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1: 36.2 Catastrophes and Canonical Integrals

… ►Special cases: $K=1$, fold catastrophe; $K=2$, cusp catastrophe; $K=3$, swallowtail catastrophe. … ►

§36.2(ii) Special Cases

… ►Addendum: For further special cases see §36.2(iv) … ►

§36.2(iv) Addendum to 36.2(ii) Special Cases

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36.2.29 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,-z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right){\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$$ .

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Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E29

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

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

2: 19.36 Methods of Computation

… ►The computation is slowest for complete cases. … ►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. … ►Also, see Todd (1975) for a special case of $K\left(k\right)$. 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). … ►These special theorems are also useful for checking computer codes. …

3: Bibliography K

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Special Functions and Perturbations of Black Holes. In Special Functions (Hong Kong, 1999), pp. 140–151.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.

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External Links:

ISSN 0002-9890, FullText, MathReview (S. L. Shukla), ZentralBlatt

Referenced by:

§16.23(iii).

Encodings:

BibTeX

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Some special cases of the generalized hypergeometric function ${}_{q+1}{}^{}F_q^{}$ . J. Comput. Appl. Math. 78 (1), pp. 79–95.

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External Links:

ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§16.7.

Encodings:

BibTeX

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4: 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

… ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. … ►(In the latter case (25.12.11) becomes (25.5.1)). …

5: 36.4 Bifurcation Sets

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Special Cases

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

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

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6: 7.8 Inequalities

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7.8.5 $x\ge 0$.

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Symbols:

$𝖬\left(x\right)$: Mills’ ratio and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

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7.8.8 $x>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $x$: real variable

Source:

Pólya (1949, (1.5))

Referenced by:

§7.8, Erratum (V1.0.17) for Equation (7.8.8)

Permalink:

http://dlmf.nist.gov/7.8.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This inequality was added. It is Pólya (1949, (1.5)). Note that it is a specialcase of (8.10.11).

Suggested by Roberto Iacono

See also:

Annotations for §7.8 and Ch.7

7: 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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The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.

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External Links:

ISSN 1095-7170, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(v).

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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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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On certain special sets of orthogonal polynomials. Proc. Amer. Math. Soc. 1, pp. 731–737.

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External Links:

ISSN 0002-9939, Document, Link, MathReview (A. C. Offord)

Referenced by:

§18.35(i).

Encodings:

BibTeX

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8: 20.11 Generalizations and Analogs

… ►In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►However, in this case $q$ is no longer regarded as an independent complex variable within the unit circle, because $k$ is related to the variable $\tau =\tau (k)$ of the theta functions via (20.9.2). … ►For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii). …

9: Bibliography N

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Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.

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External Links:

ISSN 0031-9015, Document, MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

BibTeX

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

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Mathematical Approximation of Special Functions. Nova Science Publishers Inc., Commack, NY.

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Notes:

Ten papers on Chebyshev expansions

External Links:

ISBN 1-56072-052-2, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.76(ii), §10.76(ii), §10.76(ii), §10.76(iii), 2nd item.

Encodings:

BibTeX

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

BibTeX

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10: 25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: … ►

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Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. …8, 1, $-20\le x\le 10$. … Magnify

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§25.11(v) Special Values

… ►

25.11.13 $$\zeta (0,a)=\frac{1}{2}-a.$$

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function and $a$: real or complex parameter

Keywords:

special value

Source:

Apostol (1976, p. 268)

Permalink:

http://dlmf.nist.gov/25.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

… ►For the more general case ${\zeta }^{\prime }(-m,a)$, $m=1,2,\mathrm{\dots }$, see Elizalde (1986). …