discrete analog
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1: 20.11 Generalizations and Analogs
§20.11 Generalizations and Analogs
… ►It is a discrete analog of theta functions. If both are positive, then allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): …This is the discrete analog of the Poisson identity (§1.8(iv)). …2: 11.8 Analogs to Kelvin Functions
§11.8 Analogs to Kelvin Functions
…3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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► The analogous orthonormality is
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§1.18(v) Point Spectra and Eigenfunction Expansions
… ► … ►The analog of (1.18.34) is … ►The analogs of (1.18.49)–(1.18.52) may be written in a similar fashion each now including contributions from both the discrete and continuous parts of the spectrum, as in (1.18.65). …4: 28.27 Addition Theorems
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►They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)).
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5: Alexander I. Bobenko
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► Eitner), published by Springer in 2000, and Discrete Differential Geometry: Integrable Structure (with Y.
…He is also coeditor of Discrete Integrable Geometry and Physics (with R.
Seiler), published by Oxford University Press in 1999, and Discrete Differential Geometry (with P.
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6: 18.39 Applications in the Physical Sciences
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►However, in the remainder of this section will will assume that the spectrum is discrete, and that the eigenfunctions of form a discrete, normed, and complete basis for a Hilbert space.
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►The spectrum is entirely discrete as in §1.18(v).
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►The spectrum is entirely discrete as in §1.18(v).
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►Analogous to (18.39.7) the 3D Schrödinger operator is
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►Analogous to (18.39.8) the 3D time-independent Schrödinger equation with potential is
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7: Sidebar 5.SB1: Gamma & Digamma Phase Plots
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►This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices.
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►The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.
8: Vadim B. Kuznetsov
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►Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics.
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9: 18.38 Mathematical Applications
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Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions
►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. …10: 17.17 Physical Applications
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►They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics.
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