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21: Richard A. Askey

… ►Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and related areas. … ►

5 Gamma Function

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22: Bibliography W

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Asymptotische Darstellungen der hypergeometrischen Funktion für große Parameter unterschiedlicher Größenordnung. Z. Anal. Anwendungen 5 (3), pp. 265–276 (German).

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External Links:

ISSN 0232-2064, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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External Links:

ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

Encodings:

BibTeX

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Surface Waves. In Handbuch der Physik, Vol. 9, Part 3, pp. 446–778.

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External Links:

Online edition, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§11.12.

Encodings:

BibTeX

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Asymptotic behavior of stability regions for Hill’s equation. SIAM J. Appl. Math. 47 (5), pp. 941–958.

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External Links:

ISSN 0036-1399, Document, MathReview (Ekkehard Wagenführer), ZentralBlatt

Referenced by:

§28.29(iii).

Encodings:

BibTeX

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Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

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External Links:

Document

Referenced by:

§2.11(vi), §2.11(vi).

Encodings:

BibTeX

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23: 1.2 Elementary Algebra

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1.2.40 $$⟨𝐮,𝐯⟩=\sum _{i=1}^n{u}_i\overline{v_i}={𝐯}^{\mathrm{H}}𝐮.$$

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Defines:

$⟨𝐮,𝐯⟩$: inner product over vectors

Symbols:

${𝐀}^{\mathrm{H}}$: Hermitian conjugate of matrix, $\overline{z}$: complex conjugate, $n$: nonnegative integer, $𝐯$: column vector and $𝐮$: column vector

Referenced by:

§1.18(i), §1.2(v)

Permalink:

http://dlmf.nist.gov/1.2.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.41 $$⟨𝐮,𝐯⟩=\overline{⟨𝐯,𝐮⟩},$$

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Symbols:

$\overline{z}$: complex conjugate, $⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.42 $$⟨\alpha 𝐮,\beta 𝐯⟩=\alpha \overline{\beta }⟨𝐮,𝐯⟩,$$

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Symbols:

$\overline{z}$: complex conjugate, $⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Referenced by:

§1.2(v)

Permalink:

http://dlmf.nist.gov/1.2.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.43 $$⟨𝐯,𝐯⟩=0,$$

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $𝐯$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.44 $$⟨𝐮,𝐯⟩=0.$$

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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24: Karl Dilcher

… ►Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. …

25: 26.10 Integer Partitions: Other Restrictions

… ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $m$ for which $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$. … ►It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). …

26: 1.1 Special Notation

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$x,y$	real variables.
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$⟨f,g⟩$	inner, or scalar, product for real or complex vectors or functions.
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27: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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${\sum }_{d|n}$, ${\prod }_{d|n}$	sum, product taken over divisors of $n$.
${\sum }_{\left(m,n\right)=1}$	sum taken over $m$, $1\le m\le n$ and $m$ relatively prime to $n$.
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${\sum }_p$, ${\prod }_p$	sum, product extended over all primes.
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28: 10.43 Integrals

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§10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\mathrm{\infty })$

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§10.43(iv) Integrals over the Interval ($0,\mathrm{\infty }$)

… ► … ►For collections of the Kontorovich–Lebedev transform, see Erdélyi et al. (1954b, Chapter 12), Prudnikov et al. (1986b, pp. 404–412), and Oberhettinger (1972, Chapter 5). …

29: Need Help?

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30: 11.12 Physical Applications

… ►Applications of Struve functions occur in water-wave and surface-wave problems (Hirata (1975) and Ahmadi and Widnall (1985)), unsteady aerodynamics (Shaw (1985) and Wehausen and Laitone (1960)), distribution of fluid pressure over a vibrating disk (McLachlan (1934)), resistive MHD instability theory (Paris and Sy (1983)), and optical diffraction (Levine and Schwinger (1948)). …