined by the equations 1 if x > 0 HðxÞ ¼ (219) 0 if x < 0 a relation that can be precisely stated in terms of Stieltjes integration.4

4.9 THE FOURIER TRANSFORM The Fourier transform (FT) is of great interest in the calculation of multicentre molecular integrals, in the theory of intermolecular potentials and, in its discretized form (Griffiths, 1978), in recent applications to infrared spectroscopy. The FT was first suggested by Prosser and Blanchard (1962) to simplify the computation of molecular integrals over STOs and, more extensively, by Silverstone and co-workers (Todd et al., 1982) and by the Steinborn group (Filter and Steinborn, 1978; Weniger and Steinborn, 1983; Weniger et al., 1986) in the attempt to give compact analytical forms useful in the practical calculation of multicentre molecular integrals. With the help of the FT, some six-dimensional integrals in coordinate space with nonseparable integration variables can be transformed into three-dimensional integrals in momentum space where the integration variables are separated quite easily. The same is true for the application of the FT method to the study of intermolecular potentials, as we shall see more in detail in Section 17.8 of Chapter 17 of this book. The FT of the interparticle distance in the intermolecular potential allows for a generalized expansion converging for all intermolecular separations R and for separation of angle-dependent from R-dependent factors (Koide, 1976; Knowles and Meath, 1987; Magnasco and Figari, 1989). We now proceed to introducing the essential elements of the FT. The FT of a function f(x) of a real variable x is defined as ZN dx f ðxÞexpð
ipxÞ (220) FðpÞ ¼
N

where p is a parameter, whereas 1 f ðxÞ ¼ 2p

ZN dp Fð pÞexpðipxÞ
N

is called the inverse Fourier transform or Anti-Fourier transform. 4

For the definition of a Stieltjes integral, see Sneddon (1956) p. 162.

(221)

186

CHAPTER 4 Special functions

Equation (221) is obtained as the limit as L /N of the complex Fourier expansion of a function f(x) defined in the interval (
L, L): N  pk X ak exp i x (222) f ðxÞ ¼ L k¼
N the coefficients being given by 1 ak ¼ 2L

ZL

 pk x dx f ðxÞexp
i L


L

(223)

In fact, we can rewrite expansion (222) as N  pk ZL  pk 1 X f ðxÞ ¼ x x f ðxÞ exp i dx exp
i 2L k¼
N L L

(224)


L

and change variable to p¼

pk L

(225)

p increases in steps of unity by Dp ¼ pðk þ 1Þ
pðkÞ ¼

kþ1 k p 1 1 p
p¼ 0 ¼ Dp L L L 2L 2p

(226)

Then, using these relations, expansion (224) becomes ZL N 1 X f ðxÞ ¼ Dp expðipxÞ dx expð
ipxÞf ðxÞ 2p k¼
N

(227)


L

Now, taking the limit for L / N, Dp becomes infinitesimal, so that the sum over k becomes an integral: ZN N X lim Dp ¼ dp (228) lim Dp ¼ dp; L/N

L/N

k¼
N


N

and expansion (227) becomes 1 f ðxÞ ¼ 2p

ZN dp FðpÞexpðipxÞ

(229)


N

ZN dx f ðxÞexpð
ipxÞ

FðpÞ ¼
N

(230)

4.9 The Fourier transform

187

A more symmetrical definition of FT is 1 f ðxÞ ¼ pffiffiffiffiffiffi 2p

ZN

1 FðpÞ ¼ pffiffiffiffiffiffi 2p

dp FðpÞexpðipxÞ;
N

ZN dx f ðxÞexpð
ipxÞ

(231)


N

Using Eqn (231), we can write 1 f ðxÞ ¼ 2p

ZN dp expðipxÞ
N

ZN ¼
N

ZN

dx0 f ðx0 Þexpð
ipx0 Þ


N

dx0 f ðx0 Þ

1 2p

ZN

dp exp½iðx
x0 Þp ¼


N

ZN

(232) dx0 dðx
x0 Þf ðx0 Þ


N

where use was made of the integral representation of the Dirac d-function (Section 4.8) 1 dðx
x Þ ¼ 2p 0

ZN

dp exp½iðx
x0 Þp

These formulae can be extended to a three-dimensional space (r,k): Z 1 f ðrÞ ¼
pffiffiffiffiffiffi3 dk expðik $ rÞFðkÞ 2p 1 FðkÞ ¼
pffiffiffiffiffiffi3 2p

(233)


N

(234)

Z dr expð
ik $ rÞf ðrÞ

(235)

where (k $ r) is the scalar product of vectors k and r k $ r ¼ kx x þ ky y þ kz z

(236)

and dk ¼ dkx dky dkz ;

dr ¼ dx dy dz

(237)

From these relations, we have the generalization of the integral representation of the Dirac dfunction to three dimensions: 0

dðx
x Þ ¼

1 ð2pÞ3

ZN

dk expðiðx
x0 Þ $ kÞ

(238)


N

where the three-dimensional d-function has the properties Z Z dðxÞ ¼ 0 x s 0; dx dðxÞ ¼ 1; dx dðx
yÞf ðxÞ ¼ f ðyÞ

(239)

188

CHAPTER 4 Special functions

Of importance in the applications is the fact that the FT transforms convolutions into products and vice versa. The convolution (German / Faltung) of two functions f1(x) and f2(x) is denoted by f1(x) * f2(x) and is given by the integral relation (Rossetti, 1984) ZN dx0 f1 ðx
x0 Þf2 ðx0 Þ (240) f ðxÞ ¼ f1 ðxÞ  f2 ðxÞ ¼
N

The convolutory product has the properties f1 ðxÞ  f2 ðxÞ ¼ f2 ðxÞ  f1 ðxÞ

(241)

ðf1  f2 Þ  f3 ¼ f1  ðf2  f3 Þ

(242)

and Now F½ f1  f2  ¼

pffiffiffiffiffiffi 2p F½ f1  $ F½ f2 

(243)

and 1 F½f1 $ f2  ¼ pffiffiffiffiffiffi F½f1   F½f2  2p If f (x) is a real even function, its FT is real and is given by rffiffiffiffi ZN ZN 1 2 FðpÞ ¼ pffiffiffiffiffiffi dx cosðpxÞf ðxÞ ¼ dx cosðpxÞ f ðxÞ p 2p
N

(244)

(245)

0

If f (x) is a real odd function, its FT is purely imaginary and is given by rffiffiffiffi ZN ZN i 2 FðpÞ ¼
pffiffiffiffiffiffi dx sinðpxÞ f ðxÞ ¼
i dx sinðpxÞ f ðxÞ p 2p
N

(246)

0

A few examples of FTs are given in Problems 4.4–4.5 (Rossetti, 1984).

