Debye's asymptotic expansion for x ≤ 9990
Graphical Error Analysis
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, 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.
- Handbook of Mathematical Functions, Abramowitz & Stegun, Dover Publications, fifth printing, pp366, 9.3.7.
- Table of Integrals Series and Products, Gradshteyn & Rizhik, Academic Press, 1965, p963, 8.452.
- Calculations were performed on an AMD Athlon XP 1700+, 1.47 GHz, microprocessor. Computer programs were written in Assembly (MASM 5.1).
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