c6-4 (779503), страница 2
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The asymptotic forms for x ν arer112cos x − νπ − πJν (x) ∼πx24(6.5.4)r112Yν (x) ∼sin x − νπ − ππx24In the transition region where x ∼ ν, the typical amplitudes of the Bessel functionsare on the orderJν (ν) ∼21/3132/3 Γ( 23 )1/3ν 1/3Yν (ν) ∼ −2∼131/6 Γ( 23 ) ν 1/30.4473ν 1/30.7748∼ − 1/3ν(6.5.5)Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.
To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).This section and the next one present practical algorithms for computing variouskinds of Bessel functions of integer order. In §6.7 we deal with fractional order. Infact, the more complicated routines for fractional order work fine for integer ordertoo. For integer order, however, the routines in this section (and §6.6) are simplerand faster. Their only drawback is that they are limited by the precision of theunderlying rational approximations.
For full double precision, it is best to work withthe routines for fractional order in §6.7.For any real ν, the Bessel function Jν (x) can be defined by the seriesrepresentation ν X∞(− 14 x2 )k1x(6.5.1)Jν (x) =2k!Γ(ν + k + 1).
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