4.10 THE LAPLACE TRANSFORM The Laplace transform (LT) of a function F(t) of the real variable t is defined as ZN LðsÞ ¼

dt FðtÞexpð
stÞ

(247)

0

where s is a complex variable. The integral (247) converges whenever Re(s) > 0. F(t) is called the original function and L(s) the image function. Equation (247) can be rewritten as LðsÞ ¼ L^s FðsÞ

(248)

4.10 The Laplace transform

where L^s is the integral operator with kernel exp(
st) ZN ZN ^ ^ Ls FðsÞ ¼ dt expð
stÞPst FðsÞ ¼ dt expð
stÞFðtÞ ¼ LðsÞ 0

(249)

0

Fundamental properties of the LT are   d FðtÞ ¼ sLs ½FðtÞ
Fð0Þ Ls dt and, in general

189

(250)



 n
1 X dn F ðkÞ ð0Þsn
1
k Ls n FðtÞ ¼ sn Ls ½FðtÞ
dt k¼0

where the notation for the derivatives means F

ðnÞ



dn FðtÞ ð0Þ ¼ lim t/0 dtn

(251)

 (252)

The LT and its property (251) are of interest in the solution of the linear differential equations with constant coefficients such as those derived in chemical kinetics. More details about that can be found elsewhere (Rossetti, 1984). A few LTs, taken from Abramowitz and Stegun (1965), are given in Table 4.1. They are easily proved by integration as shown in Problem 4.6. Table 4.1 Table of Laplace Transforms L(s)

F(t)

1 s 1 s2 1 ðn ¼ 1; 2; 3; /Þ sn 1 aþs 1 ðn ¼ 1; 2; 3; /Þ ða þ sÞn

1 t t n
1 ðn
1Þ! exp(eat) t n
1 expð
atÞ ðn
1Þ!

1 ðbsaÞ ða þ sÞðb þ sÞ

expð
atÞ
expð
btÞ b
a

s
1=2

1 pffiffiffiffiffi pt


 1 s
nþ 2 1 a2 þ s2

1

2n t n
2 pffiffiffiffi 1$3$5/ð2n
1Þ p 1 sinðatÞ a

190

CHAPTER 4 Special functions

4.11 SPHERICAL TENSORS A n-rank tensor is a quantity having 3n components and can be regarded as a vector in a 3n-dimensional space whose components carry a representation of the rotation group (Section 8.15 of Chapter 8). Thus, scalars can be regarded as tensors of rank zero (n¼ 0), while a vector in three-space can be regarded as a first-rank tensor (n ¼ 1) having three components. An irreducible tensor operator T of rank k is an operator with (2 kþ 1) components Tkq transforming in a well-defined way under rotation of axes (Brink and Satchler, 1993). Spherical harmonics are examples of spherical tensors. The spherical tensors Rlm(r), also called regular solid harmonics (Brink and Satchler, 1993; Stone, 1996), can be given in complex or real form. Complex spherical tensors are eigenfunctions of the square of the angular momentum operator L^2 and of the z-component of the angular momentum operator L^z, while real spherical tensors are eigenfunctions only of the square of the angular momentum operator L^2. In atomic theory use is often made of a theorem connected to spherical tensors, the Wigner–Eckart theorem (Wigner, 1959; Eckart, 1930a), which states that    0 0 (253) lmTkq l m ¼ hlkTk kl0 ihl0 m0 kqjlmi where Tkq is the q-component of a spherical tensor T of rank k, hlkTk kl0i is a reduced matrix element independent of m and q, and hl0 m0 kqjlmi is the Clebsch–Gordan coefficient (Chapter 10) describing the coupling of the angular momentum eigenvector jl0 m0 i to jkqi to give a state with resultant angular momentum jlmi. Thus, all directional properties are contained in the Clebsch– Gordan coefficients, while the dynamical properties of the system appear in the scalar factor hlkTk kl0 i.

4.11.1 Spherical tensors in complex form The spherical tensors Rlm(r) in complex form are defined in terms of the (normalized) modified spherical harmonics (Brink and Satchler, 1993), eigenfunctions of the operators L^2 and L^z : 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > ðl
mÞ! m > P ðxÞ½ð
1Þm expðim4Þ > > > ðl þ mÞ! l rffiffiffiffiffiffiffiffiffiffiffiffi > < 4p Ylm ðq; 4Þ ¼ r l Pl ðxÞ Rlm ðrÞ ¼ rl > 2l þ 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > ðl
jmjÞ! jmj > > > P ðxÞexpð
ijmj4Þ > : ðl þ jmjÞ! l

m>0 m¼0

(254)

m<0

where use is made of the so-called Condon–Shortley phase convention, and Pm l is the associated Legendre polynomial ðm ¼ jmjÞ (Abramowitz and Stegun, 1965) of Section 4.2.1


Pm l ðxÞ

m=2 lþm l
2 1
x2 d ¼ x
1 ð2lÞ!! dxlþm

with x ¼ cos q and the double factorial function is defined by Eqn (138).

(255)

4.11 Spherical tensors

191

4.11.2 Spherical tensors in real form The spherical tensors Rlm(r) in real form are defined in terms of the (normalized) modified real spherical harmonics, also called tesseral harmonics (MacRobert, 1947): 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > ðl
mÞ! m > > 2 P ðxÞcos m4 > > ðl þ mÞ! l > > rffiffiffiffiffiffiffiffiffiffiffiffi > < 4p c;s Pl ðxÞ Ylm ðq; 4Þ ¼ r l Rlm ðrÞ ¼ rl > 2l þ 1 > s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > ðl þ mÞ! jmj > > > : 2 ðl
mÞ! Pl ðxÞsinjmj4

m>0 m¼0

(256)

m<0

where m ¼
jmj

(257)

and Pm l is the associated Legendre polynomial. Spherical tensors in real form are eigenfunctions of the operator L^2 alone, but still form a complete orthogonal set with respect to integration over the angular variables. The explicit form of the first few of them (l ¼ 0, 1, 2, 3, 4) is best written in terms of Cartesian coordinates as (

R00 ¼ 1 R10 ¼ z;

R11 ¼ x;

R11 ¼ y

(258)

8  1
> R20 ¼ 3z2
r 2 > > > 2 > > > < pffiffiffi pffiffiffi R21 ¼ 3z x; R21 ¼ 3yz > > > > pffiffiffi > > pffiffiffi > : R ¼ 3
x2
y2 ; R ¼ 3xy 22 22 2

(259)

8 5z2
3r 2 > > > R30 ¼ z > > 2 > > > > rffiffiffi rffiffiffi > > >   3
2 3
2 > 2 > > R31 ¼ 5z
r x; R31 ¼ 5z
r 2 y > < 8 8 pffiffiffiffiffi > > pffiffiffiffiffi  > 15
2 2 > > R x ¼
y ¼ 15xyz z; R 32 > 32 > 2 > > > > > rffiffiffi rffiffiffi > >   > 5
2 5
2 > 2 : R33 ¼ x
3y x; R33 ¼ 3x
y2 y 8 8

(260)

192

CHAPTER 4 Special functions

8 > > > > R40 > > > > > > > > > > R41 > > > > > > > < R42 > > > > > > > > > > R43 > > > > > > > > > > > : R44

8z4 þ 3x4 þ 3y4
24z2 x2
24y2 z2 þ 6x2 y2 8 pffiffiffi 2 pffiffiffi 2 5 4z
3x2
3y2 5 4z
3x2
3y2 zx; R41 ¼ pffiffiffi yz ¼ pffiffiffi 2 2 2 2 ¼

pffiffiffi 6z2
x2
y2 pffiffiffi 6z2
x2
y2
2  x
y2 ; R42 ¼ 5 xy 5 4 4 pffiffiffiffiffi 2 pffiffiffiffiffi 35 x
3y2 35 3x2
y2 z x; R43 ¼ pffiffiffi yz ¼ pffiffiffi 2 2 2 2 ¼

¼

pffiffiffiffiffi x4 þ y4
6x2 y2 ; 35 8

R44 ¼

(261)

pffiffiffiffiffi x2
y2 xy 35 2

The STOs of valence theory are real spherical tensors multiplied by a radial decay factor exp(
cr) with c a real non-zero positive number. Figures 4.1–4.5 give the angular form in space for the AOs with l ¼ 0, 1, 2, 3, 4 and m ¼ 0.5 The plots were obtained from the function ParametricPlot3D of Mathematica 6 (Wolfram Research, 2007) using a small software due to Ottonelli.

