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1: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

BibTeX

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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The sign of Lommel’s function. Trans. Amer. Math. Soc. 163, pp. 123–129.

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External Links:

ISSN 0002-9947, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§11.9(iv).

Encodings:

BibTeX

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

Encodings:

BibTeX

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2: 9.18 Tables

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Miller (1946) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-20(.01)2;$ ${\log }_{10}\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)/\mathrm{Ai}\left(x\right)$ for $x=0(.1)25(1)75$; $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=-10(.1)2.5$; ${\log }_{10}\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)/\mathrm{Bi}\left(x\right)$ for $x=0(.1)10$; $M\left(x\right)$, $N\left(x\right)$, $\theta \left(x\right)$, $\varphi \left(x\right)$ (respectively $F(x)$, $G(x)$, $\chi (x)$, $\psi (x)$) for $x=-80(1)-30(.1)0$. Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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Zhang and Jin (1996, p. 337) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

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Rothman (1954b) tabulates ${\int }_0^x\mathrm{Ai}\left(t\right)dt$ and ${\int }_0^x\mathrm{Bi}\left(t\right)dt$ for $x=-10(.1)\mathrm{\infty }$ and $-10(.1)2$, respectively; 7D. The entries in the columns headed ${\int }_0^x\mathrm{Ai}\left(-x\right)dx$ and ${\int }_0^x\mathrm{Bi}\left(-x\right)dx$ all have the wrong sign. The tables are reproduced in Abramowitz and Stegun (1964, Chapter 10), and the sign errors are corrected in later reprintings.

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3: 18.39 Applications in the Physical Sciences

… ►The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). … ►

c) Spherical Radial Coulomb Wave Functions

… ►(where the minus sign is often omitted, as it arises as an arbitrary phase when taking the square root of the real, positive, norm of the wave function), allowing equation (18.39.37) to be rewritten in terms of the associated Coulomb–Laguerre polynomials ${𝐋}_{n+l}^{2l+1}({\rho }_n)$. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►For either sign of $Z$, and $s$ chosen such that $n+l+1+(2Z/s)>0$, $n=0,1,2,\mathrm{\dots }$, truncation of the basis to $N$ terms, with $x_i^N\in [-1,1]$, the discrete eigenvectors are the orthonormal $L^2$ functions …

4: 36.4 Bifurcation Sets

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$x=\frac{9}{20}{z}^2.$

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$x=-\frac{3}{20}{z}^2,$

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36.4.11 $$x+\mathrm{i}y=-z^2\exp \left(\frac{2}{3}\mathrm{i}\pi m\right),$$ $m=0,1,2$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $n$: integer and $x$: real parameter

Referenced by:

§36.7(iii)

Permalink:

http://dlmf.nist.gov/36.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

… ►The $+$ sign labels the cusped sheet; the $-$ sign labels the sheet that is smooth for $z\ne 0$ (see Figure 36.4.4). …

5: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

… ►and … ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at $k=n-1$ ($k\ge 0$) and $z$ is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. … ►

6: 36.5 Stokes Sets

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$K=1$. Airy Function

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36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.8 $$16u^5-\frac{Y^2}{10u}+4u^3\mathrm{sign}\left(z\right)-\frac{3}{10}|Y|\mathrm{sign}\left(z\right)+4t^5+2t^3\mathrm{sign}\left(z\right)+|Y|t^2=0,$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $t$: variable and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.9 $$t=-u+{\left(\frac{|Y|}{10u}-u^2-\frac{3}{10}\mathrm{sign}\left(z\right)\right)}^{1/2}.$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $t$: variable and $Y$: scaled coordinate

Referenced by:

§36.5(ii)

Permalink:

http://dlmf.nist.gov/36.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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7: 26.13 Permutations: Cycle Notation

… ►See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ … ►The sign of a permutation is $+$ if the permutation is even, $-$ if it is odd. …

8: 3.4 Differentiation

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$$B_{-2}^5=\frac{1}{120}(6-10t-15t^2+20t^3-5t^4),$$

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§3.4(ii) Analytic Functions

… ►With the choice $r=k$ (which is crucial when $k$ is large because of numerical cancellation) the integrand equals ${\mathrm{e}}^k$ at the dominant points $\theta =0,2\pi$, and in combination with the factor $k^{-k}$ in front of the integral sign this gives a rough approximation to $1/k!$. … ► … ►

3.4.33 $${\nabla }^4{u}_{0,0}=\frac{1}{h^4}(20u_{0,0}-8(u_{1,0}+u_{0,1}+u_{-1,0}+u_{0,-1})+2(u_{1,1}+u_{1,-1}+u_{-1,1}+u_{-1,-1})+(u_{0,2}+u_{2,0}+u_{-2,0}+u_{0,-2}))+O\left(h^2\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding and $u$: function

A&S Ref:

25.3.32

Permalink:

http://dlmf.nist.gov/3.4.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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9: 36.2 Catastrophes and Canonical Integrals

… ► ${\mathrm{\Psi }}_1$ is related to the Airy function (§9.2): … … ►

36.2.19 $${\mathrm{\Psi }}_2\left(0,y\right)=\frac{\pi }{2}\sqrt{\frac{|y|}{2}}\exp \left(-\mathrm{i}\frac{y^2}{8}\right)\left(\exp \left(\mathrm{i}\frac{\pi }{8}\right)J_{-1/4}\left(\frac{y^2}{8}\right)-\mathrm{sign}\left(y\right)\exp \left(-\mathrm{i}\frac{\pi }{8}\right)J_{1/4}\left(\frac{y^2}{8}\right)\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\mathrm{sign}x$: sign of and $y$: real parameter

Referenced by:

§36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(ii), §36.2 and Ch.36

►For the Bessel function $J$ see §10.2(ii). … ►Addendum: For further special cases see §36.2(iv) …

10: 1.11 Zeros of Polynomials

… ►A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative. ►

Descartes’ Rule of Signs

►The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity. A similar relation holds for the changes in sign of the coefficients of $f(-z)$, and hence for the number of negative zeros of $f(z)$. … ►Both polynomials have one change of sign; hence for each polynomial there is one positive zero, one negative zero, and six complex zeros. …