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1: 28.26 Asymptotic Approximations for Large $q$

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28.26.3 $$\varphi =2h\sinh z-\left(m+\frac{1}{2}\right)\arctan \left(\sinh z\right).$$

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Symbols:

$\sinh z$: hyperbolic sine function, $\arctan z$: arctangent function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.8 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E3

Encodings:

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See also:

Annotations for §28.26(i), §28.26 and Ch.28

►Then as $h\to +\mathrm{\infty }$ with fixed $z$ in $\mathrm{\Re }z>0$ and fixed $s=2m+1$, ►

28.26.4 $${\mathrm{Fc}}_m(z,h)\sim 1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.9 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E4

Encodings:

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See also:

Annotations for §28.26(i), §28.26 and Ch.28

… ►The asymptotic expansions of ${\mathrm{Fs}}_m(z,h)$ and ${\mathrm{Gs}}_m(z,h)$ in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. … ►For asymptotic approximations for ${\mathrm{M}}_{\nu }^{(3,4)}(z,h)$ see also Naylor (1984, 1987, 1989).

2: Bibliography Y

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Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.

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External Links:

MathReview (B. M. Agrawal), ZentralBlatt

Referenced by:

§35.10.

Encodings:

BibTeX

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Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.

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External Links:

Document

Referenced by:

§1.18(viii).

Encodings:

BibTeX

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3: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

… ►Cornell, Watching Dark Solitons Decay into Vortex Rings in a Bose–Einstein Condensate, Phys. Rev. Lett. 86, 2926–2929 (2001)

4: Bibliography F

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Tablicy značeniĭ funkcii $w(z)=e^{-z^2}(1+\frac{2i}{\sqrt{\pi }}{\int }_0^z{e}^{t^2}𝑑t)$ ot kompleksnogo argumenta. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).

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Notes:

In Russian. English translation, see Faddeyeva and Terent’ev (1961)

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§7.21.

Encodings:

BibTeX

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Tables of Values of the Function $w(z)=e^{-z^2}(1+2i{\pi }^{-1/2}{\int }_0^z{e}^{t^2}𝑑t)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

V. N. Faddeeva and N. M. Terent’ev (1954).

Encodings:

BibTeX

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Complex roots of $\sin z=az$, $\cos z=az$, and $\cosh z=az$ . Math. Comp. 30 (135), pp. 541–545.

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External Links:

FullText, MathReview (Gerhard Merz), ZentralBlatt

Referenced by:

§4.46.

Encodings:

BibTeX

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Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (2), pp. 313–344.

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External Links:

ISSN 0010-3616, Link, MathReview (Nikolai A. Kostov), ZentralBlatt

Referenced by:

§32.15.

Encodings:

BibTeX

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Error bounds for a uniform asymptotic expansion of the Legendre function $Q_n^{-m}(\cosh z)$ . SIAM J. Math. Anal. 21 (2), pp. 523–535.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.26.

Encodings:

BibTeX

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5: 4.14 Definitions and Periodicity

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4.14.4 $$\tan z=\frac{\sin z}{\cos z},$$

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Defines:

$\tan z$: tangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.3

Referenced by:

§4.14, §4.45(i)

Permalink:

http://dlmf.nist.gov/4.14.E4

Encodings:

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See also:

Annotations for §4.14 and Ch.4

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4.14.5 $$\csc z=\frac{1}{\sin z},$$

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Defines:

$\csc z$: cosecant function

Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.4

Permalink:

http://dlmf.nist.gov/4.14.E5

Encodings:

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See also:

Annotations for §4.14 and Ch.4

… ►The functions $\sin z$ and $\cos z$ are entire. In $ℂ$ the zeros of $\sin z$ are $z=k\pi$, $k\in \mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\frac{1}{2}\right)\pi$, $k\in \mathbb{Z}$. The functions $\tan z$, $\csc z$, $\sec z$, and $\cot z$ are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …

6: 10.29 Recurrence Relations and Derivatives

… ►With ${𝒵}_{\nu }\left(z\right)$ defined as in §10.25(ii), ►

${𝒵}_{\nu -1}\left(z\right)-{𝒵}_{\nu +1}\left(z\right)=(2\nu /z){𝒵}_{\nu }\left(z\right),$

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${𝒵}_{\nu }^{\prime }\left(z\right)={𝒵}_{\nu -1}\left(z\right)-(\nu /z){𝒵}_{\nu }\left(z\right),$

… ►For results on modified quotients of the form $z{𝒵}_{\nu \pm 1}\left(z\right)/{𝒵}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). … ►

$${\left(\frac{1}{z}\frac{d}{dz}\right)}^k(z^{\nu }{𝒵}_{\nu }\left(z\right))=z^{\nu -k}{𝒵}_{\nu -k}\left(z\right),$$

