TAILIEUCHUNG - Lập Trình C# all Chap "NUMERICAL RECIPES IN C" part 29

Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 29', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 230 Chapter 6. Special Functions Bessel Functions ofInteger Order This section and the next one present practical algorithms for computing various kinds of Bessel functions of integer order. In 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 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 . For any real u the Bessel function Jv x can be defined by the series representation J x 1 X -4 x2 k 2 J k 0 k Y v k 1 The series converges for all x but it is not computationally very useful for x 1. For u not an integer the Bessel function Yv x is given by Yv x Jv x cos vY J-v x sin v The right-hand side goes to the correct limiting value Yn x as u goes to some integer n but this is also not computationally useful. For arguments x u both Bessel functions look qualitatively like simple power laws with the asymptotic forms for 0 x u Jv x Y0 x 1 1 V x r v 1R2 J 2 . . ln x v 0 Yv x r v v 0 For x u both Bessel functions look qualitatively like sine or cosine waves whose amplitude decays as x-1 2. The asymptotic forms for x u are Jv x v X. cos x - 2ux - 4x AT 1 1 Yv x sin x--------ux------x y V xx V 2 4 In the transition region where x u the typical amplitudes of the Bessel functions are on the order Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 21 3 1 v v 32 3r 3 Vi 3 V 21 3 1 v v 3V6r 3 T3 Vv3 Bessel Functions ofInteger Order 231 Bessel functions Figure . Bessel functions Jo x through J3 x and Yo x through Y2 x . which holds asymptotically for large v. Figure plots the first few Bessel functions of each kind. The Bessel functions satisfy the recurrence relations and Jn

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