## Stephen Barnett and Paul Radmore

Print publication date: 2002

Print ISBN-13: 9780198563617

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198563617.001.0001

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# (p.228) Appendix 3 SPECIAL FUNCTIONS

Source:
Methods in Theoretical Quantum Optics
Publisher:
Oxford University Press

In this appendix, we summarize the properties of the special functions required in Chapters 2, 3, and 4.

The gamma function Г(x) is defined as

(A3.1)
where we require x > 0 for the integral to converge. We have from (A3.1), for example, Г(1) = 1 and whilst Now consider Г(x + 1); integrating by parts we obtain
(A3.2)
The integrated term is zero and hence, using (A3.1),
(A3.3)
This relationship can be used to define Г (x) for negative values of x. For example, choosing we have so that . Since it follows that Г (x) diverges at the negative integers using this definition. Suppose now that x = n, where is an integer. Then, using (A3.3), we have
(A3.4)
Since Г(1) = 1, we find
(A3.5)
In addition to the above, Г(x) satisfies the reflection formula
(A3.6)

The beta function B(p, q) is defined to be

(A3.7)
where p > 0, q > 0. An alternative form can be found from (A3.7) by making the substitution x = sin2 φ. Then, since dx = 2sin φ cos φ d φ, we have
(A3.8)
(p.229) The relationship between the beta function and the gamma function is
(A3.9)
In Chapter 3, we require the value of the integral
(A3.10)
(see (3.8.12)). Putting , we obtain
(A3.11)
using (A3.8). Then, from (A3.9) and (A3.5), we find
(A3.12)
using

The Hermite polynomials H n(x) satisfy the differential equation

(A3.13)
where n = 0, 1, 2, …. In particular
(A3.14)
The orthogonality property of the Hermite polynomials is expressed by the integral
(A3.15)
The generating function is
(A3.16)
(p.230) The Legendre polynomials P n(x) satisfy the differential equation
(A3.17)
where n = 0, 1, 2, … and . In particular
(A3.18)
The orthogonality property of the Legendre polynomials is expressed by the integral
(A3.19)
The generating function is
(A3.20)
where . The Laguerre polynomials L n(x) satisfy the differential equation
(A3.21)
where n = 0, 1, 2, … and . In particular
(A3.22)
The orthogonality property of the Laguerre polynomials is expressed by the integral
(A3.23)
The generating function is
(A3.24)
where . The Laguerre polynomial L n(x) can be obtained from the relation
(A3.25)
and expressed as the power series
(A3.26)
(p.231) The associated Laguerre polynomials are defined by
(A3.27)
and hence is identically zero for all x if k > n. If , it is straight-forward to show from (A3.26) and (A3.27) that
(A3.28)

The Bessel functions J n(x) of order n satisfy the differential equation

(A3.29)
where n = 0, 1, 2, …, and can be expressed as the power series
(A3.30)
The generating function is
(A3.31)
where . Important recurrence relations are
(A3.32)
(A3.33)
Note that . The integral form of J n(x) is
(A3.34)
This integral may be written in a number of equivalent forms. In particular
(A3.35)
(p.232) since . Further, since sin θ is periodic with period 2π, the final integral in (A3.35) can have any limits which differ by 2π. Alternatively, θ can be shifted by an arbitrary angle φ so that
(A3.36)

A number of relationships exist between the Laguerre polynomials and the Bessel function J 0. Consider first the integral

(A3.37)
Substituting the series expansion (A3.30) with we obtain
(A3.38)
Writing , we see that (A3.38) can be expressed as
(A3.39)
Finally, using (A3.25), we have the result
(A3.40)
This result can be used to evaluate the integral
(A3.41)
for p < 1, required in Chapter 4. Writing in (A3.41) we obtain
(A3.42)
Substituting the series expansion (A3.26) for L n and performing the resulting integral using (A3.40) we obtain
(A3.43)
(p.233) Again substituting the series expansion for L m and reversing the order of the two summations in (A3.43) we find
(A3.44)
where we have written mk = j. The summation over j is simply the binomial expansion of so that
(A3.45)
again using (A3.26). If p = −1, the result is found by taking the limit as p → −1 of (A3.45). The argument of the Laguerre polynomial becomes large in this limit, whilst the factor (1 + p)n becomes small. From (A3.26) we have
(A3.46)
Hence
(A3.47)
as in (4.5.32).