series%20representations
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1: 25.12 Polylogarithms
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►The cosine series in (25.12.7) has the elementary sum
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►For real or complex and the polylogarithm
is defined by
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►For each fixed complex the series defines an analytic function of for .
The series also converges when , provided that .
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Integral Representation
…2: 19.36 Methods of Computation
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►The incomplete integrals and can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to , accompanied by two quadratically convergent series in the case of ; compare Carlson (1965, §§5,6).
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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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►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals.
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►For series expansions of Legendre’s integrals see §19.5.
Faster convergence of power series for and can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12).
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3: Bibliography V
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Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms.
Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.
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Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions.
Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.
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Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms.
Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.
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Special Functions and the Theory of Group Representations.
American Mathematical Society, Providence, RI.
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Fourier series representation of Ferrers function
.
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4: Bibliography S
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The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions.
2nd edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York.
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Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation.
Adv. Quantum Chem. 72, pp. 95–127.
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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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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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5: Bibliography R
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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials.
Canad. J. Math. 33 (4), pp. 915–928.
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Integral representations for products of Airy functions.
Z. Angew. Math. Phys. 46 (2), pp. 159–170.
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Integral representations for products of Airy functions. II. Cubic products.
Z. Angew. Math. Phys. 48 (4), pp. 646–655.
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Integral representations for products of Airy functions. III. Quartic products.
Z. Angew. Math. Phys. 48 (4), pp. 656–664.
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Elliptic hypergeometric series on root systems.
Adv. Math. 181 (2), pp. 417–447.
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6: Bibliography M
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On the Representation of Meijer’s -Function in the Vicinity of Singular Unity.
In Complex Analysis and Applications ’81 (Varna, 1981),
pp. 383–398.
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An integral representation for the Bessel form.
J. Comput. Appl. Math. 57 (1-2), pp. 251–260.
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An integral representation for the Lambert function.
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On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation.
SIAM J. Numer. Anal. 3 (3), pp. 390–409.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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7: 11.6 Asymptotic Expansions
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►If the series on the right-hand side of (11.6.1) is truncated after terms, then the remainder term is .
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►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)).
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►These and higher coefficients can be computed via the representations in Nemes (2015b).
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8: Bibliography C
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On the representation of a large even integer as the sum of a prime and the product of at most two primes.
Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.
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On a Tricomi series representation for the generalized exponential integral.
Internat. J. Comput. Math. 31, pp. 257–262.
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On some series representations of the Hurwitz zeta function.
J. Comput. Appl. Math. 216 (1), pp. 297–305.
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Multi-Integral Representations for Associated Legendre and Ferrers Functions.
Symmetry 12 (10).
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Gauss hypergeometric representations of the Ferrers function of the second kind.
SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.
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9: Bibliography G
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Integral representations for computing real parabolic cylinder functions.
Numer. Math. 98 (1), pp. 105–134.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Stirling number representation problems.
Proc. Amer. Math. Soc. 11 (3), pp. 447–451.
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Bessel functions and representation theory. I.
J. Functional Analysis 22 (2), pp. 73–105.
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Representations of Integers as Sums of Squares.
Springer-Verlag, New York.
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10: Bibliography P
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An asymptotic representation for
.
Methods Appl. Anal. 4 (4), pp. 449–470.
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A new basis for the representation of the rotation group. Lamé and Heun polynomials.
J. Mathematical Phys. 14 (8), pp. 1130–1139.
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A numerical evaluator for the generalized hypergeometric series.
Comput. Phys. Comm. 77 (2), pp. 249–254.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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Fourier Series and Integral Transforms.
Cambridge University Press, Cambridge.
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