Accurate analog computer generation of Bessel functions for large ranges

Abstract

The generation of Bessel functions J_n(x) is required when simulating many systems on an analog computer. Because of an indeterminate expression at the origin, however, they have not been amenable to an accurate generation for the range 0 ≤ x ≤ X_max for large X_max.

This paper extends an idea of Van Remoortere to use an approximation for 0 ≤ x ≤ X ₁ and solve a differential equation for X₁ ≤ x ≤ X_max, combining both phases by switching. The technique described here uses Chebyshev polynomials to minimize equipment in the approximation phase and generates the function 1/x by an integration process in the differential-equation phase to extend the range. The examples given for J ₀ and J ₉ indicate excellent accuracy for at least 0 ≤ x ≤ 100.