# multivariate gamma function

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## 11—20 of 20 matching pages

##### 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 $R_{F}\left(x,a,y\right)=R_{-\frac{1}{4}}\left(\tfrac{3}{4},\tfrac{1}{2};a^{2},% xy\right),$ $a=\frac{1}{2}(x+y)$.
When the variables are real and distinct, the various cases of $R_{J}\left(x,y,z,p\right)$ 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 $R_{D}\left(x,y,a\right)=R_{-\frac{3}{4}}\left(\tfrac{5}{4},\tfrac{1}{2};a^{2},% xy\right),$ $a=\tfrac{1}{2}x+\tfrac{1}{2}y$.
##### 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 $\frac{{c}^{2}M}{2\pi}=ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos\theta)^{-1/2% }\cos\theta\mathrm{d}\theta=2ab\int_{-1}^{1}\frac{t\mathrm{d}t}{\sqrt{(1+t)(1-% t)(a_{3}-2abt)}}=2abI(\mathbf{e}_{5}),$
Application of (19.29.4) and (19.29.7) with $\alpha=1$, $a_{\beta}+b_{\beta}t=1-t$, $\delta=3$, and $a_{\gamma}+b_{\gamma}t=1$ yields …
19.34.7 $M=(2/{c}^{2})(\pi a^{2})(\pi b^{2})R_{-\frac{3}{2}}\left(\tfrac{3}{2},\tfrac{3% }{2};r_{+}^{2},r_{-}^{2}\right).$
##### 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 $\alpha,\beta,\gamma,\delta$ is any permutation of the numbers $1,2,3,4$, then … where the arguments of the $R_{D}$ function are, in order, $(a-b)(u-c)$, $(b-c)(a-u)$, $(a-b)(b-c)$. … $I(\mathbf{m})$ can be reduced to a linear combination of basic integrals and algebraic functions. … where $\alpha,\beta,\gamma$ 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.