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1: 36.9 Integral Identities
§36.9 Integral Identities
36.9.9 | Ψ ( E ) ( x , y , z ) | 2 = 8 π 2 3 2 / 3 0 0 2 π ( Ai ( 1 3 1 / 3 ( x + i y + 2 z u exp ( i θ ) + 3 u 2 exp ( - 2 i θ ) ) ) Bi ( 1 3 1 / 3 ( x - i y + 2 z u exp ( - i θ ) + 3 u 2 exp ( 2 i θ ) ) ) ) u d u d θ .
2: Bibliography O
  • I. Olkin (1959) A class of integral identities with matrix argument. Duke Math. J. 26 (2), pp. 207–213.
  • 3: Bibliography W
  • H. S. Wilf and D. Zeilberger (1992a) An algorithmic proof theory for hypergeometric (ordinary and “ q ”) multisum/integral identities. Invent. Math. 108, pp. 575–633.
  • H. S. Wilf and D. Zeilberger (1992b) Rational function certification of multisum/integral/“ q identities. Bull. Amer. Math. Soc. (N.S.) 27 (1), pp. 148–153.
  • 4: 19.1 Special Notation
    R F ( x , y , z ) , R G ( x , y , z ) , and R J ( x , y , z , p ) are the symmetric (in x , y , and z ) integrals of the first, second, and third kinds; they are complete if exactly one of x , y , and z is identically 0. …
    5: 1.4 Calculus of One Variable
    1.4.32 f 2 2 a b | f ( x ) | 2 d x < .
    6: 24.7 Integral Representations
    §24.7 Integral Representations
    §24.7(i) Bernoulli and Euler Numbers
    The identities in this subsection hold for n = 1 , 2 , . …
    §24.7(ii) Bernoulli and Euler Polynomials
    Mellin–Barnes Integral
    7: 35.10 Methods of Computation
    Other methods include numerical quadrature applied to double and multiple integral representations. See Yan (1992) for the F 1 1 and F 1 2 functions of matrix argument in the case m = 2 , and Bingham et al. (1992) for Monte Carlo simulation on O ( m ) applied to a generalization of the integral (35.5.8). Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …
    8: 36.10 Differential Equations
    §36.10 Differential Equations
    K = 1 , fold: (36.10.6) is an identity. K = 2 , cusp: … K = 3 , swallowtail: … In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( x ) satisfy the following operator equations …
    9: 25.12 Polylogarithms
    10: 35.3 Multivariate Gamma and Beta Functions
    35.3.3 B m ( a , b ) = 0 < X < I | X | a - 1 2 ( m + 1 ) | I - X | b - 1 2 ( m + 1 ) d X , ( a ) , ( b ) > 1 2 ( m - 1 ) .