On the singular values of the incomplete Beta function

On the singular values of the incomplete Beta function

A new denition of the incomplete beta function as a distribution- valued meromorphic function is given and the finite parts of it and of its partial derivatives at the singular values are calculated and compared with formulas in the literature.

___

  • J. G. van der Corput: Introduction to the neutrix calculus, J. Analyse Math., 7 (1959/60), 291–398.
  • J. Dieudonné: Eléments d’analyse III, Chap. XVI et XVII, Gauthier-Villars, Paris (1970).
  • B. Fisher, M. Lin and S. Orankitjaroen: Results on partial derivatives of the incomplete beta function, Rostock Math. Kolloq., 72 (2019/20), 3–10.
  • I. S. Gradshteyn, I. M. Ryzhik: Table of integrals, series and products, Academic Press, New York (1980).
  • W. Gröbner, N. Hofreiter: Integraltafel, 2. Teil: Bestimmte Integrale, 5th edn., Springer, Wien (1973).
  • L. Hörmander: The analysis of linear partial differential operators. Vol. I (Distribution theory and Fourier analysis), Grundlehren Math. Wiss. 256, 2nd edn., Springer, Berlin (1990).
  • J. Horváth: Finite parts of distributions. In: Linear operators and approximation (ed. by P. L. Butzer et al.), 142–158, Birkhäuser, Basel (1972).
  • S. G. Krantz: Handbook of complex variables, Birkhäuser, Boston (1999).
  • J. Lavoine: Calcul symbolique. Distributions et pseudo-fonctions, Editions du CNRS, Paris (1959).
  • N. Ortner, P. Wagner: Distribution-valued analytic functions, Tredition, Hamburg (2013).
  • N. Ortner, P. Wagner, Fundamental solutions of linear partial differential operators, Springer, New York (2015).
  • E. Özçağ, İ. Ege and H. Gürçay: An extension of the incomplete beta function for negative integers, J. Math. Anal. Appl., 338 (2008), 984–992.
  • V. P. Palamodov: Distributions and harmonic analysis. In: Commutative harmonic analysis. Vol. III (Enc. Math. Sci. Vol. 72, ed. by N.K. Nikol’skij), 1–127, Springer, Berlin (1995).
  • M. Riesz: L’intégrale de Riemann–Liouville et le problème de Cauchy, Acta Math., 81 (1948), 1–223.
  • L. Schwartz: Théorie des distributions, 2nd edn., Hermann, Paris (1966).