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1: 18.39 Applications in the Physical Sciences
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►where is a spatial coordinate, the mass of the particle with potential energy , is the reduced Planck’s constant, and a finite or infinite interval.
►Here the term represents the quantum kinetic energy of a single particle of mass , and its potential energy.
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►and , has eigenfunctions
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►, , Mohr and Taylor (2005, Table XXX, p. 71), where the relationship of to SI units is spelled out.
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►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry.
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2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►A complex linear vector space is called an inner product space if an inner product
is defined for all with the properties: (i) is complex linear in ; (ii) ; (iii) ; (iv) if then .
…Two elements and in are orthogonal if .
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►Functions for which are identified with each other.
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►, , for .
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►The adjoint of does satisfy where .
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3: Bibliography I
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IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015.
The Institute of Electrical and Electronics Engineers, Inc..
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IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017.
The Institute of Electrical and Electronics Engineers, Inc..
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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of and of Bessel functions of any real order
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Linear Algebra Appl. 194, pp. 35–70.
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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines.
Z. Angew. Math. Mech. 75 (12), pp. 917–926.
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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4: 20 Theta Functions
Chapter 20 Theta Functions
…5: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
…It can be expressed as a sum over all primes :
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6: 18.40 Methods of Computation
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►Let .
…Results of low ( to decimal digits) precision for are easily obtained for to .
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►Here is an interpolation of the abscissas , that is, , allowing differentiation by .
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►This is a challenging case as the desired on has an essential singularity at .
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►Further, exponential convergence in , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate for these OP systems on and respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a).
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7: Bibliography L
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Algorithm 917: complex double-precision evaluation of the Wright function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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On the theory of diffraction by an aperture in an infinite plane screen. I.
Phys. Rev. 74 (8), pp. 958–974.
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Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray.
Math. Comp. 17 (84), pp. 395–404.
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Integrating some infinite oscillating tails.
J. Comput. Appl. Math. 12/13, pp. 109–117.
8: 1.2 Elementary Algebra
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§1.2(iii) Partial Fractions
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1.2.41
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1.2.43
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1.2.44
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►If then, depending on , there is either no solution or there are infinitely many solutions, being the sum of a particular solution of (1.2.61) and any solution of .
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9: 25.12 Polylogarithms
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►The principal branch has a cut along the interval
and agrees with (25.12.1) when ; see also §4.2(i).
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25.12.8
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25.12.10
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