§11.15 Approximations

§11.15(i) Expansions in Chebyshev Series

Luke (1975, pp. 416–421) gives Chebyshev-series expansions for $\mathop{\mathbf{H}_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathbf{L}_{{n}}\/}\nolimits\!\left(x\right)$ , $0\leq\left|x\right|\leq 8$ , and $\mathop{\mathbf{H}_{{n}}\/}\nolimits\!\left(x\right)-\mathop{Y_{{n}}\/}% \nolimits\!\left(x\right)$ , $x\geq 8$ , for ; $\int_{0}^{x}t^{{-m}}\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(t\right)dt$ , $\int_{0}^{x}t^{{-m}}\mathop{\mathbf{L}_{{0}}\/}\nolimits\!\left(t\right)dt$ , $0\leq\left|x\right|\leq 8$ , and $\int_{0}^{x}(\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(t\right)-\mathop{Y_{{% 0}}\/}\nolimits\!\left(t\right))dt$ , $\int_{x}^{\infty}t^{{-1}}(\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(t\right)% -\mathop{Y_{{0}}\/}\nolimits\!\left(t\right))dt$ , $x\geq 8$ ; the coefficients are to 20D.
MacLeod (1993) gives Chebyshev-series expansions for $\mathop{\mathbf{L}_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathbf{L}_{{1}}\/}\nolimits\!\left(x\right)$ , $0\leq x\leq 16$ , and $\mathop{I_{{0}}\/}\nolimits\!\left(x\right)-\mathop{\mathbf{L}_{{0}}\/}% \nolimits\!\left(x\right)$ , $\mathop{I_{{1}}\/}\nolimits\!\left(x\right)-\mathop{\mathbf{L}_{{1}}\/}% \nolimits\!\left(x\right)$ , $x\geq 16$ ; the coefficients are to 20D.

Newman (1984) gives polynomial approximations for $\mathop{\mathbf{H}_{{n}}\/}\nolimits\!\left(x\right)$ for , $0\leq x\leq 3$ , and rational-fraction approximations for $\mathop{\mathbf{H}_{{n}}\/}\nolimits\!\left(x\right)-\mathop{Y_{{n}}\/}% \nolimits\!\left(x\right)$ for , $x\geq 3$ . The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.