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1: About MathML

… ►By default, DLMF will use Native support when available; You may choose how MathML is processed (Native or MathJax) at Customize DLMF. ►In rare cases, a browser lacks both MathML support and a robust enough javascript implementation capable of running MathJax; you may wish to visit the Customize DLMF page and choose the HTML+images document format. … ►Of course you are encouraged to use a modern, up-to-date browser. … ►Most modern browsers support ‘Web Fonts’, fonts that are effectively included with a web site. DLMF uses the STIX web font to provide a consistent coverage. …

2: How to Cite

How to Cite

►When citing DLMF from a formal publication, we suggest a format similar to the following: … ►Citations from other electronic media (the web, email, …), should, of course, use the appropriate means to give the site URL (https://dlmf.nist.gov/), or specific Permalinks. … ►Note the ‘E’, ‘F’ and ‘T’ used to disambiguate equations, figures and tables. … ►For convenience, the permalink can be found in the pop-up ‘Info box’ associated with each item in the site.

3: DLMF Project News

error generating summary

4: Foreword

… ►That 1046-page tome proved to be an invaluable reference for the many scientists and engineers who use the special functions of applied mathematics in their day-to-day work, so much so that it became the most widely distributed and most highly cited NIST publication in the first 100 years of the institution’s existence. The success of the original handbook, widely referred to as “Abramowitz and Stegun” (“A&S”), derived not only from the fact that it provided critically useful scientific data in a highly accessible format, but also because it served to standardize definitions and notations for special functions. … ►However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. The document you are now holding, or the Web page you are now reading, represents an effort to extend the legacy of A&S well into the 21st century. … ►Particular attention is called to the generous support of the National Science Foundation, which made possible the participation of experts from academia and research institutes worldwide. …

5: 13.5 Continued Fractions

… ►If $a,b\in ℂ$ such that $a\ne -1,-2,-3,\mathrm{\dots }$, and $a-b\ne 0,1,2,\mathrm{\dots }$, then …This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $ℂ$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). ►If $a,b\in ℂ$ such that $a\ne 0,-1,-2,\mathrm{\dots }$, and $b-a\ne 2,3,4,\mathrm{\dots }$, then …This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector . …

6: 13.17 Continued Fractions

… ►If $\kappa ,\mu \in ℂ$ such that $\mu \pm (\kappa -\frac{1}{2})\ne -1,-2,-3,\mathrm{\dots }$, then …This continued fraction converges to the meromorphic function of $z$ on the left-hand side for all $z\in ℂ$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). ►If $\kappa ,\mu \in ℂ$ such that $\mu +\frac{1}{2}\pm (\kappa +1)\ne -1,-2,-3,\mathrm{\dots }$, then …This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector . …

7: 33.21 Asymptotic Approximations for Large $|r|$

… ►We indicate here how to obtain the limiting forms of $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, and $c(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$, with $ϵ$ and $\mathrm{\ell }$ fixed, in the following cases: ►

When $r\to \pm \mathrm{\infty }$ with $ϵ>0$, Equations (33.16.4)–(33.16.7) are combined with (33.10.1).

►

When $r\to \pm \mathrm{\infty }$ with , Equations (33.16.10)–(33.16.13) are combined with

33.21.1

${\zeta }_{\mathrm{\ell }}(\nu ,r)\sim {\mathrm{e}}^{-r/\nu }{(2r/\nu )}^{\nu },$

${\xi }_{\mathrm{\ell }}(\nu ,r)\sim {\mathrm{e}}^{r/\nu }{(2r/\nu )}^{-\nu }$ , $r\to \mathrm{\infty }$,

ⓘ

Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function

Permalink:

http://dlmf.nist.gov/33.21.E1

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §33.21(i), §33.21 and Ch.33

33.21.2

${\zeta }_{\mathrm{\ell }}(-\nu ,r)\sim {\mathrm{e}}^{r/\nu }{(-2r/\nu )}^{-\nu },$

${\xi }_{\mathrm{\ell }}(-\nu ,r)\sim {\mathrm{e}}^{-r/\nu }{(-2r/\nu )}^{\nu },$ $r\to -\mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function

Permalink:

http://dlmf.nist.gov/33.21.E2

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §33.21(i), §33.21 and Ch.33

Corresponding approximations for $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ as $r\to \mathrm{\infty }$ can be obtained via (33.16.17), and as $r\to -\mathrm{\infty }$ via (33.16.18).

►

When $r\to \pm \mathrm{\infty }$ with $ϵ=0$, combine (33.20.1), (33.20.2) with §§10.7(ii), 10.30(ii).

… ►For asymptotic expansions of $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$ with $ϵ$ and $\mathrm{\ell }$ fixed, see Curtis (1964a, §6).

8: 29.7 Asymptotic Expansions

… ►As $\nu \to \mathrm{\infty }$, … ►

29.7.5 $$b_{\nu }^{m+1}\left(k^2\right)-a_{\nu }^m\left(k^2\right)=O\left({\nu }^{m+\frac{3}{2}}{\left(\frac{1-k}{1+k}\right)}^{\nu }\right),$$ $\nu \to \mathrm{\infty }$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter

Referenced by:

§29.7(i)

Permalink:

http://dlmf.nist.gov/29.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

… ►Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as $\nu \to \mathrm{\infty }$, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$ and ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$. Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§28.29(i)) that are applicable to the Lamé equation.

9: 18.40 Methods of Computation

… ►

A numerical approach to the recursion coefficients and quadrature abscissas and weights

… ►See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain $w(x)$. … ►The question is then: how is this possible given only $F_N(z)$, rather than $F(z)$ itself? $F_N(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im }z$ at a distance greater than the pole spacing of the $x_n$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_N(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. … ►The quadrature points and weights can be put to a more direct and efficient use. …

10: Bibliography G

… ►

How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.

ⓘ

External Links:

ISSN 0006-3835, Document, MathReview (G. A. Evans), ZentralBlatt

Referenced by:

§18.40(ii), §3.5(v), §9.17(iii).

Encodings:

BibTeX

… ►

A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.

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Notes:

Double-precision Fortran, maximum accuracy 18S

External Links:

Document, Code, ZentralBlatt

Referenced by:

5th item, 1st item.

Encodings:

BibTeX

… ►

A Treatise on Bessel Functions and their Applications to Physics. 2nd edition, Macmillan and Co., London.

ⓘ

Notes:

Table errata: Math. Comp. v. 24 (1970), p. 240, v. 31 (1978), pp. 806 and 1046, and v. 32 (1978), pp. 318. Reprinted by Dover Publications Inc., New York, 1966.

External Links:

MathReview, ZentralBlatt

Referenced by:

§10.21(x), §10.73(i), §10.73(i).

Encodings:

BibTeX

… ►

Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

ⓘ

External Links:

ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt

Referenced by:

§18.28(xi), §18.38(iii).

Encodings:

BibTeX

… ►

General approach to few-cycle intense laser interactions with complex atoms. Phys. Rev. A 76, pp. 053411.

ⓘ

External Links:

Document, Link

Referenced by:

§3.2(vi).

Encodings:

BibTeX

…