246 Chapter 6. Special Functions Modified Bessel Functions Steed's method does not work for modified Bessel functions because in this case CF2 is purely imaginary and we have only three relations among the four functions.Temme[3]has given a normalization condition that provides the fourth relation. The Wronskian relation is W=1K-K=- (6.7.20) The continued fraction CFl becomes 1 1 fv F元=五+2u+1/x+2u+2/z+ = (6.7.21) 81 To get CF2 and the normalization condition in a convenient form,consider the sequence of confluent hypergeometric functions zn(x)=U(w+1/2+n,2v+1,2x) (6.7.22) for fixed v.Then K(z)=x/2(2x)”er20(x) (6.7.23) RECIPES -+++(e-)周 K+1(x）_1[,1 (6.7.24) 2 K(r) Equation (6.7.23)is the standard expression for K in terms of a confluent hypergeometric function,while equation (6.7.24)follows from relations between contiguous confluent hy- Press. pergeometric functions (equations 13.4.16 and 13.4.18 in Abramowitz and Stegun).Now the functions n satisfy the three-term recurrence relation (equation 13.4.15 in Abramowitz and Stegun) Programs 2n-1(E)=bnzn()+an+12n+1 (6.7.25) with SCIENTIFIC( bn =2(n+T) am+1=-[(n+1/2)2-v2] (6.7.26) 6 Following the steps leading to equation(5.5.18),we get the continued fraction CF2 =本于 1a2 COMPUTING (ISBN 1392 20 (6.7.27) from which (6.7.24)gives K+1/K and thus K/K. 10521 Temme's normalization condition is that Numerical 2-() +1/2 431 (6.7.28) 容 uction Recipes where C.=-)”u+12+ (6.7.29) n!T(w+1/2-n) Note that the Cn's can be determined by recursion: C0=1, Cn+1 =-antiCn (6.7.30) n+1 We use the condition(6.7.28)by finding S=Cn in (6.7.31) n=1 20 Then +1/21 1+ (6.7.32)246 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). Modified Bessel Functions Steed’s method does not work for modified Bessel functions because in this case CF2 is purely imaginary and we have only three relations among the four functions. Temme [3] has given a normalization condition that provides the fourth relation. The Wronskian relation is W ≡ IνK
ν − KνI
ν = − 1 x (6.7.20) The continued fraction CF1 becomes fν ≡ I
ν Iν = ν x + 1 2(ν + 1)/x + 1 2(ν + 2)/x + ··· (6.7.21) To get CF2 and the normalization condition in a convenient form, consider the sequence of confluent hypergeometric functions zn(x) = U(ν + 1/2 + n, 2ν + 1, 2x) (6.7.22) for fixed ν. Then Kν(x) = π1/2 (2x) ν e −xz0(x) (6.7.23) Kν+1(x) Kν (x) = 1 x  ν + 1 2 + x + ν2 − 1 4 z1 z0  (6.7.24) Equation (6.7.23) is the standard expression for Kν in terms of a confluent hypergeometric function, while equation (6.7.24) follows from relations between contiguous confluent hy￾pergeometric functions (equations 13.4.16 and 13.4.18 in Abramowitz and Stegun). Now the functions zn satisfy the three-term recurrence relation (equation 13.4.15 in Abramowitz and Stegun) zn−1(x) = bnzn(x) + an+1zn+1 (6.7.25) with bn = 2(n + x) an+1 = −[(n + 1/2)2 − ν2 ] (6.7.26) Following the steps leading to equation (5.5.18), we get the continued fraction CF2 z1 z0 = 1 b1 + a2 b2 + ··· (6.7.27) from which (6.7.24) gives Kν+1/Kν and thus K
ν/Kν . Temme’s normalization condition is that ∞ n=0 Cnzn = 1 2x ν+1/2 (6.7.28) where Cn = (−1)n n! Γ(ν + 1/2 + n) Γ(ν + 1/2 − n) (6.7.29) Note that the Cn’s can be determined by recursion: C0 = 1, Cn+1 = − an+1 n + 1Cn (6.7.30) We use the condition (6.7.28) by finding S = ∞ n=1 Cn zn z0 (6.7.31) Then z0 = 1 2x ν+1/2 1 1 + S (6.7.32)