# §11.15 Approximations

## §11.15(i) Expansions in Chebyshev Series

• Luke (1975, pp. 416–421) gives Chebyshev-series expansions for $\mathbf{H}_{n}\left(x\right)$, $\mathbf{L}_{n}\left(x\right)$, $0\leq\left|x\right|\leq 8$, and $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$, $x\geq 8$, for $n=0,1$; $\int_{0}^{x}t^{-m}\mathbf{H}_{0}\left(t\right)\mathrm{d}t$, $\int_{0}^{x}t^{-m}\mathbf{L}_{0}\left(t\right)\mathrm{d}t$, $0\leq\left|x\right|\leq 8$, $m=0,1$ and $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\mathrm{d}t$, $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))% \mathrm{d}t$, $x\geq 8$; the coefficients are to 20D.

• MacLeod (1993) gives Chebyshev-series expansions for $\mathbf{L}_{0}\left(x\right)$, $\mathbf{L}_{1}\left(x\right)$, $0\leq x\leq 16$, and $I_{0}\left(x\right)-\mathbf{L}_{0}\left(x\right)$, $I_{1}\left(x\right)-\mathbf{L}_{1}\left(x\right)$, $x\geq 16$; the coefficients are to 20D.

## §11.15(ii) Rational and Polynomial Approximations

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