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multivariate gamma function

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11: Bibliography F
  • R. H. Farrell (1985) Multivariate Calculation. Use of the Continuous Groups. Springer Series in Statistics, Springer-Verlag, New York.
  • C. Ferreira, J. L. López, and E. Pérez Sinusía (2005) Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.
  • C. Ferreira and J. L. López (2001) An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.
  • J. L. Fields (1966) A note on the asymptotic expansion of a ratio of gamma functions. Proc. Edinburgh Math. Soc. (2) 15, pp. 43–45.
  • A. M. S. Filho and G. Schwachheim (1967) Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.
  • 12: 19.20 Special Cases
    In this subsection, and also §§19.20(ii)19.20(v), the variables of all R -functions satisfy the constraints specified in §19.16(i) unless other conditions are stated. …
    19.20.3 R F ( x , a , y ) = R - 1 4 ( 3 4 , 1 2 ; a 2 , x y ) , a = 1 2 ( x + y ) .
    When the variables are real and distinct, the various cases of R J ( x , y , z , p ) are called circular (hyperbolic) cases if ( p - x ) ( p - y ) ( p - z ) is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. …
    19.20.23 R D ( x , y , a ) = R - 3 4 ( 5 4 , 1 2 ; a 2 , x y ) , a = 1 2 x + 1 2 y .
    19.20.25 R - c ( b ; z ) = j = 1 n z j - b j ,
    13: Bibliography P
  • V. I. Pagurova (1963) Tablitsy nepolnoi gamma-funktsii. Vyčisl. Centr Akad. Nauk SSSR, Moscow (Russian).
  • V. I. Pagurova (1965) An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).
  • R. B. Paris (2002a) Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.
  • R. B. Paris (2002b) A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.
  • M. D. Perlman and I. Olkin (1980) Unbiasedness of invariant tests for MANOVA and other multivariate problems. Ann. Statist. 8 (6), pp. 1326–1341.
  • 14: 19.34 Mutual Inductance of Coaxial Circles
    19.34.1 c 2 M 2 π = a b 0 2 π ( h 2 + a 2 + b 2 - 2 a b cos θ ) - 1 / 2 cos θ d θ = 2 a b - 1 1 t d t ( 1 + t ) ( 1 - t ) ( a 3 - 2 a b t ) = 2 a b I ( e 5 ) ,
    Application of (19.29.4) and (19.29.7) with α = 1 , a β + b β t = 1 - t , δ = 3 , and a γ + b γ t = 1 yields …
    19.34.7 M = ( 2 / c 2 ) ( π a 2 ) ( π b 2 ) R - 3 2 ( 3 2 , 3 2 ; r + 2 , r - 2 ) .
    15: Bibliography M
  • A. J. MacLeod (1989) Algorithm AS 245. A robust and reliable algorithm for the logarithm of the gamma function. Appl. Statist. 38 (2), pp. 397–402.
  • H. R. McFarland and D. St. P. Richards (2001) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.
  • H. R. McFarland and D. St. P. Richards (2002) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.
  • C. Mortici (2013a) A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.
  • R. J. Muirhead (1982) Aspects of Multivariate Statistical Theory. John Wiley & Sons Inc., New York.
  • 16: Bibliography C
  • B. W. Char (1980) On Stieltjes’ continued fraction for the gamma function. Math. Comp. 34 (150), pp. 547–551.
  • M. A. Chaudhry, N. M. Temme, and E. J. M. Veling (1996) Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.
  • M. A. Chaudhry and S. M. Zubair (1994) Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.
  • C. K. Chui (1988) Multivariate Splines. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • A. G. Constantine (1963) Some non-central distribution problems in multivariate analysis. Ann. Math. Statist. 34 (4), pp. 1270–1285.
  • 17: Bibliography B
  • R. Barakat (1961) Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.
  • M. V. Berry (1991) Infinitely many Stokes smoothings in the gamma function. Proc. Roy. Soc. London Ser. A 434, pp. 465–472.
  • D. K. Bhaumik and S. K. Sarkar (2002) On the power function of the likelihood ratio test for MANOVA. J. Multivariate Anal. 82 (2), pp. 416–421.
  • W. G. C. Boyd (1994) Gamma function asymptotics by an extension of the method of steepest descents. Proc. Roy. Soc. London Ser. A 447, pp. 609–630.
  • T. Burić and N. Elezović (2011) Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions. J. Comput. Appl. Math. 235 (11), pp. 3315–3331.
  • 18: 19.29 Reduction of General Elliptic Integrals
    and α , β , γ , δ is any permutation of the numbers 1 , 2 , 3 , 4 , then … where the arguments of the R D function are, in order, ( a - b ) ( u - c ) , ( b - c ) ( a - u ) , ( a - b ) ( b - c ) . … I ( m ) can be reduced to a linear combination of basic integrals and algebraic functions. … where α , β , γ is any permutation of the numbers 1 , 2 , 3 , and … It depends primarily on multivariate recurrence relations that replace one integral by two or more. …
    19: Bibliography G
  • W. Gautschi (1979a) Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 482–489.
  • W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.
  • W. Gautschi (1974) A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.
  • W. Gautschi (1979b) A computational procedure for incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 466–481.
  • P. Groeneboom and D. R. Truax (2000) A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.
  • 20: Bibliography W
  • E. L. Wachspress (2000) Evaluating elliptic functions and their inverses. Comput. Math. Appl. 39 (3-4), pp. 131–136.
  • P. L. Walker (1991) Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.
  • P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
  • J. Wishart (1928) The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.
  • J. W. Wrench (1968) Concerning two series for the gamma function. Math. Comp. 22 (103), pp. 617–626.