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1: 31.12 Confluent Forms of Heun’s Equation

… ►There are four standard forms, as follows: …

2: 28.33 Physical Applications

… ►Substituting $z=\omega t$, $a=b/{\omega }^2$, and $2q=f/{\omega }^2$, we obtain Mathieu’s standard form (28.2.1). …

3: Bibliography T

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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External Links:

ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

Encodings:

BibTeX

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4: 28.2 Definitions and Basic Properties

… ►The standard form of Mathieu’s equation with parameters $(a,q)$ is ►

28.2.1 $$w^{\prime \prime }+(a-2q\cos \left(2z\right))w=0.$$

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Symbols:

$\cos z$: cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.3.1 20.1.1 (in slightly different form)

Referenced by:

§1.18(vii), Figure 28.17.1, Figure 28.17.1, §28.17, §28.17, §28.2(ii), §28.2(ii), §28.2(iii), §28.2(iv), §28.20(i), §28.29(i), §28.32(i), §28.32(i), 1st item, 2nd item, §28.33(iii), item (a), item (c), item (c), §28.5(i), §28.8(iv), §28.8(iv), §28.8(iv), §30.2(iii)

Permalink:

http://dlmf.nist.gov/28.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

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5: 12.2 Differential Equations

… ►with three distinct standard forms …

6: 1.13 Differential Equations

… ►A standard form for second order ordinary differential equations with $x\in ℝ$, and with a real parameter $\lambda$, and real valued functions $p(x),q(x),$ and $\rho (x)$, with $p(x)$ and $\rho (x)$ positive, is …

7: Guide to Searching the DLMF

… ►To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. … ►Note that the first form may match other functions $K$ than the Bessel $K$ function, so if you are sure you want Bessel $K$, you might as well enter one of the other 3 forms. … ►DLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. … ►

The following standard special functions: si, Si, ci, Ci, shi, Shi, ce, Ce, se, Se, ln, Ln, Lommels, LommelS, Jacobiphi, and the list is still growing.

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8: About the Project

… ►The Digital Library of Mathematical Functions (DLMF) Project was initiated to perform a complete revision of Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, published in 1964 by the National Bureau of Standards. …The results have been published in book form as the NIST Handbook of Mathematical Functions, by Cambridge University Press, and disseminated in the free electronic Digital Library of Mathematical Functions. …

9: Bruce R. Miller

… … ► 1953 in Houston, Texas) is on the staff of the Applied and Computational Mathematics Division of the Information Technology Laboratory in the National Institute of Standards and Technology. … ►This led to accepting a Post-doctoral position at the then National Bureau of Standards under the supervision of André Deprit. … ►He is the developer of the tools used to process the DLMF into both book and web forms. …

10: DLMF Project News

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