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31: 19.2 Definitions

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19.2.6 $$D(\varphi ,k)={\int }_0^{\varphi }\frac{{\sin }^2\theta d\theta }{\sqrt{1-k^2{\sin }^2\theta }}={\int }_0^{\sin \varphi }\frac{t^{2}dt}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}}=(F(\varphi ,k)-E(\varphi ,k))/k^2.$$

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Defines:

$D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type

Symbols:

$dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

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32: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►Of importance for special functions are infinite determinants of Hill’s type. These have the property that the double series …Hill-type determinants always converge. …

33: 18.10 Integral Representations

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§18.10(i) Dirichlet–Mehler-Type Integral Representations

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§18.10(ii) Laplace-Type Integral Representations

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34: 27.5 Inversion Formulas

… ►Other types of Möbius inversion formulas include: …

35: Bibliography P

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A Kummer-type transformation for a ${}_2{}^{}F_2^{}$ hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:

ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

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External Links:

ZentralBlatt

Referenced by:

§24.10(iii).

Encodings:

BibTeX

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36: 3.11 Approximation Techniques

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3.11.16 $$R_{k,\mathrm{\ell }}(x)=\frac{p_0+p_{1}x+\mathrm{\cdots }+p_k{x}^k}{1+q_{1}x+\mathrm{\cdots }+q_{\mathrm{\ell }}{x}^{\mathrm{\ell }}}$$

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Defines:

$R_{k,\mathrm{\ell }}(x)$: rational approximation (locally), $p_k$: coefficients (locally) and $q_{\mathrm{\ell }}$: coefficients (locally)

Referenced by:

§3.11(iii)

Permalink:

http://dlmf.nist.gov/3.11.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(iii), §3.11 and Ch.3

►of type $[k,\mathrm{\ell }]$ to $f$ on $[a,b]$ minimizes the maximum value of $\left|{ϵ}_{k,\mathrm{\ell }}(x)\right|$ on $[a,b]$, where … ►With $w(x)=1$ and 14-digit computation, we obtain the following rational approximation of type $[3,3]$ to the Bessel function $J_0\left(x\right)$ (§10.2(ii)) on the interval $0\le x\le j_{0,1}$, where $j_{0,1}$ is the first positive zero of $J_0\left(x\right)$: … ►In computer graphics a special type of spline is used which produces a Bézier curve. …A special applications area of Bézier curves is mathematical typography and the design of type fonts. …

37: 18.30 Associated OP’s

… ►They can be expressed in terms of type 3 Pollaczek polynomials (which are also associated type 2 Pollaczek polynomials) by (18.35.10). … ►The type 3 Pollaczek polynomials are the associated type 2 Pollaczek polynomials, see §18.35. …

38: Bibliography

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Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials. Comm. Math. Phys. 398 (3), pp. 1291–1348.

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External Links:

ISSN 0010-3616, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.11(iii), §8.11(iii), Erratum (V1.1.10) for Section 8.11(iii).

Encodings:

BibTeX

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:

ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.

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External Links:

ISSN 1095-7154, Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.18(iv).

Encodings:

BibTeX

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39: 29.12 Definitions

… ►There are eight types of Lamé polynomials, defined as follows: … ►The prefixes $u$, $s$, $c$, $d$, $\mathrm{𝑠𝑐}$, $\mathrm{𝑠𝑑}$, $\mathrm{𝑐𝑑}$, $\mathrm{𝑠𝑐𝑑}$ indicate the type of the polynomial form of the Lamé polynomial; compare the 3rd and 4th columns in Table 29.12.1. In the fourth column the variable $z$ and modulus $k$ of the Jacobian elliptic functions have been suppressed, and $P({\mathrm{sn}}^2)$ denotes a polynomial of degree $n$ in ${\mathrm{sn}}^2(z,k)$ (different for each type). …

40: 10.32 Integral Representations

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Mellin–Barnes Type

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Mellin–Barnes Type

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