for squares
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11: 19.31 Probability Distributions
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and occur as the expectation values, relative to a normal probability distribution in or , of the square root or reciprocal square root of a quadratic form.
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12: Bibliography Y
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-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian.
Phys. Rev. A 11 (4), pp. 1144–1156.
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13: 20.12 Mathematical Applications
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►For applications of to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143).
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14: 28.30 Expansions in Series of Eigenfunctions
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►Then every continuous -periodic function whose second derivative is square-integrable over the interval can be expanded in a uniformly and absolutely convergent series
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15: 22.11 Fourier and Hyperbolic Series
16: 18.39 Applications in the Physical Sciences
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►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being and forming a complete set.
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►where is the (squared) angular momentum operator (14.30.12).
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►with an infinite set of orthonormal eigenfunctions
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►The bound state eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the -function normalized (non-) in Chapter 33, where the solutions appear as Whittaker functions.
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►The fact that non- continuum scattering eigenstates may be expressed in terms or (infinite) sums of functions allows a reformulation of scattering theory in atomic physics wherein no non- functions need appear.
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17: 4.35 Identities
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§4.35(ii) Squares and Products
… ►The square roots assume their principal value on the positive real axis, and are determined by continuity elsewhere. …18: 4.21 Identities
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§4.21(ii) Squares and Products
…19: 4.30 Elementary Properties
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20: 10.65 Power Series
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