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1: 28.12 Definitions and Basic Properties
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 ν ^ 0 , 1 ; equivalently ν n . …
§28.12(ii) Eigenfunctions me ν ( z , q )
Two eigenfunctions correspond to each eigenvalue a = λ ν ( q ) . …
2: 28.2 Definitions and Basic Properties
A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to ν . …Therefore a nontrivial solution w ( z ) is either a Floquet solution with respect to ν , or w ( z + π ) e i ν π w ( z ) is a Floquet solution with respect to ν . … leads to a Floquet solution. …
§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?
In the Digital Library of Mathematical Functions, we have tried to provide the most accurate, carefully selected information about Special Functions possible. We have also tried to use the best technologies available in order to make this information useful and accessible. …
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    • How do I search within DLMF? See Guide to Searching the DLMF.

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  • 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. 22 2 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. … 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. … 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. …
    5: 25 Zeta and Related Functions
    6: About MathML
    , built-in 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. … Of course you are encouraged to use a modern, up-to-date browser. … DLMF uses the STIX web font to provide a consistent coverage. …
    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 summary
    9: 10.74 Methods of Computation
    It should be noted, however, that there is a difficulty in evaluating the coefficients A k ( ζ ) , B k ( ζ ) , C k ( ζ ) , and D k ( ζ ) , 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. … A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. … In the case of J n ( x ) , 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). …
    §10.74(viii) Functions of Imaginary Order
    10: 29.7 Asymptotic Expansions
    As ν , …
    29.7.5 b ν m + 1 ( k 2 ) a ν m ( k 2 ) = O ( ν m + 3 2 ( 1 k 1 + k ) ν ) , ν .
    Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as ν , 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 𝐸𝑐 ν m ( z , k 2 ) and 𝐸𝑠 ν 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.