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## 1—10 of 12 matching pages

##### 2: 19.38 Approximations
Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …
##### 3: 18.36 Miscellaneous Polynomials
They are related to Hermite–Padé approximation and can be used for proofs of irrationality or transcendence of interesting numbers. …
##### 4: 3.11 Approximation Techniques
Any five approximants arranged in the Padé table as … For further information on Padé approximations, see Baker and Graves-Morris (1996, §4.7), Brezinski (1980, pp. 9–39 and 126–177), and Lorentzen and Waadeland (1992, pp. 367–395). …
##### 5: 8.27 Approximations
• Luke (1975, §4.3) gives Padé approximation methods, combined with a detailed analysis of the error terms, valid for real and complex variables except on the negative real $z$-axis. See also Temme (1994b, §3).

• Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for $E_{1}\left(z\right)$ and $z^{-1}\int_{0}^{z}t^{-1}(1-e^{-t})\mathrm{d}t$ for complex $z$ with $\left|\operatorname{ph}z\right|\leq\pi$.

• ##### 6: 13.31 Approximations
• Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$, $\operatorname{erf}z$, $\operatorname{erfc}z$, $C\left(z\right)$, and $S\left(z\right)$; approximate errors are given for a selection of $z$-values.
• Luke (1969b, pp. 402, 410, and 415–421) gives main diagonal Padé approximations for $\mathrm{Ein}\left(z\right)$, $\mathrm{Si}\left(z\right)$, $\mathrm{Cin}\left(z\right)$ (valid near the origin), and $E_{1}\left(z\right)$ (valid for large $|z|$); approximate errors are given for a selection of $z$-values.
WKBJ approximations2.7(iii)) for $\rho>\rho_{\mathrm{tp}}\left(\eta,\ell\right)$ are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. …Seaton (1984) estimates the accuracies of these approximations. Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for $F_{0}$ and $G_{0}$ in the region inside the turning point: $\rho<\rho_{\mathrm{tp}}\left(\eta,\ell\right)$.