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1: 21.10 Methods of Computation
  • Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.

  • Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

  • 2: 21.7 Riemann Surfaces
    §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 2 , which is made compact by adding its points at infinity. …
    3: Bibliography G
  • P. Gianni, M. Seppälä, R. Silhol, and B. Trager (1998) Riemann surfaces, plane algebraic curves and their period matrices. J. Symbolic Comput. 26 (6), pp. 789–803.
  • 4: Bibliography B
  • E. Brieskorn and H. Knörrer (1986) Plane Algebraic Curves. Birkhäuser Verlag, Basel.
  • 5: 22.18 Mathematical Applications
    §22.18(i) Lengths and Parametrization of Plane Curves
    6: 19.30 Lengths of Plane Curves
    §19.30 Lengths of Plane Curves
    §19.30(i) Ellipse
    §19.30(ii) Hyperbola
    §19.30(iii) Bernoulli’s Lemniscate
    7: Philip J. Davis
    This immediately led to discussions among some of the project members about what might be possible, and the discovery that some interactive graphics work had already been done for the NIST Matrix Market, a publicly available repository of test matrices for comparing the effectiveness of numerical linear algebra algorithms. … Moreover, a cutting plane feature allows users to track curves of intersection produced as a moving plane cuts through the function surface. …
    8: 23.20 Mathematical Applications
    §23.20(ii) Elliptic Curves
    An algebraic curve that can be put either into the form …is an example of an elliptic curve22.18(iv)). … If a , b , then C intersects the plane 2 in a curve that is connected if Δ 4 a 3 + 27 b 2 > 0 ; if Δ < 0 , then the intersection has two components, one of which is a closed loop. …
    9: Bibliography S
  • T. C. Scott, G. Fee, J. Grotendorst, and W. Z. Zhang (2014) Numerics of the generalized Lambert W function. ACM Commun. Comput. Algebra 48 (2), pp. 42–56.
  • A. Sidi (2011) Asymptotic expansion of Mellin transforms in the complex plane. Int. J. Pure Appl. Math. 71 (3), pp. 465–480.
  • A. Sidi (2012a) Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp. 81 (280), pp. 2159–2173.
  • A. Sidi (2012b) Euler-Maclaurin expansions for integrals with arbitrary algebraic-logarithmic endpoint singularities. Constr. Approx. 36 (3), pp. 331–352.
  • J. H. Silverman and J. Tate (1992) Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, Springer-Verlag, New York.