infinite%20partial%20fractions
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11: 18.39 Applications in the Physical Sciences
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►with an infinite set of orthonormal eigenfunctions
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►This indicates that the Laguerre polynomials appearing in (18.39.29) are not classical OP’s, and in fact, even though infinite in number for fixed , do not form a complete set.
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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►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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►These, taken together with the infinite sets of bound states for each , form complete sets.
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12: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series.
J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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Computation of infinite integrals involving Bessel functions of arbitrary order by the -transformation.
J. Comput. Appl. Math. 78 (1), pp. 125–130.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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13: 3.4 Differentiation
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§3.4(iii) Partial Derivatives
… ► … ►The results in this subsection for the partial derivatives follow from Panow (1955, Table 10). Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives. … ►14: 36.4 Bifurcation Sets
15: 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.
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16: 25.11 Hurwitz Zeta Function
17: 10.73 Physical Applications
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10.73.1
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10.73.2
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►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
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10.73.3
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10.73.4
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18: Bibliography L
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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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Note on a method for computing infinite integrals of oscillatory functions.
Proc. Cambridge Philos. Soc. 52 (4), pp. 764–768.
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Evaluating infinite integrals involving Bessel functions of arbitrary order.
J. Comput. Appl. Math. 64 (3), pp. 217–231.
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Integrating some infinite oscillating tails.
J. Comput. Appl. Math. 12/13, pp. 109–117.
19: 5.19 Mathematical Applications
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►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions.
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►By decomposition into partial fractions (§1.2(iii))
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►Many special functions can be represented as a Mellin–Barnes
integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of , the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends.
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20: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
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22.12.13