On the Evaluation of Bessel Functions

Abstract In the present paper we describe an algorithm for the evaluation of Bessel functions J ν ( x ), Y ν ( x ) and H ( j ) ν ( x ) ( j = 1, 2) of arbitrary positive orders and arguments at a constant CPU time. The algorithm employs Taylor series, the Debye asymptotic expansions, and numerical evaluation of the Sommerfeld integral, and is based on the following two observations. (1) The Debye asymptotic expansions, contrary to what appears to be a popular belief, are not expansions in inverse powers of (large) parameter ν but turn out to be uniform expansions in inverse powers of (large) parameter g 1 = ( x - ν) x 1/3 for x > ν and (large) parameter g 2 (ν − x )/ν 1/3 for x x and ν such that both Taylor and Debye expansions do not provide a specified accuracy Bessel functions can be computed at a constant CPU time via (numerical) evaluation of the Sommerfeld integral along contours of steepest descents. In addition, in Appendix B we obtain certain new estimates concerning decay of the functions J ν ( x ) and −1/ Y ν ( x ) of fixed x and large ν, and in Appendix C we show that functions J ν ( x ) of integer ν provide the solution for a certain system of coupled harmonic oscillators.