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1: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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§8.19(ii) Graphics

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§8.19(ix) Inequalities

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§8.19(x) Integrals

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§8.19(xi) Further Generalizations

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2: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

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Polynomials

… ►Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … ►

§16.2(v) Behavior with Respect to Parameters

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3: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

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§8.21(i) Definitions: General Values

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§8.21(iv) Interrelations

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§8.21(v) Special Values

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4: 1.16 Distributions

… ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►More generally, for $\alpha :[a,b]\to [-\mathrm{\infty },\mathrm{\infty }]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ (see §1.4(v)) can be considered as a distribution: … ►More generally, if $\alpha (x)$ is an infinitely differentiable function, then … ►Friedman (1990) gives an overview of generalized functions and their relation to distributions. …

5: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(i) Definition

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Convergence Properties

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§35.8(iv) General Properties

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Confluence

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6: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

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7: Bibliography B

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Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.12.

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Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

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Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.

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External Links:

MathReview

Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §17.1, §17.4(i), Ch.17.

Encodings:

BibTeX

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Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.

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External Links:

ISSN 0025-5793, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§9.13(ii).

Encodings:

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More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Timothy S. Trudgian), ZentralBlatt

Referenced by:

§25.10(ii), §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).

Encodings:

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8: 25.10 Zeros

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

9: 8.16 Generalizations

§8.16 Generalizations

►For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). Other generalizations are considered in Guthmann (1991) and Paris (2003).

10: Software Index

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	Open Source													With Book						Commercial
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4.48(iii) General Precision	✓					✓	✓	✓				✓	✓								✓	✓	a		REDUCE
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16 Generalized Hypergeometric Functions & Meijer G-Function
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This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

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An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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