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connection with Riemann theta functions

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1: 21.10 Methods of Computation
§21.10(ii) Riemann Theta Functions Associated with a 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
    3: 20.12 Mathematical Applications
    This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …
    4: 21.9 Integrable Equations
    §21.9 Integrable Equations
    Typical examples of such equations are the Korteweg–de Vries equation …Here, and in what follows, x , y , and t suffixes indicate partial derivatives. …
    5: Bibliography J
  • D. L. Jagerman (1974) Some properties of the Erlang loss function. Bell System Tech. J. 53, pp. 525–551.
  • D. S. Jones (2001) Asymptotics of the hypergeometric function. Math. Methods Appl. Sci. 24 (6), pp. 369–389.
  • D. S. Jones (2006) Parabolic cylinder functions of large order. J. Comput. Appl. Math. 190 (1-2), pp. 453–469.
  • N. Joshi and M. D. Kruskal (1992) The Painlevé connection problem: An asymptotic approach. I. Stud. Appl. Math. 86 (4), pp. 315–376.
  • JTEM (website) Java Tools for Experimental Mathematics
  • 6: 25.12 Polylogarithms
    When z = e i θ , 0 θ 2 π , (25.12.1) becomes … The special case z = 1 is the Riemann zeta function: ζ ( s ) = Li s ( 1 ) . … Further properties include …and … In terms of polylogarithms …
    7: 1.9 Calculus of a Complex Variable
    and when z 0 , … A domain D , say, is an open set in that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. … …
    Cauchy–Riemann Equations
    Analyticity
    8: Bibliography K
  • A. A. Karatsuba and S. M. Voronin (1992) The Riemann Zeta-Function. de Gruyter Expositions in Mathematics, Vol. 5, Walter de Gruyter & Co., Berlin.
  • M. Katsurada (2003) Asymptotic expansions of certain q -series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.
  • A. Khare and U. Sukhatme (2004) Connecting Jacobi elliptic functions with different modulus parameters. Pramana 63 (5), pp. 921–936.
  • K. S. Kölbig (1970) Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.
  • K. S. Kölbig (1972a) Complex zeros of two incomplete Riemann zeta functions. Math. Comp. 26 (118), pp. 551–565.
  • 9: Bibliography
  • C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson (1985) Chapter 16 of Ramanujan’s second notebook: Theta-functions and q -series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.
  • G. Allasia and R. Besenghi (1989) Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule. In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.), IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.
  • K. Aomoto (1987) Special value of the hypergeometric function F 2 3 and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.
  • T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
  • T. M. Apostol (1985a) Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44 (169), pp. 223–232.