integrals%20with%20respect%20to%20degree
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1: 7.23 Tables
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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
2: 12.10 Uniform Asymptotic Expansions for Large Parameter
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►In this section we give asymptotic expansions of PCFs for large values of the parameter that are uniform with respect to the variable , when both and
are real.
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►where and are polynomials in of degree
, ( odd), ( even, ).
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►and the coefficients are the product of and a polynomial in of degree
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►The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when uniformly with respect to
.
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►where are given by (12.10.7), (12.10.23), respectively, and
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3: Bibliography C
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Introduction to the Theory of Fourier’s Series and Integrals.
3rd edition, Macmillan, London.
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Derivatives with respect to the degree and order of associated Legendre functions for using modified Bessel functions.
Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.
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A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: Application to Pearcey’s integral and its derivatives.
J. Phys. A 15 (4), pp. 1179–1190.
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Chazy’s second-degree Painlevé equations.
J. Phys. A 39 (39), pp. 11955–11971.
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Algorithm AS 24: From normal integral to deviate.
Appl. Statist. 18 (3), pp. 290–293.
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4: 19.36 Methods of Computation
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►For the polynomial of degree 7, for example, is
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►For both and the factor in Carlson (1995, (2.18)) is changed to
when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms:
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►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively.
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►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)).
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►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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5: 18.39 Applications in the Physical Sciences
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►For further details about the Schrödinger equation, including applications in physics and chemistry, see Gottfried and Yan (2004) and Pauling and Wilson (1985), respectively, among many others.
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►Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range.
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►Analogous to (18.39.7) the 3D Schrödinger operator is
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►Orthogonality and normalization of eigenfunctions of this form is respect to the measure .
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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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