Pad%C3%A9%20approximations

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11: 5.23 Approximations

§5.23 Approximations

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§5.23(i) Rational Approximations

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§5.23(ii) Expansions in Chebyshev Series

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§5.23(iii) Approximations in the Complex Plane

►See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\mathrm{\Gamma }\left(z\right)$. …

12: 19.38 Approximations

§19.38 Approximations

►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. … ►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). …The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

13: 3.8 Nonlinear Equations

… ►For real functions $f(x)$ the sequence of approximations to a real zero $\xi$ will always converge (and converge quadratically) if either: … ►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … ►Initial approximations to the zeros can often be found from asymptotic or other approximations to $f(z)$, or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

14: 28.8 Asymptotic Expansions for Large $q$

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§28.8(iv) Uniform Approximations

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Barrett’s Expansions

… ►It is stated that corresponding uniform approximations can be obtained for other solutions, including the eigensolutions, of the differential equations by application of the results, but these approximations are not included. ►

Dunster’s Approximations

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15: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). … ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.

16: 32.12 Asymptotic Approximations for Complex Variables

§32.12 Asymptotic Approximations for Complex Variables

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17: 25.12 Polylogarithms

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Figure 25.12.1: Dilogarithm function ${\mathrm{Li}}_2\left(x\right)$, Magnify

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

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$G_s(x)={\mathrm{Li}}_{s+1}\left({\mathrm{e}}^x\right).$

►For a uniform asymptotic approximation for $F_s(x)$ see Temme and Olde Daalhuis (1990).

18: 9.19 Approximations

§9.19 Approximations

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§9.19(i) Approximations in Terms of Elementary Functions

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Martín et al. (1992) provides two simple formulas for approximating $\mathrm{Ai}\left(x\right)$ to graphical accuracy, one for , the other for .

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§9.19(ii) Expansions in Chebyshev Series

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§9.19(iii) Approximations in the Complex Plane

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19: 6.16 Mathematical Applications

… ►Hence if $x=\pi /(2n)$ and $n\to \mathrm{\infty }$, then the limiting value of $S_n(x)$ overshoots $\frac{1}{4}\pi$ by approximately 18%. Similarly if $x=\pi /n$, then the limiting value of $S_n(x)$ undershoots $\frac{1}{4}\pi$ by approximately 10%, and so on. … ►

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Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

20: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

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$$2^{20}{\beta }_5=-527q^7-61529q^5-\mathrm{10\hspace{0.33em}43961}{q}^3-\mathrm{22\hspace{0.33em}41599}q+32m^2(5739q^5+\mathrm{1\hspace{0.33em}27550}{q}^3+\mathrm{2\hspace{0.33em}98951}q)-2048m^4(355q^3+1505q)+65536m^{6}q.$$

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§30.9(iii) Other Approximations and Expansions

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