theta functions
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21: 23.15 Definitions
22: Bernard Deconinck
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►He is the coauthor of several Maple commands to work with Riemann surfaces and the command to compute multidimensional theta functions numerically.
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23: 21.5 Modular Transformations
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§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 is determinate: … ►§21.5(ii) Riemann Theta Functions with Characteristics
… ►For explicit results in the case , see §20.7(viii).24: 22.2 Definitions
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►where and the theta functions are defined in §20.2(i).
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22.2.7
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22.2.9
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►The six functions containing the letter in their two-letter name are odd in ; the other six are even in .
►In terms of Neville’s theta functions (§20.1)
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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
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►Tables of Neville’s theta functions
, , , (see §20.1) and their logarithmic -derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for , where (in radian measure) , and is defined by (20.15.1).
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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 , with … ►For the scaled Riemann theta functions depicted in Figures 21.4.2–21.4.5 … ► ►29: 21.6 Products
§21.6 Products
… ►Then …On using theta functions with characteristics, it becomes …Many identities involving products of theta functions can be established using these formulas. … ►For addition formulas for classical theta functions see §20.7(ii).30: 21.7 Riemann Surfaces
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