Numerische Mathematik

, Volume 104, Issue 4, pp 445–456 | Cite as

A stable recurrence for the incomplete gamma function with imaginary second argument

  • Joris Van Deun
  • Ronald CoolsEmail author


Even though the two term recurrence relation satisfied by the incomplete gamma function is asymptotically stable in at least one direction, for an imaginary second argument there can be a considerable loss of correct digits before stability sets in. We present an approach to compute the recurrence relation to full precision, also for small values of the arguments, when the first argument is negative and the second one is purely imaginary. A detailed analysis shows that this approach works well for all values considered.

Mathematics Subject Classification (2000)

65D20 65Q05 


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  1. 1.
    Abramowitz M., Stegun I.A. (1964) Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, vol. 55 of Applied Mathematics Series. National Bureau of Standards, WashingtonGoogle Scholar
  2. 2.
    Berry M.V. (1989) Uniform asymptotic smoothing of Stokes’ discontinuities. Proc. R. Soc. Lond. A 422, 7–21zbMATHMathSciNetGoogle Scholar
  3. 3.
    Chirikjian G.S. (1996) Fredholm integral equations on the Euclidean motion group. Inverse Probl. 12, 579–599zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Conway J.T. (2000) Analytical solutions for the newtonian gravitational field induced by matter within axisymmetric boundaries. Mon. Not. R. Astron. Soc. 316, 540–554CrossRefGoogle Scholar
  5. 5.
    Davis A.M.J. (1992) Drag modifications for a sphere in a rotational motion at small non-zero Reynolds and Taylor numbers: wake interference and possible coriolis effects. J. Fluid Mech. 237, 13–22zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Davis S. (2001) Scalar field theory and the definition of momentum in curved space. Class. Quant. Grav. 18, 3395–3425zbMATHCrossRefGoogle Scholar
  7. 7.
    Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G. (1953) Higher Transcendental Functions, volume II of California Institute of Technology, Bateman Manuscript Project. McGraw-Hill Book Company, Inc., New YorkGoogle Scholar
  8. 8.
    Gaspard R., Alonso Ramirez D. (1992) Ruelle classical resonances and dynamical chaos: the three- and four-disk scatterers. Phys. Rev. A 45(12): 8383–8397MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gautschi, W.: The computation of special functions by linear difference equations. In: Elaydi, S., Győri, I., Ladas, G. (eds.) Proceedings of the Second International Conference on Difference Equations, pp. 213–243. Gordon and Breach Science Publishers, New York (1995)Google Scholar
  10. 10.
    Groote S., Körner J.G., Pivovarov A.A. (1999) On the evaluation of sunset-type Feynman diagrams. Nucl. Phys. B 542, 515–547zbMATHCrossRefGoogle Scholar
  11. 11.
    Holmes M.J., Kashyap R., Wyatt R. (1999) Physical properties of optical fiber sidetap grating filters: free space model. IEEE J. Sel. Topics in Quant. Electron. 5(5): 1353–1365CrossRefGoogle Scholar
  12. 12.
    Lotter T., Benien C., Vary P. (2003) Multichannel direction-independent speech enhancement using spectral amplitude estimation. EURASIP J. Appl. Signal Process. 11, 1147–1156zbMATHCrossRefGoogle Scholar
  13. 13.
    Miller, G.F.: Tables of generalized exponential integrals, vol. 3 of Mathematical Tables. National Physics Laboratory, Her Majesty’s Stationary Office, London (1960)Google Scholar
  14. 14.
    Olver F.W.J. (1991) Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral. SIAM J. Math. Anal. 22, 1460–1474zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Olver, F.W.J.: The generalized exponential integral. In: Zahar, R.V.M. (eds.) Approximation and Computation: A Festschrift in Honor of Walter Gautschi, vol. 119 of Internat. Ser. Num. Math. pp. 497–510. Birkhaüser (1994)Google Scholar
  16. 16.
    Paris R.B. (1994) An asymptotic representation for the Riemann zeta function on the critical line. Proc. R. Soc. Lond. A 446, 565–587zbMATHMathSciNetGoogle Scholar
  17. 17.
    Roesset J.M. (1998) Nondestructive dynamic testing of soils and pavements. Tamkang J. Sci. Eng. 1(2): 61–81Google Scholar
  18. 18.
    Stone H.A. McConnell H.M. (1995) Hydrodynamics of quantized shape transitions of lipid domains. R. Soc. Lond. Proc. Ser. A 448, 97–111CrossRefGoogle Scholar
  19. 19.
    Tanzosh J., Stone H.A. (1994) Motion of a rigid particle in a rotating viscous flow: an integral equation approach. J. Fluid Mech. 275, 225–256zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Temme N.M. (1975) Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function. Math. Comp. 29(132): 1109–1114zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Temme N.M. (1979) The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10(4): 757–766zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Temme N.M. (1996) Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters. Methods Appl. Anal. 3(3): 335–344zbMATHMathSciNetGoogle Scholar
  23. 23.
    Van Deun J., Cools R. (2005) Integrating products of Bessel functions using the incomplete Gamma function. In: Simos T.E., Psihoyios G., Tsitouras Ch. (eds) International Conference on Numerical Analysis and Applied Mathematics 2005. Wiley–VCH, New York, pp. 668–671Google Scholar
  24. 24.
    Van Deun, J., Cools, R.: Algorithm 8XX: Computing infinite range integrals of an arbitrary product of Bessel functions. ACM Trans. Math. Softw. (To appear)Google Scholar
  25. 25.
    Van Deun J., Cools R. A Matlab implementation of an algorithm for computing integrals of products of Bessel functions. In: Takayama N., Iglesias A., Gutierrez J., (eds) Proceedings of ICMS 2006, vol. 4151 of Lecture Notes in Computer Science, pp. 289–300Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Computer ScienceK.U. LeuvenLeuvenBelgium

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