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1: 21 Multidimensional Theta Functions
Chapter 21 Multidimensional Theta Functions
2: Bernard Deconinck
He is the coauthor of several Maple commands to work with Riemann surfaces and the command to compute multidimensional theta functions numerically. …
  • 3: 5.14 Multidimensional Integrals
    §5.14 Multidimensional Integrals
    5.14.7 1 ( 2 π ) n [ - π , π ] n 1 j < k n | e i θ j - e i θ k | 2 b d θ 1 d θ n = Γ ( 1 + b n ) ( Γ ( 1 + b ) ) n , b > - 1 / n .
    4: 21.1 Special Notation
    The function Θ ( ϕ | B ) = θ ( ϕ / ( 2 π i ) | B / ( 2 π i ) ) is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
    5: 20.2 Definitions and Periodic Properties
    §20.2(i) Fourier Series
    6: 20.11 Generalizations and Analogs
    Multidimensional theta functions with characteristics are defined in §21.2(ii) and their properties are described in §§21.3(ii), 21.5(ii), and 21.6. …
    7: 3.5 Quadrature
    For a comprehensive survey of quadrature of highly oscillatory integrals, including multidimensional integrals, see Iserles et al. (2006). … For integrals in higher dimensions, Monte Carlo methods are another—often the only—alternative. …
    8: 35.8 Generalized Hypergeometric Functions of Matrix Argument
    Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions F q p and F p p + 1 of matrix argument. …These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case m = 1 . …
    9: Software Index
    10: 2.5 Mellin Transform Methods
    The first reference also contains explicit expressions for the error terms, as do Soni (1980) and Carlson and Gustafson (1985). The Mellin transform method can also be extended to derive asymptotic expansions of multidimensional integrals having algebraic or logarithmic singularities, or both; see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 7), and McClure and Wong (1987). …