FIGURE 4.1 Three-dimensional representation in space of the s AO with l ¼ m ¼ 0 (For colour version of this figure, the reader is referred to the online version of this book.)

The form of complex and real spherical tensors is identical for m ¼ 0.

5

4.11 Spherical tensors

193

FIGURE 4.2 Three-dimensional representation in space of the p AO with l ¼ 1 m ¼ 0 (For colour version of this figure, the reader is referred to the online version of this book.)

FIGURE 4.3 Three-dimensional representation in space of the d AO with l ¼ 2 m ¼ 0 (For colour version of this figure, the reader is referred to the online version of this book.)

194

CHAPTER 4 Special functions

FIGURE 4.4 Three-dimensional representation in space of the f AO with l ¼ 3 m ¼ 0 (For colour version of this figure, the reader is referred to the online version of this book.)

FIGURE 4.5 Three-dimensional representation in space of the g AO with l ¼ 4 m ¼ 0 (For colour version of this figure, the reader is referred to the online version of this book.)

4.12 Orthogonal polynomials

195

4.11.3 Generalized spherical tensors The generalized spherical tensors in k-space (Magnasco and Figari, 1989) are spherical tensors whose rl radial part is replaced by (2lþ 1)!!jl(rk): rffiffiffiffiffiffiffiffiffiffiffiffi 4p Ylm ðq; 4Þ (262) Rlm ðr; kÞ ¼ ð2l þ 1Þ!!jl ðrkÞ 2l þ 1 where jl(rk) is a spherical Bessel function (Abramowitz and Stegun, 1965). They are related to the ordinary spherical tensors Rlm(r) by the relation   Rlm ðr; kÞ 1 vl ¼ R ðr; kÞ (263) Rlm ðrÞ ¼ lim lm k/0 kl l! vkl k¼0 The spherical Bessel functions were discussed in Section 4.6.3 and (2 l þ 1)!! is the double factorial function defined by Eqn (138). We then have rffiffiffiffiffiffiffiffiffiffiffiffi  r lþ2 klþ2 r lþ4 klþ4 4p l l þ
/ Ylm ðq; 4Þ (264) Rlm ðr; kÞ ¼ r k
2ð2l þ 3Þ 8ð2l þ 3Þð2l þ 5Þ 2l þ 1 an equation which is obtained from the representation (139) of jl(rk) as the infinite series jl ðrkÞ ¼

N X

ð
1Þp ðrkÞlþ2p ð2pÞ!!ð2p þ 2l þ 1Þ!! p¼0

(265)

4.12 ORTHOGONAL POLYNOMIALS We have seen in Section 4.2.3 that any regular function f(x), defined in the interval jxj < 1, can be expanded in series of Legendre polynomials. They form a complete set of orthogonal functions so that the expansion is unique. However, they are not the only possible set of orthogonal polynomials. Given a set of powers of the variable, say {xn}, it is always possible by Schmidt orthogonalization (Chapter 1) to obtain a complete set of polynomials {Pn(x)} of degree n, orthogonal and normalizable in the finite interval (a,b) and with respect to any weight function p(x), such that Zb ðPn ; Pm Þ ¼

dx Pn ðxÞPm ðxÞpðxÞ ¼ dnm

(266)

a

Any power xn can then be expressed in terms of the first (n þ 1) orthogonal polynomials, each polynomial being orthogonal to any polynomial of lower degree. It can be shown that the expansion is unique and the following important theorem exists (Rossetti, 1984): the zeros of a polynomial Pn(x) belonging to a system of polynomials {Pn(x)} orthogonal in L2 are all simple and lie inside the interval (a,b). The importance of orthogonal polynomials lies in the fact that the n-th reduced sum in the expansion of a continuous function f(x) allows for the interpolation of the function in at least (n þ 1) points within the interval (a,b).

196

CHAPTER 4 Special functions

In other words, if we have the expansion f ðxÞ ¼

N X

ck Pk ðxÞ

(267)

k¼0

where the polynomials are orthogonal according to Eqn (266), then the n-th reduced sum Sn ðxÞ ¼

n X

ck Pk ðxÞ

(268)

k¼0

coincides with the function f(x) in at least (n þ 1) points within the interval (a,b). This property is of great importance in practical applications, such as those involving numerical quadratures, e.g. the Gauss–Legendre quadrature. If X(x) is at most a quadratic polynomial with real roots, then it can be shown (Rossetti, 1984) that the Rodriguez formula holds: Pn ðxÞ ¼

1 1 dn ½ pðxÞX n ðxÞ Kn pðxÞ d xn

(269)

where p(x) is the weight function, which must be real and integrable in (a,b), and Kn is a constant factor. Possible forms of p(x) are pðxÞ ¼ ðb
xÞa ðx
aÞb

(270)

pðxÞ ¼ expð
xÞðx
aÞa

(271)


 pðxÞ ¼ exp
x2

(272)

with a and b real and >
1; with a >
1;

In Eqn (269), the constant factor Kn, called the standardization factor, defines in a unique way each different class of polynomials. The majority of orthogonal polynomials entering practical applications are given by the Rodriguez formula (269). They are called classical orthogonal polynomials, and correspond to three classes according to the form chosen from Eqn (270–272) for the weight function p(x). Most common is to choose the interval (a,b) as (
1,1),6 in which case XðxÞ ¼ 1
x2

(273)

and the weight function p(x) is pðxÞ ¼ ð1
xÞa ð1 þ xÞb

a; b >
1

(274)

Such polynomials are called Jacobi polynomials. In the interval (0,N), X(x) is given by XðxÞ ¼ x 6

It is always possible to reduce any interval (a,b) to (
1,1) by an appropriate linear transformation.

(275)

4.13 Pade´ approximants

197

with the weight factor pðxÞ ¼ expð
xÞxa

a >
1

(276)

Such polynomials are called Laguerre polynomials. They were treated in detail in Section 4.3. In the interval (
N,N), X(x) is given by with the weight factor

XðxÞ ¼ 1

(277)


 pðxÞ ¼ exp
x2

(278)

Such polynomials are called Hermite polynomials. They were treated in detail in Section 4.4. Specific values given to a and b originate particular polynomials. For instance, the choice a¼b¼l


1 2

l>


1 2

(279)

gives the Gegenbauer (or ultraspherical) polynomials, denoted by Cnl ðxÞ. Among these polynomials, particular importance have the Tchebichef polynomials (of first and second kind) which are obtained by choosing l ¼ 0, 1 and, perhaps the most important of all, the Legendre polynomials, obtained by choosing l ¼ 1/2. These polynomials were treated in detail in Section 4.2. So, we see that all polynomials found in the previous sections of this chapter, obtained as regular solutions of the appropriate differential equations of Chapter 3, can be reassigned to particular classes of the more general definition of orthogonal polynomials. Much more than this can be found in Chapter IV of Rossetti (1984).

