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inverse incomplete beta function

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1: 8.18 Asymptotic Expansions of I x ( a , b )
Inverse Function
2: Bibliography T
  • N. M. Temme (1992b) Asymptotic inversion of the incomplete beta function. J. Comput. Appl. Math. 41 (1-2), pp. 145–157.
  • 3: Bibliography D
  • A. R. DiDonato and A. H. Morris (1986) Computation of the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 12 (4), pp. 377–393.
  • A. R. DiDonato and A. H. Morris (1987) Algorithm 654: Fortran subroutines for computing the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 13 (3), pp. 318–319.
  • A. R. DiDonato and A. H. Morris (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.
  • T. M. Dunster (2006) Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2), pp. 339–353.
  • J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.
  • 4: 19.1 Special Notation
    (For other notation see Notation for the Special Functions.) … All derivatives are denoted by differentials, not by primes. … of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals … In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by K ( α ) , E ( α ) , Π ( n \ α ) , F ( ϕ \ α ) , E ( ϕ \ α ) , and Π ( n ; ϕ \ α ) , where α = arcsin k and n is the α 2 (not related to k ) in (19.1.1) and (19.1.2). … The first three functions are incomplete integrals of the first, second, and third kinds, and the cel function includes complete integrals of all three kinds.
    5: 19.7 Connection Formulas
    Provided the functions in these identities are correctly analytically continued in the complex β -plane, then the identities will also hold in the complex β -plane. … There are three relations connecting Π ( ϕ , α 2 , k ) and Π ( ϕ , ω 2 , k ) , where ω 2 is a rational function of α 2 . …
    19.7.8 Π ( ϕ , α 2 , k ) + Π ( ϕ , ω 2 , k ) = F ( ϕ , k ) + c R C ( ( c - 1 ) ( c - k 2 ) , ( c - α 2 ) ( c - ω 2 ) ) , α 2 ω 2 = k 2 .
    19.7.9 ( k 2 - α 2 ) Π ( ϕ , α 2 , k ) + ( k 2 - ω 2 ) Π ( ϕ , ω 2 , k ) = k 2 F ( ϕ , k ) - α 2 ω 2 c - 1 R C ( c ( c - k 2 ) , ( c - α 2 ) ( c - ω 2 ) ) , ( 1 - α 2 ) ( 1 - ω 2 ) = 1 - k 2 .
    19.7.10 ( 1 - α 2 ) Π ( ϕ , α 2 , k ) + ( 1 - ω 2 ) Π ( ϕ , ω 2 , k ) = F ( ϕ , k ) + ( 1 - α 2 - ω 2 ) c - k 2 R C ( c ( c - 1 ) , ( c - α 2 ) ( c - ω 2 ) ) , ( k 2 - α 2 ) ( k 2 - ω 2 ) = k 2 ( k 2 - 1 ) .
    6: 19.16 Definitions
    §19.16(ii) R - a ( b ; z )
    The R -function is often used to make a unified statement of a property of several elliptic integrals. …where B ( x , y ) is the beta function5.12) and … Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …
    7: Bibliography K
  • S. H. Khamis (1965) Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).
  • D. A. Kofke (2004) Comment on “The incomplete beta function law for parallel tempering sampling of classical canonical systems” [J. Chem. Phys. 120, 4119 (2004)]. J. Chem. Phys. 121 (2), pp. 1167.
  • K. S. Kölbig (1970) Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.
  • K. S. Kölbig (1972a) Complex zeros of two incomplete Riemann zeta functions. Math. Comp. 26 (118), pp. 551–565.
  • K. S. Kölbig (1972b) On the zeros of the incomplete gamma function. Math. Comp. 26 (119), pp. 751–755.
  • 8: Bibliography C
  • L. Carlitz (1963) The inverse of the error function. Pacific J. Math. 13 (2), pp. 459–470.
  • B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.
  • 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.
  • R. Cicchetti and A. Faraone (2004) Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics. IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.
  • 9: Bibliography R
  • Ju. M. Rappoport (1979) Tablitsy modifitsirovannykh funktsii Besselya K 1 2 + i β ( x ) . “Nauka”, Moscow (Russian).
  • H. E. Rauch and A. Lebowitz (1973) Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.
  • I. S. Reed, D. W. Tufts, X. Yu, T. K. Truong, M. T. Shih, and X. Yin (1990) Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.
  • R. Reynolds and A. Stauffer (2021) Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).
  • W. Rudin (1973) Functional Analysis. McGraw-Hill Book Co., New York.
  • 10: 3.10 Continued Fractions
    For example, by converting the Maclaurin expansion of arctan z (4.24.3), we obtain a continued fraction with the same region of convergence ( | z | 1 , z ± i ), whereas the continued fraction (4.25.4) converges for all z except on the branch cuts from i to i and - i to - i . … For applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967). … For elementary functions, see §§ 4.9 and 4.35. For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). …