Date of Publication :30th November 2023

Abstract: This paper outlines an original algorithm for Bessel functions of complex values. Bessel functions of the first kind, with an integer index, J_n (x), when using the standard math or complex headers in C++, are only defined for real numbers. My algorithm uses both the power series representation of J_n (x), for x near the origin, otherwise, a Taylor series of J_n (x). For the Taylor series ofJ_n (x) about x_0, the kth derivative, J_n^((k) ) (x_0 ), is written as (a_0+∑_(m=1)^k▒a_m x_0^(-m) ) J_(n-1) (x_0 )+(b_0+∑_(m=1)^k▒b_m x_0^(-m) ) J_n (x_0 ), where the coefficients can be calculated from the coefficients from lower derivatives.

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