4.13 PADE´ APPROXIMANTS Even if strictly pertinent to numerical analysis, a few elements of the Pade´ approximants technique (Baker and Gammel, 1975) may be of interest here. They are of importance, for instance, in assessing the form of the exchange-correlation functional (Vosko et al., 1980) in density functional theory, in Rayleigh–Schroedinger (RS) (Wilson et al., 1977) and Brillouin–Wigner (BW) (Bendazzoli et al., 1970) perturbation theories, and in the power expansion of time-dependent correlation functions (Paul, 1980). The Pade´ approximants technique deals with the possibility of reproducing in an efficient way a given function f(x) in terms of polynomials and it is an improvement over the Taylor approximants. We first recall a theorem due to Weierstrass which states that if a function f(x) is continuous at a point x0, interior to the domain of definition of the function, for any arbitrary positive ε it is possible to find a polynomial: n X ak ðx
x0 Þk (280) Pn ðx
x0 Þ ¼ k¼0

such that for any point x j f ðxÞ
Pn ðx
x0 Þj < ε

(281)

The degree of the polynomial and the values of its coefficients are flexible parameters that must be properly fixed so as to satisfy the required accuracy ε.

198

CHAPTER 4 Special functions

We examine first the Taylor approximant. The Taylor expansion of a function f(x) is the power series f ðxÞ ¼

N X

ak ðx
x0 Þk

(282)

k¼0

with ðkÞ

f ak ¼ 0 ; k!

ðkÞ f0

 ¼

dk f d xk

 (283) x¼x0

The degree of a Taylor polynomial can be unlimited as far as the function admits unlimited derivatives. Taylor’s expansion requires rather regular functions, having in x0 continuous derivatives up to the order corresponding to its polynomial approximant. Stopping the expansion at a finite degree n, we have a truncation error in the Taylor approximation, namely, we have a remainder Rn(x) which we can estimate, for instance, in the Lagrange form: Rn ðxÞ ¼ f ðxÞ


ðkÞ n X f0 f ðnþ1Þ ðxÞ ðx
x0 Þk ¼ ðx
x0 Þnþ1 k! ðn þ 1Þ! k¼0

(284)

for x ˛ ðx0 ; xÞ

(285)

The difference at the point x0 between the function and its polynomial approximant is therefore proportional to the value assumed, at a point x lying in-between x0 and x, by the derivative successive to the last derivative included in the polynomial. It is clear that if f (nþ1)(x) is limited in absolute value at the interior of the interval (x0,x), the error has an upper bound which can be evaluated exactly. We can see that f (nþ1)(x) is the factor to be included in the remainder if we write f ðxÞ


ðkÞ n X f0 ðx
x0 Þnþ1 ðx
x0 Þk
D ¼0 k! ðn þ 1Þ! k¼0

(286)

Then, using repeatedly Rolle’s theorem, it can be shown that f ðnþ1Þ ðxnþ1 Þ
D ¼ 0

(287)

(nþ1)

(xnþ1) where xnþ1 is a value (denoted by x in Eqn (284) for the so that D coincides with f remainder) lying in the interval (x0,x). The remainder is an indicator of the convergence properties of the series, which will be reproduced exactly if lim Rn ðxÞ ¼ 0

n/N

(288)

Let us now turn to the Pade´ approximants. In this case, the values of the function f(x) are estimated as the ratio [m,n] between two polynomials: f ðxÞ ¼

Qm ðx
x0 Þ ¼ ½m; n Tn ðx
x0 Þ

(289)

4.14 Green’s functions

where7 Qm ðx
x0 Þ ¼

m X

bk ðx
x0 Þk

199

(290)

k¼0

Tn ðx
x0 Þ ¼

n X

ck ðx
x0 Þk

ðc0 ¼ 1Þ

(291)

k¼0

The coefficients of the Pade´ expansion are obtained by the comparison with the corresponding Taylor expansion of the function ðkÞ N X f Qm ðx
x0 Þ ; ak ¼ 0 (292) ak ðx
x0 Þk ¼ k! Tn ðx
x0 Þ k¼0 N X

ak ðx
x0 Þk $

k¼0

n X

cl ðx
x0 Þl ¼

m X

bm ðx
x0 Þm

(293)

m¼0

l¼0

Since coefficients of the same power of (x
x0) must be identical, this gives m X bm ðm ¼ 0; 1; 2; /; mÞ am
l cl ¼ 0 ðm ¼ m þ 1; m þ 2; /; m þ nÞ

(294)

l¼0

cl ¼ 0

for l > n

(295)

In this way, we obtain an algebraic system where the m þ n þ 1 unknown parameters b0, b1,/, bm, c1, c2,/, cn are put in relation to the a0,a1,/,amþn parameters of the corresponding Taylor series. The connection between Pade´ and Taylor expansions is such that their difference depends on powers higher than m þ n: Xm XN Xn m N b ðx
x Þ X a c ðx
x0 Þk m 0 m¼0 k k¼mþnþ1 l¼0 k
l l Xn ak ðx
x0 Þ
Xn ¼ (296) c ðx
x0 Þl c ðx
x0 Þl k¼0 l¼0 l l¼0 l It is not always possible to associate a Pade´ approximant with given values of m and n to the Taylor expansion of a function, because sometimes system (294) does not admit a solution. However, if the Pade´ approximant exists, then it is a more flexible instrument than the Taylor series, which must be regarded as a particular case of Pade´ approximant having n ¼ 0 in the denominator of Eqn (289). Unfortunately, there does not exist any rule giving the best balance of the powers m and n of the two polynomials so as to give the most efficient approximation, but it is generally true that m ¼ n and m ¼ n  1 are appropriate choices.

4.14 GREEN’S FUNCTIONS Green’s functions originate in the study of differential equations and even if they occur in many problems of theoretical and mathematical physics (Morse and Fesbach, 1953; Courant and Hilbert, Choosing c0 ¼ 1 is not a constraint, since it corresponds to dividing each term of the fraction by c0 leaving the fraction unchanged. 7

200

CHAPTER 4 Special functions

1989) they are sensibly less popular in quantum chemistry, except perhaps for their use in propagator theory (Thouless, 1961; McWeeny, 1992). For a conceptually simple introduction to Green’s functions theory we follow here the presentation given by Mathews and Walker (1970). The theory is strictly connected with the solution of the inhomogeneous differential equations of mathematical physics arising when an eigenvalue problem is modified by some point field disturbance and rests on the fundamental statement that the series expansion of a regular function in the complete set of eigenfunctions of a Hermitian operator L^ converges absolutely and uniformly in the domain D of definition of the function. We assume that we know the whole set of orthogonal eigenfunctions {uk(x)} and eigenvalues lk of the operator L^ obtained as solutions of the eigenvalue equation ^ k ðxÞ ¼ lk uk ðxÞ; huk juk0 i ¼ dkk0 (297) Lu We want to solve for the unknown function u(x) the inhomogeneous differential equation: ^ LuðxÞ
luðxÞ ¼ f ðxÞ

