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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 $\widehat{\nu }\ne 0,1$; equivalently $\nu \ne n$. … ►To complete the definition we require … ►

§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ► … ►

2: 28.2 Definitions and Basic Properties

… ►A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to $\nu$. …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 $-\nu$. … ►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?

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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. 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. …

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How to Cite

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7: DLMF Project News

error generating summary

8: 25 Zeta and Related Functions

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9: 29.7 Asymptotic Expansions

… ►As $\nu \to \mathrm{\infty }$, … ►

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

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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:

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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.

10: 10.74 Methods of Computation

… ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►Temme (1997) shows how to overcome this difficulty by use of the Maclaurin expansions for these coefficients or by use of auxiliary functions. … ►Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions. … ►

§10.74(viii) Functions of Imaginary Order

►For the computation of the functions ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ defined by (10.45.2) see Temme (1994c) and Gil et al. (2002d, 2003a, 2004b).