distributional completeness
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1: 14.30 Spherical and Spheroidal Harmonics
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Distributional Completeness
…2: 1.17 Integral and Series Representations of the Dirac Delta
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►Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv).
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3: 29.3 Definitions and Basic Properties
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►For each pair of values of and there are four infinite unbounded sets of real eigenvalues for which equation (29.2.1) has even or odd solutions with periods or .
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►To complete the definitions, is positive and is negative.
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§29.3(ii) Distribution
… ►In this table the nonnegative integer corresponds to the number of zeros of each Lamé function in , whereas the superscripts , , or correspond to the number of zeros in . … ►4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►Such orthonormal sets are called complete.
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►of the Dirac delta distribution.
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►and completeness implies
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►and completeness relation
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►The formal completeness relation is now
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5: Errata
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►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions.
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6: 27.2 Functions
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27.2.1
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►Tables of primes (§27.21) reveal great irregularity in their distribution.
They tend to thin out among the large integers, but this thinning out is not completely regular.
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27.2.3
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7: 22.4 Periods, Poles, and Zeros
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§22.4(i) Distribution
… ►Figure 22.4.1 illustrates the locations in the -plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices , , , . … ►For the distribution of the -zeros of the Jacobian elliptic functions see Walker (2009). … ►Figure 22.4.2 depicts the fundamental unit cell in the -plane, with vertices , , , . The set of points , , comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by , where again . …8: Bibliography J
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Tafeln höherer Funktionen (Tables of Higher Functions).
7th edition, B. G. Teubner, Stuttgart (Bilingual).
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Distributions of matrix variates and latent roots derived from normal samples.
Ann. Math. Statist. 35 (2), pp. 475–501.
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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles.
Eurandom Preprint Series
Technical Report 14, Eurandom, Eindhoven, The Netherlands.
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Continuous Univariate Distributions.
2nd edition, Vol. I, John Wiley & Sons Inc., New York.
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Continuous Univariate Distributions.
2nd edition, Vol. II, John Wiley & Sons Inc., New York.
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9: 2.6 Distributional Methods
§2.6 Distributional Methods
… ►Motivated by the definition of distributional derivatives, we can assign them the distributions defined by … ►The Dirac delta distribution in (2.6.17) is given by … ►These equations again hold only in the sense of distributions. … ►§2.6(iv) Regularization
…10: Philip J. Davis
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►He returned to Harvard after the war and completed a Ph.
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►He also had a big influence on the development of the NBS Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (A&S), which became one of the most widely distributed and highly cited publications in NIST’s history.
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►Decades later, Olver became Editor-in-Chief and Mathematics Editor of the NIST Digital Library of Mathematical Functions (DLMF), a complete revision of A&S that was publicly released in 2010.
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