(298)

where L^ is a linear Hermitian operator, l is a parameter, and u(x) and f(x) are regular functions defined over a domain D and subjected to the usual boundary conditions. We expand u(x) and the inhomogeneity f(x) in the eigenfunction (297) of the operator L^ as Z X (299) ak uk ðxÞ; ak ¼ d x0 uk ðx0 Þuðx0 Þ ¼ huk jui uðxÞ ¼ k

f ðxÞ ¼

X

Z bk uk ðxÞ;

bk ¼

d x0 uk ðx0 Þf ðx0 Þ ¼ huk jf i

(300)

k

so that Eqn (298) becomes

X

ak ðlk
lÞuk ðxÞ ¼

X

k

X

namely,

bk uk ðxÞ

(301)

k

ak ½ðlk
lÞ
bk uk ðxÞ ¼ 0

(302)

k

Since the set {uk(x)} is linearly independent, we obtain ak ½ðlk
lÞ
bk  ¼ 0 and, provided (lk
l) s 0 ak ¼

bk ¼ lk
l

Z

d x0

uk ðx0 Þf ðx0 Þ ¼ ðlk
lÞ
1 huk j f i lk
l

Therefore X

X

Z

uk ðx0 Þf ðx0 Þ lk
l k k Z Z X uk ðxÞ d x0 uk ðx0 Þf ðx0 Þ ¼ d x0 Gðx; x0 Þf ðx0 Þ ¼
l l k k

uðxÞ ¼

ak uk ðxÞ ¼

uk ðxÞ

(303)

(304)

d x0

D

(305)

4.14 Green’s functions

201

where Gðx; x0 Þ ¼

X uk ðxÞu ðx0 Þ k

k

lk
l

¼

X juk ðxÞ >< uk ðx0 Þj k

lk
l

(306)

is called the Green’s function of the operator L^. To emphasize the dependence of G on the value of the parameter l, the Green’s function is sometimes written as Gðx; x0 ; lÞ. It can be seen that the Green’s ^ function (306) is the kernel of the integral operator GðxÞ whose action on a function f(x) is Z ^ GðxÞf ðxÞ ¼ dx0 Gðx; x0 Þf ðx0 Þ ¼ uðxÞ (307) If f(x) is the Dirac’s d-function f ðxÞ ¼ dðx
x0 Þ then the solution of Eqn (298) can be written as Z uðxÞ ¼ dx0 Gðx; x0 Þdðx0
x0 Þ ¼ Gðx; x0 Þ

(308)

(309)

D

so that

Gðx; x0 Þ

is the solution of the differential equation ^ LGðx; x0 Þ
lGðx; x0 Þ ¼ dðx
x0 Þ

(310)

In other words, the physical meaning of the Green’s function is that of being the solution of problem (298) when the disturbance is the unit point source f ðxÞ ¼ dðx
x0 Þ. The problems arising when l equals one of the eigenvalues of L^ are avoided by imposing the orthogonality condition Z (311) dx uk ðxÞf ðxÞ ¼ 0 These considerations are readily extended to three-dimensional space simply by replacing x by r and dx by dr. In the RS perturbation theory of Chapter 1, the first-order equation ðH^0
E0 Þj1 þ ðV
E1 Þj0 ¼ 0

(312)

which must be solved under the orthogonality constraint hj0 jj1 i ¼ 0

(313)

j1 ¼
ðH^0
E0 Þ
1 ðV
E1 Þj0

(314)

has the formal solution Expanding j1 in the eigenstates fjk g of H^0 : Z X jk ðrÞ Z 0  0 0 j1 ¼
dr jk ðr ÞðV
E1 Þj0 ðr Þ ¼
dr0 Gðr; r0 ÞðV
E1 Þj0 ðr0 Þ
E E k 0 kðs 0Þ

(315)

202

CHAPTER 4 Special functions

the Green’s function Gðr; r0 Þ ¼

X jk ðrÞj ðr0 Þ k
E0 E k kðs 0Þ

(316)

is called the Rayleigh–Schroedinger resolvent of Eqn (312) (Courant and Hilbert, 1989; Hubac et al., 2000). Unfortunately, Eqns (314) and (316) are purely formal since, as already said in Chapter 1, the correct application of Eqn (315) requires consideration of the whole discrete spectrum belonging to the bounded eigenstates of H^0 as well as of its continuous spectrum belonging to the ionized state. The Green’s function in spherical coordinates for the ground state of the hydrogen atom was obtained by Hameka (1967) following earlier work by Meixner (1933), and resulted in extremely complicated sectorialized formulae involving products of confluent hypergeometric functions that cannot be reported here. Extension of the method to the ground state of helium (Hameka, 1968a) and lithium (Hameka, 1968b) atoms resulted in purely formal results not practical for numerical estimates. Molecular applications to the second-order induction energy of a hydrogen atom perturbed by a proton a distance R apart were done by Pan and Hameka (1968) and by Singh et al. (1970), who used a modified form of the Green’s functions obtained as solutions of differential equations rather than by direct summation of their spectral expansion (316). Cha1asinski and Jeziorski (1974) were the first who succeeded in calculating exactly the second-order induction interaction between the atoms in the hydrogen molecule using the Green’s functions of the hydrogen atom as explained in detail in Section 17.3.3 of Chapter 17 of this book. In the opinion of the author, the method of linear pseudostates presented in Section 1.3.2.4 of Chapter 1 of this book seems more valuable for obtaining practical approximations, even very accurate, to second-order quantities, as shown for the second-order energy in the hydrogen-like perturbation theory of the ground state of the helium atom (Byron and Joachain, 1967; Magnasco et al., 1992a), for the damping coefficients in the non-expanded second-order induction energy for Hþ 2 (Magnasco and Figari, 1987a), and from the early calculations of C6 dispersion coefficients for the long-range H–H interaction (Magnasco and Figari, 1987b) up to the recently very accurate results for the dispersion coefficients of simple atomic and molecular systems obtained by use of reduced pseudospectra (Magnasco and Figari, 2009).

4.15 PROBLEMS 4 4.1. Find the quadratic equation in q occurring in the Legendre’s equation expressed in the form of descending powers of the variable. Answer: qðq þ 1Þ
nðn þ 1Þ ¼ ðq
nÞðq þ n þ 1Þ ¼ 0 4.2. Solve the Bessel’s equation of integral order. Answer:   xn 1 2 Jn ðxÞ ¼ n 0 F1 n þ 1;
x 2 n! 4 Hint: Solve the differential Eqn (117) by the usual series expansion in the variable x.

4.16 Solved problems

203

4.3. Prove properties (1)–(4) of the gamma function. Hint: Use the definition of gamma function and the rule of integration by parts. 4.4. Find the FT of the exponential and the Gaussian function. Answer: 1. The exponential function  1=2 2 c FðpÞ ¼ p c 2 þ p2 2. The Gaussian function  2 2 c c p FðpÞ ¼ pffiffiffi exp
4 2 Hint: Evaluate the integrals defining the FT of these functions. 4.5. Find the Fourier and the anti-FT of the Coulombic potential. Answer: Z 1 expðik $ rÞ 1 dk ¼ FðrÞ ¼ 2 2p k2 r FðkÞ ¼

Z

1 ð2pÞ3=2

dr expð
ik $ rÞ

1 ¼ r

 1=2 2 1 p k2

Hint: Evaluate the integrals defining the FT and the anti-FT of 1/r. 4.6. Prove the LTs of Table 4.1. Hint: Use the definition of LT and ordinary integration rules.

