# (p.228) Appendix 3 SPECIAL FUNCTIONS

# (p.228) Appendix 3 SPECIAL FUNCTIONS

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

*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

*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

*x*) satisfies the reflection formula

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

*p*> 0,

*q*> 0. An alternative form can be found from (A3.7) by making the substitution

*x*= sin

^{2}

*φ*. Then, since d

*x*= 2sin

*φ*cos

*φ d φ*, we have

The Hermite polynomials *H* _{n}(*x*) satisfy the differential equation

*n*= 0, 1, 2, …. In particular

*P*

_{n}(

*x*) satisfy the differential equation

*n*= 0, 1, 2, … and . In particular

*L*

_{n}(

*x*) satisfy the differential equation

*L*

_{n}(

*x*) can be obtained from the relation

*x*if

*k*>

*n*. If , it is straight-forward to show from (A3.26) and (A3.27) that

The Bessel functions *J* _{n}(*x*) of order *n* satisfy the differential equation

*n*= 0, 1, 2, …, and can be expressed as the power series

*J*

_{n}(

*x*) is

*θ*is periodic with period 2

*π*, the final integral in (A3.35) can have any limits which differ by 2

*π*. Alternatively,

*θ*can be shifted by an arbitrary angle

*φ*so that

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

*p*< 1, required in Chapter 4. Writing in (A3.41) we obtain

*L*

_{n}and performing the resulting integral using (A3.40) we obtain

*L*

_{m}and reversing the order of the two summations in (A3.43) we find

*m*−

*k*=

*j*. The summation over

*j*is simply the binomial expansion of so that

*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*)

*becomes small. From (A3.26) we have*

^{n}