leading%20coefficients

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11: 28.6 Expansions for Small $q$

… ►Leading terms of the power series for $a_m\left(q\right)$ and $b_m\left(q\right)$ for $m\le 6$ are: … ►Leading terms of the of the power series for $m=7,8,9,\mathrm{\dots }$ are: … ►For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2). … ►Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: … ►Leading terms of the power series for the normalized functions are: …

12: 20 Theta Functions

Chapter 20 Theta Functions

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13: Mark J. Ablowitz

… ►Their similarity solutions lead to special ODEs which have the Painlevé property; i. …

14: 7.24 Approximations

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Cody (1969) provides minimax rational approximations for $\mathrm{erf}x$ and $\mathrm{erfc}x$. The maximum relative precision is about 20S.

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi }\mathrm{erf}x$ and ${\mathrm{e}}^{x^2}F\left(x\right)$ for $-3\le x\le 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi }x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ and $2xF\left(x\right)$ for $x\ge 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

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Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ for $x\ge 3$ (15D).

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Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x){\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $(0,\mathrm{\infty })$ (22D).

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15: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

§26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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§26.4(i) Definitions

… ► $M_1$ is the multinominal coefficient (26.4.2): … ►

§26.4(ii) Generating Function

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§26.4(iii) Recurrence Relation

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16: 10.73 Physical Applications

… ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►The analysis of the current distribution in circular conductors leads to the Kelvin functions $\mathrm{ber}x$, $\mathrm{bei}x$, $\ker x$, and $\mathrm{kei}x$. …

17: 19.36 Methods of Computation

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19.36.2 $$1-\frac{3}{14}{E}_2+\frac{1}{6}{E}_3+\frac{9}{88}{E}_2^2-\frac{3}{22}{E}_4-\frac{9}{52}{E}_2{E}_3+\frac{3}{26}{E}_5-\frac{1}{16}{E}_2^3+\frac{3}{40}{E}_3^2+\frac{3}{20}{E}_2{E}_4+\frac{45}{272}{E}_2^2{E}_3-\frac{9}{68}(E_3{E}_4+E_2{E}_5).$$

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Symbols:

$E_s(𝐳)$: symmetric function

Referenced by:

§19.19

Permalink:

http://dlmf.nist.gov/19.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.36(i), §19.36 and Ch.19

… ►If (19.25.9) is used when $0\le k^2\le 1$, cancellations may lead to loss of significant figures when $k^2$ is close to 1 and $\varphi >\pi /4$, as shown by Reinsch and Raab (2000). … ►The step from $n$ to $n+1$ is an ascending Landen transformation if $\theta =1$ (leading ultimately to a hyperbolic case of $R_C$) or a descending Gauss transformation if $\theta =-1$ (leading to a circular case of $R_C$). … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

18: 30.8 Expansions in Series of Ferrers Functions

… ►Then the set of coefficients $a_{n,k}^m({\gamma }^2)$, $k=-R,-R+1,-R+2,\mathrm{\dots }$ is the solution of the difference equation … ►The coefficients $a_{n,k}^m({\gamma }^2)$ satisfy (30.8.4) for all $k$ when we set $a_{n,k}^m({\gamma }^2)=0$ for . For $k\ge -R$ they agree with the coefficients defined in §30.8(i). …The set of coefficients $a_{}^{\prime }{}_{n,k}{}^m({\gamma }^2)$, $k=-N-1,-N-2,\mathrm{\dots }$, is the recessive solution of (30.8.4) as $k\to -\mathrm{\infty }$ that is normalized by …It should be noted that if the forward recursion (30.8.4) beginning with $f_{-N-1}=0$, $f_{-N}=1$ leads to $f_{-R}=0$, then $a_{n,k}^m({\gamma }^2)$ is undefined for and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ does not exist. …

19: 18.15 Asymptotic Approximations

… ►For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term in a related expansion see Wong and Zhao (2003). … ►The leading coefficients are given by … ►The leading coefficients are given by $A_0(\xi )=1$ and … ►The leading coefficients are given by $E_0(\zeta )=1$ and … ►The coefficients $u_m(x)$ are polynomials in $x$, and $u_0(x)=1$, $u_1(x)=\frac{1}{6}{x}^3$. …

20: 6.20 Approximations

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Cody and Thacher (1968) provides minimax rational approximations for $E_1\left(x\right)$, with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\mathrm{Ei}\left(x\right)$, with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$, with accuracies up to 20S.

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Clenshaw (1962) gives Chebyshev coefficients for $-E_1\left(x\right)-\ln |x|$ for $-4\le x\le 4$ and ${\mathrm{e}}^x{E}_1\left(x\right)$ for $x\ge 4$ (20D).

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Luke (1969b, pp. 321–322) covers $\mathrm{Ein}\left(x\right)$ and $-\mathrm{Ein}\left(-x\right)$ for $0\le x\le 8$ (the Chebyshev coefficients are given to 20D); $E_1\left(x\right)$ for $x\ge 5$ (20D), and $\mathrm{Ei}\left(x\right)$ for $x\ge 8$ (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

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