4.16 SOLVED PROBLEMS 4.1. The quadratic equation occurring in Legendre’s equation. The recurrence relation gives for 2k ¼
2 b0 ¼

ðq þ 2Þðq þ 1Þ qðq þ 1Þ
nðn þ 1Þ b
2 0 b
2 ¼ b0 qðq þ 1Þ
nðn þ 1Þ ðq þ 2Þðq þ 1Þ

(317)

Since b0 s 0, the constraint b
2 ¼ 0 necessarily implies qðq þ 1Þ
nðn þ 1Þ ¼ 0

(318)

204

CHAPTER 4 Special functions

a quadratic equation in q whose solutions are q ¼ n and q ¼
n
1, and which can be written in terms of these solutions in the form: ðq
nÞðq þ n þ 1Þ ¼ 0

(319)

4.2. The Bessel’s functions of integral order. For n positive integer, the Bessel’s differential equation of the second order is
 x2 y00 ðxÞ þ xy0 ðxÞ þ x2
n2 yðxÞ ¼ 0

(320)

The solution near the singular point x ¼ 0 suggests the power expansion: N X yðxÞ ¼ ak xkþa

(321)

k¼0

where a is a constant to be determined by solution of the indicial equation. We have for the derivatives N N X X ðk þ aÞak xkþa
1 ; y00 ðxÞ ¼ ðk þ aÞðk þ a
1Þak xkþa
2 (322) y0 ðxÞ ¼ k¼0

k¼0

so that substituting in the differential equation N N N X X X ðk þ aÞðk þ a
1Þak xkþa þ ðk þ aÞak xkþa þ ak xkþaþ2
n2 ak xkþa ¼ 0

N X k¼0

k¼0

k¼0

(323)

k¼0

we obtain for the coefficient of xkþa the equation

h i ðk þ aÞðk þ a
1Þak þ ðk þ aÞak
n2 ak ¼ 0 0 ðk þ aÞ2
n2 ak ¼ 0

giving as solution of the indicial equation the two roots 
k ¼ 0 0 a2
n2 a0 ¼ 0 0 a ¼ n

(324)

(325)

since a0 s 0. The same result could have been obtained using Eqn (104) of Section 3.5 of the previous chapter by noting that p1 ðxÞ ¼ q1 ðxÞ ¼ 1;

r1 ðxÞ ¼
n2

for x0 ¼ 0

giving the same quadratic equation in a as before. Hence for a ¼ n the first solution of the Bessel’s equation can be written as 8 y ðxÞ ¼ xn FðxÞ > > < 1 
y01 ðxÞ ¼ xn F 0 þ nx
1 F > > : y00 ðxÞ ¼ xn F 00 þ 2nx
1 F 0 þ
n2
nF 

(326)

(327)

1

By substituting Eqn (327) into the Bessel’s Eqn (320) we obtain the differential equation determining the unknown function F(x): x2 F 00 þ ð2n þ 1ÞxF 0 þ x2 F ¼ 0

(328)

4.16 Solved problems

an equation which we solve by the usual power expansion as N N N X X X ak xk ; F 0 ðxÞ ¼ kak xk
1 ; F 00 ðxÞ ¼ kðk
1Þak xk
2 FðxÞ ¼ k¼0

k¼1

205

(329)

k¼2

therefore obtaining N X

kðk
1Þak xk þ ð2n þ 1Þ

k¼2

N X

kak xk þ

k¼1

N X

ak xkþ2 ¼ 0

(330)

k¼0

The coefficient of xk in expansion (330) is kðk
1Þak þ ð2n þ 1Þkak þ ak
2 ¼ 0

(331)

giving the recurrence relation ak ¼


ak
2 kð2n þ kÞ

(332)

Since series (329) starts with the term a0 s 0, we must take a1 ¼ 0 and, in order the equation above may be satisfied for all k  2, we must take 8 > < a2kþ1 ¼ 0 (333) k ¼ 1; 2; 3; / a2k
2 > : a2k ¼
2kð2n þ 2kÞ so that only even coefficients survive in the expansion. To relate the general coefficient a2k to the non-zero coefficient a0, we write down the first few coefficients which are given explicitly by a0 (334) k ¼ 1 a2 ¼
2 $ 1ð2n þ 2Þ k¼2 k¼3

a4 ¼


a2 ð
1Þ2 a0 ¼ 2 $ 2ð2n þ 4Þ 2 $ 2ð2n þ 4Þ $ 2 $ 1ð2n þ 2Þ

a4 ð
1Þ3 a0 ¼ a6 ¼
2 $ 3ð2n þ 6Þ 2 $ 3ð2n þ 6Þ $ 2 $ 2ð2n þ 4Þ $ 2 $ 1ð2n þ 2Þ

(335) (336)

Therefore the (2k)-th coefficient is given in terms of a0 as a2k ¼ ¼

ð
1Þk a0 2kð2n þ 2kÞ $ 2ðk
1Þð2n þ 2k
2Þ $ 2ðk
2Þð2n þ 2k
4Þ/2 $ 1ð2n þ 2Þ ð
1Þk a0 ½2kð2k
2Þð2k
4Þ/2 $ 1½ð2n þ 2kÞð2n þ 2k
2Þð2n þ 2k
4Þ/ð2n þ 2Þ

ð
1Þk a0 ¼ k ½2 kðk
1Þðk
2Þ/1½2k ðn þ kÞðn þ k
1Þðn þ k
2Þ/ðn þ 2Þðn þ 1Þ  k ð
1Þk a0 a0 1 ¼ ¼ 2k
4 2 k!½ðn þ 1Þðn þ 2Þ/ðn þ k
1Þðn þ kÞ k!ðn þ 1Þk where in the last expression we have introduced the Pochhammer’ symbol (96).

(337)

206

CHAPTER 4 Special functions

Taking a0 ¼¼ 1/(2nn!), we obtain for the general coefficient   k  1 1 1
k ¼ 0; 1; 2; / a2k ¼ n 2 n! k!ðn þ 1Þk 4

(338)

finally obtaining the solution for a ¼ n in the form of the infinite series in even powers of x  2 k N N X xn X 1 x y1 ðxÞ ¼ xn
a2k x2k ¼ n (339) 2 n! 4 k!ðn þ 1Þ k k¼0 k¼0 Writing explicitly the first few terms of the series N X a2k x2k ¼ a0 þ a2 x2 þ a4 x4 þ / k¼0

"

1 ð
1Þ2 x2 þ x4
/ ¼ a0 1
2ð2n þ 2Þ 2 $ 2ð2n þ 4Þ $ 2 $ 1ð2n þ 2Þ   2  2 2  1 x 1 x þ ¼ a0 1 þ


/ 4 4 nþ1 1 $ 2ðn þ 1Þðn þ 2Þ

# (340)

we see that the series in Eqn (340) can be expressed in terms of the hypergeometric function 0 F1 ðc; zÞ containing just the single parameter c: 1 1 (341) zþ z2 þ / 0 F1 ðc; zÞ ¼ 1 þ 1$c 1 $ 2cðc þ 1Þ which for c ¼ n þ 1 and z ¼
x2/4 becomes    2  2 2 x2 1 x 1 x n þ 1;
¼ 1 þ þ F þ/

1 0 4 4 4 1 $ ðn þ 1Þ 1 $ 2ðn þ 1Þðn þ 2Þ so that y1 ðxÞ ¼

  xn x2 ¼ Jn ðxÞ n þ 1;
F 1 2n n! 0 4

(342)

(343)

which is Eqn (118) of the main text. Noting that ðn þ kÞ! ¼ ðn þ kÞðn þ k
1Þðn þ k
2Þ/½n þ k
ðk
2Þ½n þ k
ðk
1Þðn þ k
kÞ! ¼ ðn þ kÞðn þ k
1Þðn þ k
2Þ/ðn þ 2Þðn þ 1Þn! ¼ ðn þ 1Þk n!

