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1: Software Index

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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2: Bibliography S

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

ⓘ

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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3: Bibliography W

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A drop of water becomes a gateway into the world of catastrophe optics. Scientific American 261, pp. 120–123.

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Referenced by:

§36.14(ii).

Encodings:

BibTeX

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Surface Waves. In Handbuch der Physik, Vol. 9, Part 3, pp. 446–778.

ⓘ

External Links:

Online edition, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§11.12.

Encodings:

BibTeX

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4: 10.16 Relations to Other Functions

… ►Principal values on each side of these equations correspond. … ►

10.16.5 $$J_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }{\mathrm{e}}^{\mp \mathrm{i}z}}{\mathrm{\Gamma }\left(\nu +1\right)}M(\nu +\frac{1}{2},2\nu +1,\pm 2\mathrm{i}z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.69 (modified)

Referenced by:

§10.16, §10.39

Permalink:

http://dlmf.nist.gov/10.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

►

10.16.6 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(z\right)\\ H_{\nu }^{(2)}\left(z\right)\end{array}\}=\mp 2{\pi }^{-\frac{1}{2}}\mathrm{i}{\mathrm{e}}^{\mp \nu \pi \mathrm{i}}{(2z)}^{\nu }{\mathrm{e}}^{\pm \mathrm{i}z}U(\nu +\frac{1}{2},2\nu +1,\mp 2\mathrm{i}z).$$

ⓘ

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.16, §10.17(v)

Permalink:

http://dlmf.nist.gov/10.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

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10.16.7 $$J_{\nu }\left(z\right)=\frac{{\mathrm{e}}^{\mp (2\nu +1)\pi \mathrm{i}/4}}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)}{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(\pm 2\mathrm{i}z\right),$$ $2\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.16, §10.16

Permalink:

http://dlmf.nist.gov/10.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

… ►In all cases principal branches correspond at least when $|\mathrm{ph}z|\le \frac{1}{2}\pi$. …

5: Bibliography M

… ►

Hypergeometric Functions.

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Notes:

Unpublished Lecture Notes, University of London. An online version exists.

External Links:

FullText

Referenced by:

§35.7(ii), §35.8(iii).

Encodings:

BibTeX

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Catastrophe optics of spheroidal drops and generalized rainbows. J. Quantit. Spec. and Rad. Trans. 63, pp. 341–351.

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External Links:

Document

Referenced by:

§36.14(ii).

Encodings:

BibTeX

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6: Possible Errors in DLMF

… ►Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …

7: Errata

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In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

Reported by Nico Temme on 2021-06-01

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Confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

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The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

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Originally a minus sign was missing in the entries for $\mathrm{cd}u$ and $\mathrm{dc}u$ in the second column (headed $z+K+i{K}^{\prime }$). The correct entries are $-k^{-1}\mathrm{ns}z$ and $-k\mathrm{sn}z$. Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\mathrm{sn},\mathrm{cn},\mathrm{dn}$, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}{K}^{\prime }$	$z+\mathrm{i}{K}^{\prime }$	$z+2K$	$z+2K+2\mathrm{i}{K}^{\prime }$	$z+2\mathrm{i}{K}^{\prime }$
$\mathrm{⋮}$
$\mathrm{cd}u$	$-\mathrm{sn}z$	$-k^{-1}\mathrm{ns}z$	$k^{-1}\mathrm{dc}z$	$-\mathrm{cd}z$	$-\mathrm{cd}z$	$\mathrm{cd}z$
$\mathrm{⋮}$
$\mathrm{dc}u$	$-\mathrm{ns}z$	$-k\mathrm{sn}z$	$k\mathrm{cd}z$	$-\mathrm{dc}z$	$-\mathrm{dc}z$	$\mathrm{dc}z$
$\mathrm{⋮}$

Reported 2014-02-28 by Svante Janson.

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Bibliographic citations were added in §§3.5(iv), 4.44, 8.22(ii), 22.4(i), and minor clarifications were made in §§19.12, 20.7(vii), 22.9(i). In addition, several minor improvements were made affecting only ancilliary documents and links in the online version.

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8: 26.13 Permutations: Cycle Notation

… ►They are often dropped from the cycle notation. …

9: 10.20 Uniform Asymptotic Expansions for Large Order

… ►The eye-shaped closed domain in the uncut $z$-plane that is bounded by $B{P}_1{E}_1$ and $B{P}_2{E}_2$ is denoted by $𝐊$; see Figure 10.20.3. …

10: How to Cite

… ►The direct correspondence between the reference numbers in the printed Handbook and the permalinks used online in the DLMF enables readers of either version to cite specific items and their readers to easily look them up again — in either version! ►The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. …