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1: 28.12 Definitions and Basic Properties
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►The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict $\widehat{\nu}\ne 0,1$; equivalently $\nu \ne n$.
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§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu}(z,q)$
►Two eigenfunctions correspond to each eigenvalue $a={\lambda}_{\nu}\left(q\right)$. … ► … ►2: 28.2 Definitions and Basic Properties
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►A solution with the pseudoperiodic property (28.2.14) is called a Floquet
solution with respect to
$\nu $.
…Therefore a nontrivial solution $w(z)$ is either a Floquet solution with respect to
$\nu $, or $w(z+\pi ){\mathrm{e}}^{\mathrm{i}\nu \pi}w(z)$ is a Floquet solution with respect to
$\nu $.
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►leads to a Floquet solution.
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§28.2(vi) Eigenfunctions
►Table 28.2.2 gives the notation for the eigenfunctions corresponding to the eigenvalues in Table 28.2.1. …3: Need Help?
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►In the Digital Library of Mathematical Functions, we have tried to provide the most accurate, carefully selected information about Special Functions possible.
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4: Foreword
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►That 1046page tome proved to be an invaluable reference for the many scientists and engineers who use the special functions of applied mathematics in their daytoday 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.
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D. R. Lide (ed.), A Century of Excellence in Measurement, Standards, and Technology,
CRC Press, 2001. 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.
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►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.
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►This could not have been done without the cooperation of many mathematicians, information technologists, and physical scientists both within NIST and externally.
…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.
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5: 25 Zeta and Related Functions
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6: About MathML
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►, builtin to the browser) support for MathML is growing, (see Browsers supporting 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.
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7: 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. … ►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! … ►Note the ‘E’, ‘F’ and ‘T’ used to disambiguate equations, figures and tables. …8: DLMF Project News
error generating summary9: 10.74 Methods of Computation
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►It should be noted, however, that there is a difficulty in evaluating the coefficients ${A}_{k}(\zeta )$, ${B}_{k}(\zeta )$, ${C}_{k}(\zeta )$, and ${D}_{k}(\zeta )$, from the explicit expressions (10.20.10)–(10.20.13) when $z$ is close to
$1$ owing to severe cancellation.
Temme (1997) shows how to overcome this difficulty by use of the Maclaurin expansions for these coefficients or by use of auxiliary functions.
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►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods.
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►In the case of ${J}_{n}\left(x\right)$, the need for initial values can be avoided by application of Olver’s algorithm (§3.6(v)) in conjunction with Equation (10.12.4) used as a normalizing condition, or in the case of noninteger orders, (10.23.15).
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§10.74(viii) Functions of Imaginary Order
…10: 29.7 Asymptotic Expansions
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►As $\nu \to \mathrm{\infty}$,
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29.7.5
$${b}_{\nu}^{m+1}\left({k}^{2}\right){a}_{\nu}^{m}\left({k}^{2}\right)=O\left({\nu}^{m+\frac{3}{2}}{\left(\frac{1k}{1+k}\right)}^{\nu}\right),$$
$\nu \to \mathrm{\infty}$.
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►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 ${\mathrm{\mathit{E}\mathit{c}}}_{\nu}^{m}(z,{k}^{2})$ and ${\mathrm{\mathit{E}\mathit{s}}}_{\nu}^{m}(z,{k}^{2})$.
Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§28.29(i)) that are applicable to the Lamé equation.