Research Article
BibTex RIS Cite

On the singular values of the incomplete Beta function

Year 2022, Volume: 5 Issue: 2, 93 - 104, 15.06.2022
https://doi.org/10.33205/cma.1086298

Abstract

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.

References

  • 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).
Year 2022, Volume: 5 Issue: 2, 93 - 104, 15.06.2022
https://doi.org/10.33205/cma.1086298

Abstract

References

  • 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).
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Norbert Ortner

Peter Wagner

Publication Date June 15, 2022
Published in Issue Year 2022 Volume: 5 Issue: 2

Cite

APA Ortner, N., & Wagner, P. (2022). On the singular values of the incomplete Beta function. Constructive Mathematical Analysis, 5(2), 93-104. https://doi.org/10.33205/cma.1086298
AMA Ortner N, Wagner P. On the singular values of the incomplete Beta function. CMA. June 2022;5(2):93-104. doi:10.33205/cma.1086298
Chicago Ortner, Norbert, and Peter Wagner. “On the Singular Values of the Incomplete Beta Function”. Constructive Mathematical Analysis 5, no. 2 (June 2022): 93-104. https://doi.org/10.33205/cma.1086298.
EndNote Ortner N, Wagner P (June 1, 2022) On the singular values of the incomplete Beta function. Constructive Mathematical Analysis 5 2 93–104.
IEEE N. Ortner and P. Wagner, “On the singular values of the incomplete Beta function”, CMA, vol. 5, no. 2, pp. 93–104, 2022, doi: 10.33205/cma.1086298.
ISNAD Ortner, Norbert - Wagner, Peter. “On the Singular Values of the Incomplete Beta Function”. Constructive Mathematical Analysis 5/2 (June 2022), 93-104. https://doi.org/10.33205/cma.1086298.
JAMA Ortner N, Wagner P. On the singular values of the incomplete Beta function. CMA. 2022;5:93–104.
MLA Ortner, Norbert and Peter Wagner. “On the Singular Values of the Incomplete Beta Function”. Constructive Mathematical Analysis, vol. 5, no. 2, 2022, pp. 93-104, doi:10.33205/cma.1086298.
Vancouver Ortner N, Wagner P. On the singular values of the incomplete Beta function. CMA. 2022;5(2):93-104.