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21: 23.15 Definitions
§23.15 Definitions
23.15.6 λ ( τ ) = θ 2 4 ( 0 , q ) θ 3 4 ( 0 , q ) ;
23.15.7 J ( τ ) = ( θ 2 8 ( 0 , q ) + θ 3 8 ( 0 , q ) + θ 4 8 ( 0 , q ) ) 3 54 ( θ 1 ( 0 , q ) ) 8 ,
22: Bernard Deconinck
He is the coauthor of several Maple commands to work with Riemann surfaces and the command to compute multidimensional theta functions numerically. …
  • 23: 21.5 Modular Transformations
    §21.5(i) Riemann Theta Functions
    Equation (21.5.4) is the modular transformation property for Riemann theta functions. The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which ξ ( Γ ) is determinate: …
    §21.5(ii) Riemann Theta Functions with Characteristics
    For explicit results in the case g = 1 , see §20.7(viii).
    24: 22.2 Definitions
    where k = 1 - k 2 and the theta functions are defined in §20.2(i). …
    22.2.7 sd ( z , k ) = θ 3 2 ( 0 , q ) θ 2 ( 0 , q ) θ 4 ( 0 , q ) θ 1 ( ζ , q ) θ 3 ( ζ , q ) = 1 ds ( z , k ) ,
    22.2.9 sc ( z , k ) = θ 3 ( 0 , q ) θ 4 ( 0 , q ) θ 1 ( ζ , q ) θ 2 ( ζ , q ) = 1 cs ( z , k ) .
    The six functions containing the letter s in their two-letter name are odd in z ; the other six are even in z . In terms of Neville’s theta functions20.1) …
    25: 20.5 Infinite Products and Related Results
    §20.5 Infinite Products and Related Results
    Jacobi’s Triple Product
    §20.5(iii) Double Products
    26: 20.15 Tables
    §20.15 Tables
    Theta functions are tabulated in Jahnke and Emde (1945, p. 45). …
    20.15.1 sin α = θ 2 2 ( 0 , q ) / θ 3 2 ( 0 , q ) = k .
    Tables of Neville’s theta functions θ s ( x , q ) , θ c ( x , q ) , θ d ( x , q ) , θ n ( x , q ) (see §20.1) and their logarithmic x -derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for ε , α = 0 ( 5 ) 90 , where (in radian measure) ε = x / θ 3 2 ( 0 , q ) = π x / ( 2 K ( k ) ) , and α is defined by (20.15.1). …
    27: 20.10 Integrals
    §20.10 Integrals
    §20.10(i) Mellin Transforms with respect to the Lattice Parameter
    §20.10(ii) Laplace Transforms with respect to the Lattice Parameter
    For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193). … For further integrals of theta functions see Erdélyi et al. (1954a, pp. 61–62 and 339), Prudnikov et al. (1990, pp. 356–358), Prudnikov et al. (1992a, §3.41), and Gradshteyn and Ryzhik (2000, pp. 627–628).
    28: 21.4 Graphics
    §21.4 Graphics
    Figure 21.4.1 provides surfaces of the scaled Riemann theta function θ ^ ( z | Ω ) , with … For the scaled Riemann theta functions depicted in Figures 21.4.221.4.5
    See accompanying text
    Figure 21.4.4: A real-valued scaled Riemann theta function: θ ^ ( i x , i y | Ω 1 ) , 0 x 4 , 0 y 4 . … Magnify 3D Help
    See accompanying text
    Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: θ ^ ( x + i y , 0 , 0 | Ω 2 ) , 0 x 1 , 0 y 3 . … Magnify 3D Help
    29: 21.7 Riemann Surfaces
    §21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
    In almost all applications, a Riemann theta function is associated with a compact Riemann surface. … is a Riemann matrix and it is used to define the corresponding Riemann theta function. …
    §21.7(ii) Fay’s Trisecant Identity
    30: 19.10 Relations to Other Functions
    §19.10(i) Theta and Elliptic Functions
    For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …