representation via plane algebraic curve
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11: Bibliography B
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Integral representations and summations of the modified Struve function.
Acta Math. Hungar. 141 (3), pp. 254–281.
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A new asymptotic representation for and quantum spectral determinants.
Proc. Roy. Soc. London Ser. A 437, pp. 151–173.
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From Klein to Painlevé via Fourier, Laplace and Jimbo.
Proc. London Math. Soc. (3) 90 (1), pp. 167–208.
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Plane Algebraic Curves.
Birkhäuser Verlag, Basel.
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Riemann zeta function: Rapidly converging series and integral representations.
Appl. Math. Lett. 5 (2), pp. 83–88.
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12: 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 representations for Lamé and other special functions.
SIAM J. Math. Anal. 11 (4), pp. 702–723.
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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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Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales.
Trans. Amer. Math. Soc. 66 (1), pp. 93–134 (French).
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13: 19.30 Lengths of Plane Curves
§19.30 Lengths of Plane Curves
►§19.30(i) Ellipse
… ►§19.30(ii) Hyperbola
… ►For other plane curves with arclength representable by an elliptic integral see Greenhill (1892, p. 190) and Bowman (1953, pp. 32–33).14: 21.7 Riemann Surfaces
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§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
… ►Belokolos et al. (1994, §2.1)), they are obtainable from plane algebraic curves (Springer (1957), or Riemann (1851)). …Equation (21.7.1) determines a plane algebraic curve in , which is made compact by adding its points at infinity. … ►If a local coordinate is chosen on the Riemann surface, then the local coordinate representation of these holomorphic differentials is given by … ►These are Riemann surfaces that may be obtained from algebraic curves of the form …15: 15.17 Mathematical Applications
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►The quotient of two solutions of (15.10.1) maps the closed upper half-plane
conformally onto a curvilinear triangle.
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§15.17(iii) Group Representations
… ►These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …16: 4.13 Lambert -Function
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►Explicit representations for the are given in Kalugin and Jeffrey (2011).
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4.13.5_1
, .
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►See Jeffrey and Murdoch (2017) for an explicit representation for the in terms of associated Stirling numbers.
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►For these and other integral representations of the Lambert -function see Kheyfits (2004), Kalugin et al. (2012) and Mező (2020).
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17: 14.26 Uniform Asymptotic Expansions
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►The uniform asymptotic approximations given in §14.15 for and for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986).
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►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.
18: 25.11 Hurwitz Zeta Function
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§25.11(iii) Representations by the Euler–Maclaurin Formula
… ►§25.11(iv) Series Representations
… ►§25.11(vii) Integral Representations
… ►§25.11(viii) Further Integral Representations
… ►§25.11(x) Further Series Representations
…19: 16.17 Definition
20: Bibliography K
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Series expansions for the third incomplete elliptic integral via partial fraction decompositions.
J. Comput. Appl. Math. 207 (2), pp. 331–337.
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Closed-form representations of the Lambert function.
Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.
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Algebraic Aspects of Cryptography.
Springer-Verlag, Berlin.
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Meixner-Pollaczek polynomials and the Heisenberg algebra.
J. Math. Phys. 30 (4), pp. 767–769.
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Compact quantum groups and -special functions.
In Representations of Lie Groups and Quantum Groups,
Pitman Res. Notes Math. Ser., Vol. 311, pp. 46–128.
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