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Barnes’ beta integral

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1: 5.13 Integrals
BarnesBeta Integral
2: 16.15 Integral Representations and Integrals
§16.15 Integral Representations and Integrals
16.15.1 F 1 ( α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( α ) Γ ( γ α ) 0 1 u α 1 ( 1 u ) γ α 1 ( 1 u x ) β ( 1 u y ) β d u , α > 0 , ( γ α ) > 0 ,
16.15.2 F 2 ( α ; β , β ; γ , γ ; x , y ) = Γ ( γ ) Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ β ) Γ ( γ β ) 0 1 0 1 u β 1 v β 1 ( 1 u ) γ β 1 ( 1 v ) γ β 1 ( 1 u x v y ) α d u d v , γ > β > 0 , γ > β > 0 ,
For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8). …
3: 35.8 Generalized Hypergeometric Functions of Matrix Argument
Euler Integral
35.8.13 𝟎 < 𝐗 < 𝐈 | 𝐗 | a 1 1 2 ( m + 1 ) | 𝐈 𝐗 | b 1 a 1 1 2 ( m + 1 ) F q p ( a 2 , , a p + 1 b 2 , , b q + 1 ; 𝐓 𝐗 ) d 𝐗 = 1 B m ( b 1 a 1 , a 1 ) F q + 1 p + 1 ( a 1 , , a p + 1 b 1 , , b q + 1 ; 𝐓 ) , ( b 1 a 1 ) , ( a 1 ) > 1 2 ( m 1 ) .
§35.8(v) Mellin–Barnes Integrals
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 . …
4: Bibliography N
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.
  • G. Nemes (2014a) Error bounds and exponential improvement for the asymptotic expansion of the Barnes G -function. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2172), pp. 20140534, 14.
  • E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.
  • E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.
  • 5: Errata
  • Equation (5.17.5)
    5.17.5 Ln G ( z + 1 ) 1 4 z 2 + z Ln Γ ( z + 1 ) ( 1 2 z ( z + 1 ) + 1 12 ) ln z ln A + k = 1 B 2 k + 2 2 k ( 2 k + 1 ) ( 2 k + 2 ) z 2 k

    For consistency we have replaced Ln z by ln z .

  • Paragraph Starting from Invariants (in §23.22(ii))

    The statements “If c and d are real” and “If c and d are not both real” have been further clarified (suggested by Alan Barnes on 2021-03-26).

  • Paragraph Mellin–Barnes Integrals (in §8.6(ii))

    The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at s = 0 . The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at s = a for γ ( a , z ) . In the case of (8.6.12), it separates the poles of the gamma function from the poles at s = 0 , 1 , 2 , .

    Reported 2017-07-10 by Kurt Fischer.

  • Paragraph Case III: V ( x ) = 𝟏 𝟐 x 𝟐 + 𝟏 𝟒 β x 𝟒 (in §22.19(ii))

    Two corrections have been made in this paragraph. First, the correct range of the initial displacement a is 1 / β | a | < 2 / β . Previously it was 1 / β | a | 2 / β . Second, the correct period of the oscillations is 2 K ( k ) / η . Previously it was given incorrectly as 4 K ( k ) / η .

    Reported 2014-05-02 by Svante Janson.

  • Equation (17.13.3)
    17.13.3 0 t α 1 ( t q α + β ; q ) ( t ; q ) d t = Γ ( α ) Γ ( 1 α ) Γ q ( β ) Γ q ( 1 α ) Γ q ( α + β )

    Originally the differential was identified incorrectly as d q t ; the correct differential is d t .

    Reported 2011-04-08.

  • 6: Bibliography P
  • R. B. Paris and D. Kaminski (2001) Asymptotics and Mellin-Barnes Integrals. Cambridge University Press, Cambridge.
  • R. B. Paris (1992b) Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. J. Comput. Appl. Math. 41 (1-2), pp. 117–133.
  • P. I. Pastro (1985) Orthogonal polynomials and some q -beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.
  • K. Pearson (Ed.) (1968) Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.
  • H. N. Phien (1990) A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.