6.7 Bessel Functions of Fractional Order 241 You can easily derive it from the three-term recurrence relation for Bessel functions:Start with equation (6.5.6)and use equation (5.5.18).Forward evaluation of the continued fraction by one of the methods of $5.2 is essentially equivalent to backward recurrence of the recurrence relation.The rate of convergence of CFI is determined by the position of the turning point p=v(+1),beyond which the Bessel functions become oscillatory.Ifp, convergence is very rapid.If,then each iteration of the continued fraction effectively increases by one until p;thereafter rapid convergence sets in.Thus the number of iterations of CFI is of order x for large x.In the routine bessjy we set the maximum allowed number of iterations to 10,000.For larger you can use the usual asymptotic expressions for Bessel functions. One can show that the sign of is the same as the sign of the denominator of CFI once it has converged. 8 The complex continued fraction CF2 is defined by J+iy +i+/22-23/22-w2 1 学影 nted for p十gq三 一2 (6.7.3) Jvtirv x2(x+)+2(x+2i)+ (We sketch the derivation of CF2 in the analogous case of modified Bessel functions in the next subsection.)This continued fraction converges rapidly forrp,while convergence fails as z0.We have to adopt a special method for small r,which we describe below.For x not too small,we can ensure that p by a stable recurrence of andJ downwards to a value=,thus yielding the ratio fu at this lower value of v.This is the stable (North direction for the recurrence relation.The initial values for the recurrence are J=arbitrary, J=fuJv, (6.7.4) 。o with the sign of the arbitrary initial value of chosen to be the sign of the denominator of CF1.Choosing the initial value of/,very small minimizes the possibility of overflow during the recurrence.The recurrence relations are 9 Programs -1=二+ 元 (6.7.5) SCIENTIFIC U-1w-1- -1= 6 Once CF2 has been evaluated at=,then with the Wronskian(6.7.1)we have enough relations to solve for all four quantities.The formulas are simplified by introducing the quantity y≡Pfu (6.7.6) Then 10621 1/2 Numerica =±(g+p-Jm (6.7.7) 务是2 uction 431086 Ju=fuJu (6.7.8) Ya=YJμ (6.7.9) =(+)》 (6.7.10) North Software. The sign of in (6.7.7)is chosen to be the same as the sign of the initial in(6.7.4) Once all four functions have been determined at the value v=u,we can find them at the original value of v.For/and J,simply scale the values in (6.7.4)by the ratio of (6.7.7)to the value found after applying the recurrence(6.7.5).The quantities Y and Y can be found by starting with the values in(6.7.9)and (6.7.10)and using the stable upwards recurrence Y+1= _2yY-Y-1 (6.7.11) together with the relation Yo=Lyo-Yuti (6.7.12)6.7 Bessel Functions of Fractional Order 241 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). You can easily derive it from the three-term recurrence relation for Bessel functions: Start with equation (6.5.6) and use equation (5.5.18). Forward evaluation of the continued fraction by one of the methods of §5.2 is essentially equivalent to backward recurrence of the recurrence relation. The rate of convergence of CF1 is determined by the position of the turning point xtp =
ν(ν + 1) ≈ ν, beyond which the Bessel functions become oscillatory. If x <∼ xtp, convergence is very rapid. If x >∼ xtp, then each iteration of the continued fraction effectively increases ν by one until x <∼ xtp; thereafter rapid convergence sets in. Thus the number of iterations of CF1 is of order x for large x. In the routine bessjy we set the maximum allowed number of iterations to 10,000. For larger x, you can use the usual asymptotic expressions for Bessel functions. One can show that the sign of Jν is the same as the sign of the denominator of CF1 once it has converged. The complex continued fraction CF2 is defined by p + iq ≡ J
ν + iY
ν Jν + iYν = − 1 2x + i + i x (1/2)2 − ν2 2(x + i) + (3/2)2 − ν2 2(x + 2i) + ··· (6.7.3) (We sketch the derivation of CF2 in the analogous case of modified Bessel functions in the next subsection.) This continued fraction converges rapidly for x >∼ xtp, while convergence fails as x → 0. We have to adopt a special method for small x, which we describe below. For x not too small, we can ensure that x >∼ xtp by a stable recurrence of Jν and J
ν downwards to a value ν = µ <∼ x, thus yielding the ratio fµ at this lower value of ν. This is the stable direction for the recurrence relation. The initial values for the recurrence are Jν = arbitrary, J
ν = fν Jν , (6.7.4) with the sign of the arbitrary initial value of Jν chosen to be the sign of the denominator of CF1. Choosing the initial value of Jν very small minimizes the possibility of overflow during the recurrence. The recurrence relations are Jν−1 = ν xJν + J
ν J
ν−1 = ν − 1 x Jν−1 − Jν (6.7.5) Once CF2 has been evaluated at ν = µ, then with the Wronskian (6.7.1) we have enough relations to solve for all four quantities. The formulas are simplified by introducing the quantity γ ≡ p − fµ q (6.7.6) Then Jµ = ± W q + γ(p − fµ) 1/2 (6.7.7) J
µ = fµJµ (6.7.8) Yµ = γJµ (6.7.9) Y
µ = Yµ p + q γ (6.7.10) The sign of Jµ in (6.7.7) is chosen to be the same as the sign of the initial Jν in (6.7.4). Once all four functions have been determined at the value ν = µ, we can find them at the original value of ν. For Jν and J
ν , simply scale the values in (6.7.4) by the ratio of (6.7.7) to the value found after applying the recurrence (6.7.5). The quantities Yν and Y
ν can be found by starting with the values in (6.7.9) and (6.7.10) and using the stable upwards recurrence Yν+1 = 2ν x Yν − Yν−1 (6.7.11) together with the relation Y
ν = ν xYν − Yν+1 (6.7.12)