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1: 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 .
2: 3.5 Quadrature
For a comprehensive survey of quadrature of highly oscillatory integrals, including multidimensional integrals, see Iserles et al. (2006). … The standard Monte Carlo method samples points uniformly from the integration region to estimate the integral and its error. …
3: 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). …
4: 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 integrals5.19(ii)) in the special case m = 1 . …
5: Bibliography C
  • J. N. L. Connor (1973) Evaluation of multidimensional canonical integrals in semiclassical collision theory. Molecular Phys. 26 (6), pp. 1371–1377.
  • 6: Software Index
    7: 21.1 Special Notation
    g , h positive integers.
    a ω line integral of the differential ω over the cycle a .
    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).
    8: 20.11 Generalizations and Analogs
    As in §20.11(ii), the modulus k of elliptic integrals19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in q -series via (20.9.1). … 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. …