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
S.F. Boys,Prog.Roy.Soc.A 200 (1950)542.
H.Taketa, S.Huzinaga and K.O-ohata,J.Phys.Soc.Jpn 21 (1966)2312.
D.Davis and E.Clementi,J.Comput.Phys.1 (1966)223.
H.F. King and M. Dupuis,J.Comput.Phys.21 (1976)144.
L.E. McMurchie and E.R. Davidson,J.Comput.Phys.26 (1978)218.
S.Obara and A.Saika,J.Comput.Phys.84 (1986)3963.
K.Ishida,J.Comput.Chem.25 (2004)739.
Tables of the Generalized Exponential Integral Functions,Harvard University,Vol.31,(Har-vard Univ.Press, Cambridge,MA,1949).
F.G. Tricomi,Ann.Mat.Pura Appl.4 (1950)263.
K.Pearson,Tables of the Incomplete Gamma Function (Cambridge, Cambridge University Press,1951).
W.Gautschi,SIAM Rev.9 (1967)24.
M. Abramowitz and I.A. Stegun,Handbook of Mathematical Functions (Dover, New York, 1972).
J.H. McCabe,Math.Comp.28 (1974)811.
N.M. Temme,J.Math.Anal.Appl.51 (1975)678.
N.M. Temme,J.Comput.Phys.3 (1975)324.
D.A. Field,Math.Comp.31 (1977)495.
W.Gautschi,ACM Trans.Math.Soft.5 (1979)482.
D.E. Amos,ACM Trans.Math.Soft.6 (1980)365.
W.H. Press, S.A. Flannery, S.A. Teukolsky and W.T. Vetterling,Numerical Recipes (Fortran Version.)(Cambridge University Press, Cambridge,1989).
D.E. Amos,ACM Trans.Math.Soft.16 (1990)178.
D.E. Amos,ACM Trans.Math.Soft.16 (1990)169.
W.J. Cody and L. Stoltz,ACM Trans.Math.Soft.17 (1992)55.
W.H. Press, S.A. Teukolsky, W.T. Vetterling and S.A. Flannery,Numerical Recipes in Fortran, 2nd ed.(Cambridge University Press, Cambridge,1992).
W.Gautschi,ACM Trans.Math.Soft.25 (1999)101.
D.M. Smith,ACM Trans.Math.Soft.27 (2001)377.
M.O. Taha,Nucl.Phys.B 10 (1969)656.
L.T. Fan and J.S. Shastry,J.Polymer Sci.7 (1973)155.
H.E. De Meyer and G.V. Berghe,J.Phys.A:Math.Gen.11 (1978)697.
J. Dunning-Davies,Eur.J.Phys.7 (1986)160.
F. Zamani-Noor D.S. Onley,Comput.Phys.Commun.48 (1988)241.
D.H. Wilkinson,Nucl.Instrum.Meth.Phys.Res.Sect.A:Accelerators,Spectrometers,Detec-tors Assoc.Equip.322 (1992)93.
P. Erhart, A.M. Portis, B. Senning and F. Waldner.J.Phys.:Condens.Matter 6 (1994)2881.
V.Magnasco, A.Rapallo and M.Casanova,Int.J.Quantum Chem.73 (1999)333.
V.I. Starikov and S.N. Mikhailenko,J.Phys.B:At.Mol.Opt.Phys.33 (2000)2141.
W. Gautschi, F.E. Harris and N.M. Temme,Appl.Math.Let.16 (2003)1095.
I.I. Guseinov and B.A. Mamedov,J.Mol.Model.8 (2002)272.
I.I. Guseinov and B.A. Mamedov,Theor.Chem.Acc.108 (2002)21.
I.I. Guseinov and B.A. Mamedov,J.Theor.Comput.Chem.1 (2002)17.
H.A. Howlader and U. Balasooriya,Biometr.J.45 (2003)901.
I.I. Guseinov, R. Aydn and B.A. Mamedov,J.Mol.Model.9 (2003)325.
F.E. Harris,Int.J.Quantum Chem.100 (2004)142.
I.S. Gradshteyn and I.M. Ryzhik,Table of Integrals,Series and Products (Academic Press, New York,1980).
I.I. Guseinov, A. Özmen, Ü. Atav and H. Yüksel,Int.J.Quantum Chem.67 (1998)199.
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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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DOI: https://doi.org/10.1023/B:JOMC.0000044521.18885.d3