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7: 25.16 Mathematical Applications

… ► $H\left(s\right)$ is the special case $H(s,1)$ of the function ►

25.16.11 $$H(s,z)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}\sum _{m=1}^n\frac{1}{m^z},$$ $\mathrm{\Re }\left(s+z\right)>1$,

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Symbols:

$H(s,z)$: generalized Euler sums, $\mathrm{\Re }$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

Euler sum, infinite series, series representation

Source:

Apostol and Vu (1984, (1), p. 86)

Permalink:

http://dlmf.nist.gov/25.16.E11

Encodings:

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See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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25.16.12 $$H(s,z)+H(z,s)=\zeta \left(s\right)\zeta \left(z\right)+\zeta \left(s+z\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $H(s,z)$: generalized Euler sums, $s$: complex variable and $z$: complex variable

Keywords:

addition formula, connection formula

Source:

Apostol and Vu (1984, (11), p. 91)

Permalink:

http://dlmf.nist.gov/25.16.E12

Encodings:

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See also:

Annotations for §25.16(ii), §25.16 and Ch.25

►when both $H(s,z)$ and $H(z,s)$ are finite. ►For further properties of $H(s,z)$ see Apostol and Vu (1984). …

8: 4.28 Definitions and Periodicity

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4.28.9 $$\cos \left(\mathrm{i}z\right)=\cosh z,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.8

Permalink:

http://dlmf.nist.gov/4.28.E9

Encodings:

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See also:

Annotations for §4.28, §4.28 and Ch.4

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4.28.11 $$\csc \left(\mathrm{i}z\right)=-\mathrm{i}\mathrm{csch}z,$$

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Symbols:

$\csc z$: cosecant function, $\mathrm{csch}z$: hyperbolic cosecant function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.10

Permalink:

http://dlmf.nist.gov/4.28.E11

Encodings:

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See also:

Annotations for §4.28, §4.28 and Ch.4

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4.28.12 $$\sec \left(\mathrm{i}z\right)=\mathrm{sech}z,$$

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Symbols:

$\mathrm{sech}z$: hyperbolic secant function, $\mathrm{i}$: imaginary unit, $\sec z$: secant function and $z$: complex variable

A&S Ref:

4.5.11

Permalink:

http://dlmf.nist.gov/4.28.E12

Encodings:

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See also:

Annotations for §4.28, §4.28 and Ch.4

… ►The functions $\sinh z$ and $\cosh z$ have period $2\pi \mathrm{i}$, and $\tanh z$ has period $\pi \mathrm{i}$. The zeros of $\sinh z$ and $\cosh z$ are $z=\mathrm{i}k\pi$ and $z=\mathrm{i}\left(k+\frac{1}{2}\right)\pi$, respectively, $k\in \mathbb{Z}$.

9: 10.51 Recurrence Relations and Derivatives

… ►Let $f_n(z)$ denote any of ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, or ${𝗁}_n^{(2)}\left(z\right)$. … ►

$${\left(\frac{1}{z}\frac{d}{dz}\right)}^m(z^{n+1}{f}_n(z))=z^{n-m+1}{f}_{n-m}(z),$$ $m=0,1,\mathrm{\dots },n$,

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$${\left(\frac{1}{z}\frac{d}{dz}\right)}^m(z^{-n}{f}_n(z))={(-1)}^m{z}^{-n-m}{f}_{n+m}(z),$$ $m=0,1,\mathrm{\dots }.$

… ►Let $g_n(z)$ denote ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, or ${(-1)}^n$ ${𝗄}_n\left(z\right)$. Then …

10: Bibliography W

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Asymptotische Entwicklungen der hypergeometrischen Funktion $F(a,b,c,z)$ für $|c|\to \mathrm{\infty }$ und konstante Werte $a,b$ und $z$ . Demonstratio Math. 21 (2), pp. 441–458 (German).

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External Links:

ISSN 0420-1213, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§15.12(ii), §15.12(ii).

Encodings:

BibTeX

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Shadows cast on the bottom of a pool are not like other shadows. Why?. Scientific American 259, pp. 86–89.

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Referenced by:

§36.14(ii).

Encodings:

BibTeX

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The Airy transform. Amer. Math. Monthly 86 (4), pp. 271–277.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (A. G. Ramm), ZentralBlatt

Referenced by:

(9.10.13), §9.10(ix), §9.10(v).

Encodings:

BibTeX

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An asymptotic expansion of $W_{k,m}(z)$ with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

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External Links:

FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§13.19.

Encodings:

BibTeX

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

Encodings:

BibTeX

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