On the numerical evaluation of Legendre’s chi-function

Abstract:

Legendre’s chi-function, ${\chi _n}(z) = \Sigma _{k = 0}^\infty {z^{2k + 1}}/{(2k + 1)^n}$, is reexpanded in a power series in powers of $\log z$. The expansion obtained is well suited for the computation of ${\chi _n}(z)$ in the two cases of real z close to 1, and $z = {e^{i\alpha }},\alpha \in \mathbb {R}$. For $n = 2$ and $n = 3$, the present computational procedure is shown to be superior to the procedure recently proposed by Dempsey, Liu, and Dempsey, which uses Plana’s summation formula.