mixed base Heine-type transformations
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1: 1.14 Integral Transforms
§1.14 Integral Transforms
►§1.14(i) Fourier Transform
… ►§1.14(iii) Laplace Transform
… ►Fourier Transform
… ►Laplace Transform
…2: 17.9 Further Transformations of Functions
§17.9 Further Transformations of Functions
… ►F. H. Jackson’s Transformations
… ►Transformations of -Series
… ►Sears–Carlitz Transformation
… ►Mixed-Base Heine-Type Transformations
…3: 18.11 Relations to Other Functions
4: 18.7 Interrelations and Limit Relations
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§18.7(i) Linear Transformations
… ►§18.7(ii) Quadratic Transformations
… ► See §18.11(ii) for limit formulas of Mehler–Heine type.5: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►These are based on the Liouville normal form of (1.13.29).
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§1.18(viii) Mixed Spectra and Eigenfunction Expansions
… ► It is to be noted that if any of the have degenerate sub-spaces, that is subspaces of orthogonal eigenfunctions with identical eigenvalues, that in the expansions below all such distinct eigenfunctions are to be included. … ► See §18.39 for discussion of Schrödinger equations and operators. … …6: 18.39 Applications in the Physical Sciences
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►The properties of determine whether the spectrum, this being the set of eigenvalues of , is discrete, continuous, or mixed, see §1.18.
…Also presented are the analytic solutions for the , bound state, eigenfunctions and eigenvalues of the Morse oscillator which also has analytically known non-normalizable continuum eigenfunctions, thus providing an example of a mixed spectrum.
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►Brief mention of non-unit normalized solutions in the case of mixed spectra appear, but as these solutions are not OP’s details appear elsewhere, as referenced.
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►The spectrum is mixed, as in §1.18(viii), the positive energy, non-, scattering states are the subject of Chapter 33.
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►Namely for fixed the infinite set labeled by describe only the
bound states for that single , omitting the continuum briefly mentioned below, and which is the subject of Chapter 33, and so an unusual example of the mixed spectra of §1.18(viii).
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7: 1.13 Differential Equations
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Transformation of the Point at Infinity
… ►Liouville Transformation
… ►Assuming that satisfies un-mixed boundary conditions of the form … ►Transformation to Liouville normal Form
… ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, ; (ii) the corresponding (real) eigenfunctions, and , have the same number of zeros, also called nodes, for as for ; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …8: Bibliography S
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Adaptive Quasi-Monte Carlo Integration Based on MISER and VEGAS.
In Monte Carlo and Quasi-Monte Carlo Methods 2002,
pp. 393–406.
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A code to evaluate modified Bessel functions based on the continued fraction method.
Comput. Phys. Comm. 105 (2-3), pp. 263–272.
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Special Functions: A Unified Theory Based on Singularities.
Oxford Mathematical Monographs, Oxford University Press, Oxford.
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Mixed Boundary Value Problems in Potential Theory.
North-Holland Publishing Co., Amsterdam.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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