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Methods in Theoretical Quantum Optics$

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

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

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

(A3.7) Appendix 3 Special Functions
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) Appendix 3 Special Functions
(p.229) The relationship between the beta function and the gamma function is
(A3.9) Appendix 3 Special Functions
In Chapter 3, we require the value of the integral
(A3.10) Appendix 3 Special Functions
(see (3.8.12)). Putting Appendix 3 Special Functions, we obtain
(A3.11) Appendix 3 Special Functions
using (A3.8). Then, from (A3.9) and (A3.5), we find
(A3.12) Appendix 3 Special Functions
using Appendix 3 Special Functions

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

(A3.13) Appendix 3 Special Functions
where n = 0, 1, 2, …. In particular
(A3.14) Appendix 3 Special Functions
The orthogonality property of the Hermite polynomials is expressed by the integral
(A3.15) Appendix 3 Special Functions
The generating function is
(A3.16) Appendix 3 Special Functions
(p.230) The Legendre polynomials P n(x) satisfy the differential equation
(A3.17) Appendix 3 Special Functions
where n = 0, 1, 2, … and Appendix 3 Special Functions. In particular
(A3.18) Appendix 3 Special Functions
The orthogonality property of the Legendre polynomials is expressed by the integral
(A3.19) Appendix 3 Special Functions
The generating function is
(A3.20) Appendix 3 Special Functions
where Appendix 3 Special Functions. The Laguerre polynomials L n(x) satisfy the differential equation
(A3.21) Appendix 3 Special Functions
where n = 0, 1, 2, … and Appendix 3 Special Functions. In particular
(A3.22) Appendix 3 Special Functions
The orthogonality property of the Laguerre polynomials is expressed by the integral
(A3.23) Appendix 3 Special Functions
The generating function is
(A3.24) Appendix 3 Special Functions
where Appendix 3 Special Functions. The Laguerre polynomial L n(x) can be obtained from the relation
(A3.25) Appendix 3 Special Functions
and expressed as the power series
(A3.26) Appendix 3 Special Functions
(p.231) The associated Laguerre polynomials Appendix 3 Special Functions are defined by
(A3.27) Appendix 3 Special Functions
and hence Appendix 3 Special Functions is identically zero for all x if k > n. If Appendix 3 Special Functions, it is straight-forward to show from (A3.26) and (A3.27) that
(A3.28) Appendix 3 Special Functions

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

(A3.29) Appendix 3 Special Functions
where n = 0, 1, 2, …, and can be expressed as the power series
(A3.30) Appendix 3 Special Functions
The generating function is
(A3.31) Appendix 3 Special Functions
where Appendix 3 Special Functions. Important recurrence relations are
(A3.32) Appendix 3 Special Functions
(A3.33) Appendix 3 Special Functions
Note that Appendix 3 Special Functions. The integral form of J n(x) is
(A3.34) Appendix 3 Special Functions
This integral may be written in a number of equivalent forms. In particular
(A3.35) Appendix 3 Special Functions
(p.232) since Appendix 3 Special Functions. 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) Appendix 3 Special Functions

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

(A3.37) Appendix 3 Special Functions
Substituting the series expansion (A3.30) with Appendix 3 Special Functions we obtain
(A3.38) Appendix 3 Special Functions
Writing Appendix 3 Special Functions, we see that (A3.38) can be expressed as
(A3.39) Appendix 3 Special Functions
Finally, using (A3.25), we have the result
(A3.40) Appendix 3 Special Functions
This result can be used to evaluate the integral
(A3.41) Appendix 3 Special Functions
for p < 1, required in Chapter 4. Writing Appendix 3 Special Functions in (A3.41) we obtain
(A3.42) Appendix 3 Special Functions
Substituting the series expansion (A3.26) for L n and performing the resulting integral using (A3.40) we obtain
(A3.43) Appendix 3 Special Functions
(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) Appendix 3 Special Functions Appendix 3 Special Functions
where we have written mk = j. The summation over j is simply the binomial expansion of Appendix 3 Special Functions so that
(A3.45) Appendix 3 Special Functions
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) Appendix 3 Special Functions
Hence
(A3.47) Appendix 3 Special Functions
as in (4.5.32).