230 Chapter 6.Special Functions 6.5 Bessel Functions of Integer Order This section and the next one present practical algorithms for computing various kinds of Bessel functions of integer order.In $6.7 we deal with fractional order.In fact,the more complicated routines for fractional order work fine for integer order too.For integer order,however,the routines in this section (and 86.6)are simpler and faster.Their only drawback is that they are limited by the precision of the underlying rational approximations.For full double precision,it is best to work with the routines for fractional order in 86.7. For any real v,the Bessel function J(z)can be defined by the series representation 菲 J(x) (-7x2) (6.5.1) 3 The series converges for all z,but it is not computationally very useful for >1. For v not an integer the Bessel function Y()is given by Y(z)= J(z)cos(vT)-Jv() (6.5.2) sin(wT)】 The right-hand side goes to the correct limiting value Yn()as v goes to some integer n,but this is also not computationally useful. For arguments x<v,both Bessel functions look qualitatively like simple 3SO power laws,with the asymptotic forms for 0<zv 1 J(~w+2 v≥0 6回~2m(回 (6.5.3) T -() -V v>0 Numerica 10621 For >v,both Bessel functions look qualitatively like sine or cosine waves whose amplitude decays as x-1/2.The asymptotic forms forvare 43126 2 1 (outside (6.5.4) 2. 71 1 In the transition region wherev,the typical amplitudes of the Bessel functions are on the order 21/31 0.4473 四）~3亦~ v1/3 21/3 (6.5.5) 3VT方~-0.7748 1 Y(w) v1/3230 Chapter 6. Special Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). 6.5 Bessel Functions of Integer Order This section and the next one present practical algorithms for computing various kinds of Bessel functions of integer order. In §6.7 we deal with fractional order. In fact, the more complicated routines for fractional order work fine for integer order too. For integer order, however, the routines in this section (and §6.6) are simpler and faster. Their only drawback is that they are limited by the precision of the underlying rational approximations. For full double precision, it is best to work with the routines for fractional order in §6.7. For any real ν, the Bessel function Jν(x) can be defined by the series representation Jν (x) =
1 2 x ν ∞ k=0 (−1 4x2)k k!Γ(ν + k + 1) (6.5.1) The series converges for all x, but it is not computationally very useful for x 1. For ν not an integer the Bessel function Yν (x) is given by Yν(x) = Jν(x) cos(νπ) − J−ν(x) sin(νπ) (6.5.2) The right-hand side goes to the correct limiting value Y n(x) as ν goes to some integer n, but this is also not computationally useful. For arguments x<ν, both Bessel functions look qualitatively like simple power laws, with the asymptotic forms for 0 < x  ν Jν(x) ∼ 1 Γ(ν + 1)
1 2 x ν ν ≥ 0 Y0(x) ∼ 2 π ln(x) Yν (x) ∼ −Γ(ν) π
1 2 x −ν ν > 0 (6.5.3) For x>ν, both Bessel functions look qualitatively like sine or cosine waves whose amplitude decays as x−1/2. The asymptotic forms for x ν are Jν(x) ∼  2 πx cos
x − 1 2 νπ − 1 4 π Yν(x) ∼  2 πx sin
x − 1 2 νπ − 1 4 π (6.5.4) In the transition region where x ∼ ν, the typical amplitudes of the Bessel functions are on the order Jν (ν) ∼ 21/3 32/3Γ( 2 3 ) 1 ν1/3 ∼ 0.4473 ν1/3 Yν (ν) ∼ − 21/3 31/6Γ( 2 3 ) 1 ν1/3 ∼ −0.7748 ν1/3 (6.5.5)