Chebyshev expansions for the Bessel function 𝐽_{𝑛}(𝑧) in the complex plane

Abstract

Polynomial-based approximations for J 0 ( z ) {J_0}(z) and J 1 ( z ) {J_1}(z) are presented. The first quadrant of the complex plane is divided into six sectors, and separate approximations are given for | z | ⩽ 8 |z| \leqslant 8 and for | z | ⩾ 8 |z| \geqslant 8 on each sector. Each approximation is based on a Chebyshev expansion in which the argument of the Chebyshev polynomials is real on the central ray of the sector. The errors involved in extrapolation off the central ray are discussed. The approximation obtained for | z | ⩾ 8 |z| \geqslant 8 can also be used to evaluate the Bessel functions Y 0 ( z ) {Y_0}(z) and Y 1 ( z ) {Y_1}(z) and the Hankel functions of the first and second kinds.