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Integral representations for computing real parabolic cylinder functions

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Summary.

Integral representations are derived for the parabolic cylinder functions U(a,x), V(a,x) and W(a,x) and their derivatives. The new integrals will be used in numerical algorithms based on quadrature. They follow from contour integrals in the complex plane, by using methods from asymptotic analysis (saddle point and steepest descent methods), and are stable starting points for evaluating the functions U(a,x), V(a,x) and W(a,x) and their derivatives by quadrature rules. In particular, the new representations can be used for large parameter cases. Relations of the integral representations with uniform asymptotic expansions are also given. The algorithms will be given in a future paper.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs and mathematical tables. Nat. Bur. Standards Appl. Series, 55, U.S. Government Printing Office, Washington, D.C. (paperback edition published by Dover, New York) (1964)

  2. Gil, A., Segura, J., Temme, N.M.: On non-oscillating integrals for computing inhomogeneous Airy functions. Math. Comput. 70, 1183–1194 (2001)

    Article  MathSciNet  Google Scholar 

  3. Lozier, D.W., Olver, F.W.J.: Numerical evaluation of special functions, in W. Gautschi (ed.) Mathematics of computation 1943–1993: A half-century of computational mathematics. PSAM, AMS, Providence (1994)

  4. Miller, J.C.P.: On the choice of standard solutions to Weber’s equation. Proc. Cambridge Philos. Soc. 48, 428–435 (1952)

    MathSciNet  Google Scholar 

  5. Miller, J.C.P.: Tables of Weber parabolic cylinder functions. H.M. Stationary Office, London (1955)

  6. Olver, F.W.J.: Uniform asymptotic expansions for Weber parabolic cylinder functions of large order. J. Research NBS 63B, 131–169 (1959)

    Google Scholar 

  7. Olver, F.W.J.: (1974 & 1997) Asymptotics and Special Functions. Academic Press, New York. Reprinted in 1997 by A.K. Peters

  8. Temme, N.M.: Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1, 14–24 (1994)

    MathSciNet  Google Scholar 

  9. Temme, N.M.: Special functions: An introduction to the classical functions of mathematical physics. John Wiley and Sons, New York (1996)

  10. Temme, N.M.: Numerical and asymptotic aspects of parabolic cylinder functions. J. Comp. Appl. Math. 121, 221–246 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wong, R.: Asymptotic approximations of integrals. Academic Press, New York Reprinted in 2001 by SIAM (1989)

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Authors and Affiliations

  1. Departamento de Matemáticas, Estadística y Computación, U. Cantabria, 39005, Santander, Spain

    Amparo Gil & Javier Segura

  2. CWI, P.O. Box 94079, 1090, GB Amsterdam, The Netherlands

    Nico M. Temme

Authors
  1. Amparo Gil
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  2. Javier Segura
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  3. Nico M. Temme
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Correspondence to Nico M. Temme.

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Mathematics Subject Classification (2000): 33C15, 41A60, 65D20

Revised version received July 25, 2003

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Gil, A., Segura, J. & Temme, N. Integral representations for computing real parabolic cylinder functions. Numer. Math. 98, 105–134 (2004). https://doi.org/10.1007/s00211-004-0517-x

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  • DOI: https://doi.org/10.1007/s00211-004-0517-x

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