§11.15 Approximations

Luke (1975, pp. 416–421) gives Chebyshev-series expansions for ${\mathbf{H}}_n\left(x\right)$, ${\mathbf{L}}_n\left(x\right)$, $0\le \left|x\right|\le 8$, and ${\mathbf{H}}_n\left(x\right)-Y_n\left(x\right)$, $x\ge 8$, for $n=0,1$; ${\int }_0^x{t}^{-m}{\mathbf{H}}_0\left(t\right)dt$, ${\int }_0^x{t}^{-m}{\mathbf{L}}_0\left(t\right)dt$, $0\le \left|x\right|\le 8$, $m=0,1$ and ${\int }_0^x({\mathbf{H}}_0\left(t\right)-Y_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}({\mathbf{H}}_0\left(t\right)-Y_0\left(t\right))dt$, $x\ge 8$; the coefficients are to 20D.

MacLeod (1993) gives Chebyshev-series expansions for ${\mathbf{L}}_0\left(x\right)$, ${\mathbf{L}}_1\left(x\right)$, $0\le x\le 16$, and $I_0\left(x\right)-{\mathbf{L}}_0\left(x\right)$, $I_1\left(x\right)-{\mathbf{L}}_1\left(x\right)$, $x\ge 16$; the coefficients are to 20D.

Newman (1984) gives polynomial approximations for ${\mathbf{H}}_n\left(x\right)$ for $n=0,1$, $0\le x\le 3$, and rational-fraction approximations for ${\mathbf{H}}_n\left(x\right)-Y_n\left(x\right)$ for $n=0,1$, $x\ge 3$. The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.