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1: 25.11 Hurwitz Zeta Function

… ►Most references treat real $a$ with . … ►

$a$-Derivative

… ►When $a=1$, (25.11.35) reduces to (25.2.3). … ►

§25.11(xii) $a$-Asymptotic Behavior

… ►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

2: 15.10 Hypergeometric Differential Equation

… ►It has regular singularities at $z=0,1,\mathrm{\infty }$, with corresponding exponent pairs $\{0,1-c\}$, $\{0,c-a-b\}$, $\{a,b\}$, respectively. … ►Moreover, in (15.10.9) and (15.10.10) the symbols $a$ and $b$ are interchangeable. ►(c) If the parameter $c$ in the differential equation equals $2-n=0,-1,-2,\mathrm{\dots }$, then fundamental solutions in the neighborhood of $z=0$ are given by $z^{n-1}$ times those in (a) and (b), with $a$ and $b$ replaced throughout by $a+n-1$ and $b+n-1$, respectively. … ►(e) Finally, if $a-b+1$ equals $n=1,2,3,\mathrm{\dots }$, or $2-n=0,-1,-2,\mathrm{\dots }$, then fundamental solutions in the neighborhood of $z=\mathrm{\infty }$ are given by $z^{-a}$ times those in (a), (b), and (c) with $b$ and $z$ replaced by $a-c+1$ and $1/z$, respectively. … ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …

3: 36.4 Bifurcation Sets

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36.4.6 $$27x^2=-8y^3.$$

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Symbols:

$y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.4.E6

Encodings:

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See also:

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

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

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

… ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid … ►

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:

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See also:

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

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4: Bibliography N

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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

… ►

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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Coulomb Functions for Large Values of the Parameter $\eta$ . Technical report Atomic Energy of Canada Limited, Chalk River, Ontario.

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Notes:

Rep. CRT-526

External Links:

MathReview (A. Erdélyi)

Referenced by:

§33.7.

Encodings:

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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

… ►

Special Functions of Mathematical Physics: A Unified Introduction with Applications. Birkhäuser Verlag, Basel.

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Notes:

Translated from the Russian and with a preface by Ralph P. Boas, with a foreword by A. A. Samarskiĭ

External Links:

ISBN 3-7643-3183-6, MathReview, ZentralBlatt

Referenced by:

§18.18(i), Erratum (V1.2.0) Section 18.39.

Encodings:

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5: 12.11 Zeros

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12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

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See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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6: Bibliography D

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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

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… ►

Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.

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External Links:

ISSN 0308-2105, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.20(i), §14.20(ix), §14.20(ix), §14.20(viii), §14.23.

Encodings:

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Uniform asymptotic expansions for oblate spheroidal functions I: Positive separation parameter $\lambda$ . Proc. Roy. Soc. Edinburgh Sect. A 121 (3-4), pp. 303–320.

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External Links:

ISSN 0308-2105, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

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Uniform asymptotic expansions for oblate spheroidal functions II: Negative separation parameter $\lambda$ . Proc. Roy. Soc. Edinburgh Sect. A 125 (4), pp. 719–737.

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External Links:

ISSN 0308-2105, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

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7: 20.10 Integrals

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§20.10(i) Mellin Transforms with respect to the Lattice Parameter

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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

… ►Then ►

20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

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See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

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See also:

Annotations for §20.10(ii), §20.10 and Ch.20

…

8: 36.2 Catastrophes and Canonical Integrals

… ►

36.2.28 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-z\right)}=2\pi \sqrt{\frac{\pi z}{27}}\exp \left(\frac{2}{27}\mathrm{i}{z}^3\right)\left(J_{-1/6}\left(\frac{2}{27}{z}^3\right)+\mathrm{i}{J}_{1/6}\left(\frac{2}{27}{z}^3\right)\right),$$ $z\ge 0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential 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.E28

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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

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9: Bibliography F

… ►

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Uniform asymptotic expansions of certain classes of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.

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External Links:

Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§16.22.

Encodings:

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Uniform asymptotic expansions of a class of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.

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External Links:

ISSN 0036-1410, Document, MathReview (Roop N. Kesarwani), ZentralBlatt

Referenced by:

§16.22.

Encodings:

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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External Links:

ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

Encodings:

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

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10: 20.7 Identities

… ►Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices $2,3,4$. … ►Addendum: For a companion equation see (20.7.34). … ►

20.7.15 $$A\equiv A(\tau )=1/{\theta }_4\left(0\right|2\tau ),$$

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Defines:

$A$ (locally)

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\equiv$: equals by definition and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.7.E15

Encodings:

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See also:

Annotations for §20.7(vi), §20.7 and Ch.20

… ►See Lawden (1989, pp. 19–20). … ►

§20.7(viii) Transformations of Lattice Parameter

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