Bessel Function of the First Kind  J10000(x)

Debye's asymptotic expansion for  x ≤ 9990
Graphical Error Analysis

D. Baruth,

Applying Debye's asymptotic expansions for Bessel functions of large orders[1,2], I found that using only the first three terms of the expansion yields much better results than using four or five terms.  Fully utilizing the x87 FPU's 80-bit registers[3], J10000(x ≤ 9990) is evaluated with a comparatively small relative error  ε ≤ 5*10-14.  Execution time at the convergence-point  x ≈ 9990  is 0.000063 seconds.


  1. Handbook of Mathematical Functions, Abramowitz & Stegun, Dover Publications, fifth printing, pp366, 9.3.7.
  2. Table of Integrals Series and Products, Gradshteyn & Rizhik, Academic Press, 1965, p963, 8.452.
  3. Calculations were performed on an AMD Athlon XP 1700+, 1.47 GHz, microprocessor.  Computer programs were written in Assembly (MASM 5.1).

Copyright Dan Baruth © 2006.  All rights reserved.