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Evaluation of Incomplete Gamma Functions Using Downward Recursion and Analytical Relations

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Abstract

New downward recursion and analytical approaches are presented for the calculation of incomplete gamma functions γ(α,x) and Γ(α,x). The efficiency of calculation with an adequate interval of these expressions is compared with those of other methods and great numerical stability and good rate of convergence is obtained for wide range of integral parameters. It is shown that the downward recursive relations are stable for all values of α and x of γ(α,x).

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References

  1. S.F. Boys,Prog.Roy.Soc.A 200 (1950)542.

  2. H.Taketa, S.Huzinaga and K.O-ohata,J.Phys.Soc.Jpn 21 (1966)2312.

    Google Scholar 

  3. D.Davis and E.Clementi,J.Comput.Phys.1 (1966)223.

  4. H.F. King and M. Dupuis,J.Comput.Phys.21 (1976)144.

    Google Scholar 

  5. L.E. McMurchie and E.R. Davidson,J.Comput.Phys.26 (1978)218.

    Google Scholar 

  6. S.Obara and A.Saika,J.Comput.Phys.84 (1986)3963.

    Google Scholar 

  7. K.Ishida,J.Comput.Chem.25 (2004)739.

  8. Tables of the Generalized Exponential Integral Functions,Harvard University,Vol.31,(Har-vard Univ.Press, Cambridge,MA,1949).

  9. F.G. Tricomi,Ann.Mat.Pura Appl.4 (1950)263.

  10. K.Pearson,Tables of the Incomplete Gamma Function (Cambridge, Cambridge University Press,1951).

  11. W.Gautschi,SIAM Rev.9 (1967)24.

  12. M. Abramowitz and I.A. Stegun,Handbook of Mathematical Functions (Dover, New York, 1972).

    Google Scholar 

  13. J.H. McCabe,Math.Comp.28 (1974)811.

  14. N.M. Temme,J.Math.Anal.Appl.51 (1975)678.

    Google Scholar 

  15. N.M. Temme,J.Comput.Phys.3 (1975)324.

  16. D.A. Field,Math.Comp.31 (1977)495.

  17. W.Gautschi,ACM Trans.Math.Soft.5 (1979)482.

  18. D.E. Amos,ACM Trans.Math.Soft.6 (1980)365.

    Google Scholar 

  19. W.H. Press, S.A. Flannery, S.A. Teukolsky and W.T. Vetterling,Numerical Recipes (Fortran Version.)(Cambridge University Press, Cambridge,1989).

    Google Scholar 

  20. D.E. Amos,ACM Trans.Math.Soft.16 (1990)178.

    Google Scholar 

  21. D.E. Amos,ACM Trans.Math.Soft.16 (1990)169.

    Google Scholar 

  22. W.J. Cody and L. Stoltz,ACM Trans.Math.Soft.17 (1992)55.

    Google Scholar 

  23. W.H. Press, S.A. Teukolsky, W.T. Vetterling and S.A. Flannery,Numerical Recipes in Fortran, 2nd ed.(Cambridge University Press, Cambridge,1992).

    Google Scholar 

  24. W.Gautschi,ACM Trans.Math.Soft.25 (1999)101.

  25. D.M. Smith,ACM Trans.Math.Soft.27 (2001)377.

    Google Scholar 

  26. M.O. Taha,Nucl.Phys.B 10 (1969)656.

  27. L.T. Fan and J.S. Shastry,J.Polymer Sci.7 (1973)155.

    Google Scholar 

  28. H.E. De Meyer and G.V. Berghe,J.Phys.A:Math.Gen.11 (1978)697.

    Google Scholar 

  29. J. Dunning-Davies,Eur.J.Phys.7 (1986)160.

    Google Scholar 

  30. F. Zamani-Noor D.S. Onley,Comput.Phys.Commun.48 (1988)241.

    Google Scholar 

  31. D.H. Wilkinson,Nucl.Instrum.Meth.Phys.Res.Sect.A:Accelerators,Spectrometers,Detec-tors Assoc.Equip.322 (1992)93.

  32. P. Erhart, A.M. Portis, B. Senning and F. Waldner.J.Phys.:Condens.Matter 6 (1994)2881.

    Google Scholar 

  33. V.Magnasco, A.Rapallo and M.Casanova,Int.J.Quantum Chem.73 (1999)333.

    Google Scholar 

  34. V.I. Starikov and S.N. Mikhailenko,J.Phys.B:At.Mol.Opt.Phys.33 (2000)2141.

    Google Scholar 

  35. W. Gautschi, F.E. Harris and N.M. Temme,Appl.Math.Let.16 (2003)1095.

    Google Scholar 

  36. I.I. Guseinov and B.A. Mamedov,J.Mol.Model.8 (2002)272.

    Google Scholar 

  37. I.I. Guseinov and B.A. Mamedov,Theor.Chem.Acc.108 (2002)21.

    Google Scholar 

  38. I.I. Guseinov and B.A. Mamedov,J.Theor.Comput.Chem.1 (2002)17.

    Google Scholar 

  39. H.A. Howlader and U. Balasooriya,Biometr.J.45 (2003)901.

    Google Scholar 

  40. I.I. Guseinov, R. Aydn and B.A. Mamedov,J.Mol.Model.9 (2003)325.

    Google Scholar 

  41. F.E. Harris,Int.J.Quantum Chem.100 (2004)142.

  42. I.S. Gradshteyn and I.M. Ryzhik,Table of Integrals,Series and Products (Academic Press, New York,1980).

    Google Scholar 

  43. I.I. Guseinov, A. Özmen, Ü. Atav and H. Yüksel,Int.J.Quantum Chem.67 (1998)199.

    Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Faculty of Arts and Sciences, Onsekiz Mart University, Çanakkale, Turkey;

    I.I. Guseinov

  2. Department of Physics, Faculty of Arts and Sciences, Gaziosmanpaşa University, Tokat, Turkey

    B.A. Mamedov

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  1. I.I. Guseinov
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  2. B.A. Mamedov
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Correspondence to I.I. Guseinov.

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Guseinov, I., Mamedov, B. Evaluation of Incomplete Gamma Functions Using Downward Recursion and Analytical Relations. Journal of Mathematical Chemistry 36, 341–346 (2004). https://doi.org/10.1023/B:JOMC.0000044521.18885.d3

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