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1: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
… ►§8.19(ii) Graphics
… ►§8.19(ix) Inequalities
… ►§8.19(x) Integrals
… ►§8.19(xi) Further Generalizations
…2: 16.2 Definition and Analytic Properties
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§16.2(i) Generalized Hypergeometric Series
… ► … ►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
…3: 8.21 Generalized Sine and Cosine Integrals
§8.21 Generalized Sine and Cosine Integrals
►§8.21(i) Definitions: General Values
… ►§8.21(iv) Interrelations
… ►§8.21(v) Special Values
… ►4: 1.16 Distributions
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is called a distribution, or generalized function, if it is a continuous linear functional on , that is, it is a linear functional and for every in ,
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►More generally, for a nondecreasing function the corresponding Lebesgue–Stieltjes measure (see §1.4(v)) can be considered as a distribution:
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►More generally, if is an infinitely differentiable function, then
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►Friedman (1990) gives an overview of generalized functions and their relation to distributions.
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5: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
►§35.8(i) Definition
… ►Convergence Properties
… ►§35.8(iv) General Properties
… ►Confluence
…6: Wadim Zudilin
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►Robinson Award of the Canadian Mathematical Society.
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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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Transformations of generalized hypergeometric series.
Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
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Generalized Hypergeometric Series.
Stechert-Hafner, Inc., New York.
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Uniform asymptotic smoothing of Stokes’s discontinuities.
Proc. Roy. Soc. London Ser. A 422, pp. 7–21.
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Pi and the AGM, A Study in Analytic Number Theory and Computational Complexity.
Canadian Mathematical Society Series of Monographs and
Advanced Texts, John Wiley & Sons Inc., New York.
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8: 19.2 Definitions
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§19.2(i) General Elliptic Integrals
…9: 1.4 Calculus of One Variable
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►A simple discontinuity of at occurs when and exist, but .
…For an example, see Figure 1.4.1
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Stieltjes Measure with Discontinuous
►The utility of the generalization implicit in the Stieltjes measure appears when is not everywhere continuous, but has discontinuous jumps at specific values of , say . …10: Bibliography S
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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series.
J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
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On certain integrals and expansions involving Weber’s parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 4, pp. 158–166.
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Quelques intégrales définies discontinues contenant des fonctions de Mathieu.
Acad. Roy. Belg. Bull. Cl. Sci. (5) 56 (5), pp. 475–491 (French).
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On the relative extrema of the Hermite orthogonal functions.
J. Indian Math. Soc. (N.S.) 15, pp. 129–134.
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