(344)

we can write for Jn(x) the equivalent expressions Jn ðxÞ ¼ ¼

 2 k  X  2 k N xn X 1 x x n N 1 x

¼ 2n n! k¼0 k!ðn þ 1Þk 4 4 2 k¼0 k!ðn þ kÞ! N X

ð
1Þk xnþ2k nþ2k k!ðn þ kÞ! 2 k¼0

(345)

4.16 Solved problems

207

Because of relation (125), for n integer, the two solutions Jn(x) and J
n(x) are not linearly independent. Relation (125) of the main text is most easily derived in terms of the coefficients (Bessel’s coefficients of order n) of tn and t
n in the symmetrical series expansion of the exponential, which gives (Sneddon, 1956)    N x  x X x 1 t
¼ exp t exp
t
1 ¼ Jn ðxÞtn (346) exp 2 t 2 2 n¼
N For n ¼ n not an integer, we simply replace factorials by gamma functions obtaining   xn x2 n þ 1;
Jn ðxÞ ¼ n F 1 2 Gðn þ 1Þ 0 4

(347)

The two solutions Jn(x) and J
n(x) are now linearly independent and we can write the general solution of Eqn (320) in the form yðxÞ ¼ AJn ðxÞ þ BJ
n ðxÞ

(348)

with A, B arbitrary constants. It can be shown that the series obtained in this way are convergent and differentiable for any value of x, so that our formal solutions are the solution of Bessel’s differential Eqn (117) of the main text. Generalizing what we have seen so far, we give below some current definitions of the particular solutions of the Bessel’s Eqn (320) with n ¼ n (Sneddon, 1956): Jn ðxÞ Bessel’s functions ðBFsÞ of the first kind of order v

(349)

Yn ðxÞ BFs of the second kind of order v ðor Weber’s BFsÞ

(350)

where Yn(x) is a rather complicated function containing a logarithmic part, with g the Euler’s constant of Section 4.7.3 8      n
1 > 2 1 1 X ðn
k
1Þ! 2 n
2k > > g þ ln x Jn ðxÞ
Yn ðxÞ ¼ > > < p 2 p k¼0 k! x (351) nþ2k > N k k  X X > 1 ð
1Þ 1 1 > > > x ½4ðn þ kÞ þ 4ðkÞ 0 4ðkÞ ¼ :
p k!ðn þ kÞ! 2 s s¼1 k¼0 8 < Hnð1Þ ðxÞ ¼ Jn ðxÞ þ iYn ðxÞ (352) BFs of the third kind of order v ðor Hankel’s BFsÞ : H ð2Þ ðxÞ ¼ J ðxÞ
iY ðxÞ n

n

n

ð1Þ

ð2Þ

where i is the imaginary unit. Jn(x) and Yn(x), as well as Hn ðxÞ and Hn ðxÞ, are functions linearly independent for any value of n. Expressions for the functions J0(x), Y0(x) of order zero are given in Sneddon (1956). Y0(x) is also called Neumann’s Bessel function of the second kind and zero order. Much the same can be said for the SBFs which are solutions of the differential equation

 (353) x2 j00n ðxÞ þ 2xj0n ðxÞ þ x2
nðn þ 1Þ jn ðxÞ ¼ 0 with n a non-negative integer.

208

CHAPTER 4 Special functions

Particular solutions are rffiffiffiffiffi p J 1 ðxÞ SBFs of the first kind of order n 2x nþ 2 rffiffiffiffiffi p Y 1 ðxÞ SBFs of the second kind of order n yn ðxÞ ¼ 2x nþ 2 8 rffiffiffiffiffi > p ð1Þ ð1Þ > > > < hn ðxÞ ¼ jn ðxÞ þ iyn ðxÞ ¼ 2x Hnþ 12 SBFs of the third kind of order n rffiffiffiffiffi > > p ð2Þ ð2Þ > > hn ðxÞ ¼ jn ðxÞ
iyn ðxÞ ¼ H 1 : 2x nþ 2 jn ðxÞ ¼

(354) (355)

(356)

4.3. Properties (1)–(4) of the gamma function. Properties (1)–(4) can be proved by integration. We have 1. n ¼ 1 ZN

ZN d x expð
xÞ ¼


Gð1Þ ¼ 0

0 d½expð
xÞ ¼
expð
xÞjN 0 ¼ expð
xÞjN ¼ 1

0

Reminding that dðuvÞ ¼ u dv þ v du 0 u dv ¼ dðuvÞ
v du where u is the finite factor and dv the differential factor, by integrating both members follows the rule of integration by parts: Z

Z u dv ¼ uv


v du

Then, integrating by parts 2. ZN Gðn þ 1Þ ¼

d x expð
xÞx

ðnþ1Þ
1

0

¼
xn expð
xÞjN 0 þn

ZN

ZN ¼

ZN d x expð
xÞx ¼
n

0

d x expð
xÞxn
1 ¼ nGðnÞ 0

xn d½expð
xÞ 0

4.16 Solved problems

209

By repeated integration by parts 3.

ZN Gðn þ 1Þ ¼

ZN d x expð
xÞx ¼


x d½expð
xÞ ¼
x

n

0

n

n

expð
xÞjN 0

ZN þn

0

ZN ¼
n

dx expð
xÞxn
1 0

xn
1 d½expð
xÞ ¼
nxn
1 expð
xÞjN 0 þn

0

ZN


expð
xÞd xn
1

0

ZN

ZN ¼ nðn
1Þ

n
2

dx expð
xÞx 0

2

¼ nðn
1Þðn
2Þ/4


ZN

¼ / ¼ nðn
1Þðn
2Þ/

dx expð
xÞxn
ðn
2Þ 0

3 x2 d½expð
xÞ5

0

8 9 ZN < N
2 = 2 ¼ nðn
1Þðn
2Þ/
x expð
xÞ0 þ expð
xÞd x : ; 0

ZN ¼ nðn
1Þðn
2Þ/2

2

d x expð
xÞx ¼ nðn
1Þðn
2Þ/24


0

2 ¼ nðn
1Þðn
2Þ/24
x expð
xÞjN 0 þ

ZN

3

ZN

3 xd½expð
xÞ5

0

d x expð
xÞ $ 15

0

2 ¼ nðn
1Þðn
2Þ/24


ZN

3 d½expð
xÞ5 ¼ nðn
1Þðn
2Þ/2 $ 1 ¼ n!

0

1 2   ZN ZN 1 1
1 2 d x expð
xÞx ¼ d x expð
xÞx
1=2 ¼ G 2

4. n ¼

0

0

Now, change variable to x ¼ u2 ;

d x ¼ 2u du;

u ¼ x1=2

the interval of definition of the new variable being the same as that of x. Then pffiffiffiffi   ZN ZN
2 
1
 1 p pffiffiffiffi 2u du exp
u u ¼ 2 du exp
u2 ¼ 2 ¼ G ¼ p 2 2 0

0

210

CHAPTER 4 Special functions

where use was made of the integral over Gaussian functions ZN


 ðn
1Þ!! du exp
cu2 un ¼ sðnÞ nþ1 ð2cÞ 2 0 rffiffiffiffi p for n ¼ even; sðnÞ ¼ 1 sðnÞ ¼ 2

for n ¼ odd

and (n
1)!! is the double factorial function defined by Eqn (138). Therefore, for n ¼ 0, c ¼ 1 rffiffiffiffi pffiffiffiffi ZN
2 p 1 1 p du exp
u ¼ pffiffiffi sð0Þ ¼ pffiffiffi ¼ 2 2 2 2 0   pffiffiffiffi 1 ¼ p follows. and G 2 By repeated integration by parts it is also possible to derive the duplication formula (5). The recurrence property (2) allows one to obtain the G functions whose argument is a negative fraction. For example, 7.

   1 1   G
þ1   G 2 2 1 1 G
¼   ¼   ¼
2G 1 1 2 2

2 2       3 1 1   G
þ1   G
G 3 4 1 2 2 2        G
¼ ¼ ¼ ¼ G 3 3 3 1 2 3 2



2 2 2 2 

This can be shown by use  of theprevious variable transformation followed by integration by 1 parts. We prove this for G
. In fact 2 1 2   ZN 1 d x expð
xÞx
3=2 ¼ G
2 n¼


0

x¼

u2 ;

d x ¼ 2u du;

x
3=2 ¼ u
3

4.16 Solved problems

211

Then, taking v ¼ exp(
u2) as finite factor and d(u
1) as differential factor   ZN ZN

 
 1 G
¼ 2 du exp
u2 u
2 ¼
2 exp
u2 d u
1 2 0

N
 ¼
2 exp
u2 u
1 0 þ 2

ZN

0


 u
1 d exp
u2

0

Now, the first term vanishes at infinity, and also at zero by the l’Hoˆpital’s rule, since
2  
1


2 
1  d exp
u du lim exp
u u ¼ lim u/0 u/0 du du


2  ¼ lim exp
u ð
2uÞ=1 ¼ 0 u/0

so that we obtain 

1 G
2



ZN ¼2

u


1




ZN



exp
u ð
2u duÞ ¼
4 2

0




du exp
u

2



  1 ¼
2G 2

0

as it must be. 4.4. Find the FT of the exponential and the Gaussian function. 1. The exponential function Let f ðxÞ ¼ expð
cjxjÞ

(357)

be the exponential function where c is a real positive parameter. Then its FT is 1 Fð pÞ ¼ pffiffiffiffiffiffi 2p

ZN d x expð
ipxÞexpð
cjxjÞ
N

2 0 3   Z ZN 1 4 1 1 1 þ d x exp½ðc
ipÞx þ d x exp½ðc þ ipÞx5 ¼ pffiffiffiffiffiffi ¼ pffiffiffiffiffiffi 2p 2p c
ip c þ ip
N

 1=2 2 c ¼ p c2 þ p 2

0

(358) 2. The Gaussian function Let


f ðxÞ ¼ exp
x2 =c2

(359)

212

CHAPTER 4 Special functions

be the Gaussian function where c is a real parameter. Then its FT is still a Gaussian function: 1 Fð pÞ ¼ pffiffiffiffiffiffi 2p

ZN


 d x expð
ipxÞexp
x2 =c2


N

 2 2  ZN  h   2 2 1 c p x x cp 2 i c c p p ffiffiffiffiffiffi p ffiffi ffi d ¼ exp
exp
¼ exp
þ i 4 4 c c 2 2p 2

(360)


N

4.5. The FT and the anti-FT of the Coulombic potential. With reference to Figure 4.6, choosing r as polar axis, we have in spherical coordinates k$r ¼ kr cos qk

(361)

dk ¼ k dk sin qk dqk d4k

(362)

2

Putting x ¼ cos qk ;

z ¼ kr;

dz ¼ r dk

(363)

we evaluate the integral extended to the whole space: 8 Z expðik $ rÞ > > I ¼ dk > > > k2 > > > > > Z1 Z2p ZN ZN > > > expðikrxÞ expðikrÞ
expð
ikrÞ < ¼ 2 dk k dx d4k ¼ 2p dk k2 ikr > 0
1 0 0 > > > > > ZN > > > 4p sin z 4p p 2p2 > > ¼ $ ¼ dz ¼ > > r r z r 2 : 0

FIGURE 4.6 The spherical coordinates ðk; qk ; 4k Þ of the wave vector k

(364)

4.16 Solved problems

Therefore, we obtain 1 1 ¼ 2 r 2p

Z dk

expðik $ rÞ k2

213

(365)

which is the form of the FT of r
1 (Koide, 1976). Even if it does not exist in the strict sense of ordinary analysis, the theory of generalized functions (Gel’fand and Shilov, 1964) shows that the divergent integral arising in the inverse FT of 1/r can be treated as well and is given by (Weniger and Steinborn, 1983):  1=2 Z 1 1 2 1 dr expð
ik $ rÞ (366) ¼ FðkÞ ¼ 2 3=2 r p k ð2pÞ 4.6. The LTs of Table 4.1. The first six integrals of Table 4.1 are immediately evaluated by recalling the well-known general formula ZN d x xn expð
axÞ ¼

n! anþ1

0

with n a non-negative integer and a a real positive number different from zero. Then, we obtain for the third integral in the table 1 LðsÞ ¼ ðn
1Þ!

ZN dt tn
1 expð
stÞ ¼

1 ðn
1Þ! 1 ¼ n $ ðn
1Þ! sn s

0

The fourth integral is evident, since ZN

ZN dt expð
atÞexpð
stÞ ¼

LðsÞ ¼ 0

dt exp½
ða þ sÞt ¼

1 aþs

0

For the sixth integral we have 1 LðsÞ ¼ b
a

ZN dt½expð
atÞ
expð
btÞexpð
stÞ 0

2 1 4 ¼ b
a

ZN dt exp½
ða þ sÞt
0

¼

ZN

3 dt exp½
ðb þ sÞt5 ¼

0

1 bþs
a
s 1 $ ¼ b
a ða þ sÞðb þ sÞ ða þ sÞðb þ sÞ

as it must be.

  1 1 1
b
a aþs bþs

214

CHAPTER 4 Special functions   1 The seventh integral is obtained much in the same way as we did before for the G function. 2 In fact 1 LðsÞ ¼ pffiffiffiffi p

ZN

dt t
1=2 expð
stÞ

0

Changing variable to st ¼ u2 ;

s dt ¼ 2u du;

u¼

pffiffiffiffi st;

t
1=2 ¼ s1=2 u
1

we have 1 LðsÞ ¼ pffiffiffiffi p

ZN 0


2u 1=2
1 2 s u exp
u2 ¼ pffiffiffiffiffiffi du s ps

ZN




du exp
u 0

2



pffiffiffiffi 2 1 p ¼ pffiffiffiffiffiffi$ ¼ pffiffi s ps 2

The last integral in the table is a little more complicated. It is, however, easily shown that the integrand is the primitive of the function f ðtÞ ¼


expð
stÞ½s sinðatÞ þ a cosðatÞ a2 þ s 2

as we can see by calculating the first derivative of f(t). Taking the definite integral, we then have  ZN expð
stÞ½s sinðatÞ þ a cosðatÞ dt sinðatÞexpð
stÞ ¼
 a2 þ s2 t¼N 0  expð
stÞ½s sinðatÞ þ a cosðatÞ a þ ¼ 2  a2 þ s2 a þ s2 t¼0 and the integral of the table follows.