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Positive T-fraction expansions for a family of special functions

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1406)

Abstract

Positive T-fractions are studied for analytic functions of the form

$$F(n,w): = \int_0^w {\frac{{du}}{{1 + u^n }},n = 1,2,3, \ldots .}$$

It is shown that such functions can be expressed as Stieltjes transforms and that the related moments can be computed by means of recurrence relations. The positive T-fraction coefficients are then computed using quotient-difference relations and the moments. Special attention is given to the approximation and computation in the complex plane of the two functions

$$F(1,w) = Log(1 + w) and F(2,w) = Arctan(w)$$

by approximants f m (w) of the corresponding positive T-fraction. The rational functions f m (w) are two-point Padé approximants, and numerical experiments are given using various choices for the two points of interpolation. Contour maps of the number of significant digits S D(f m (w)) in the approximations f m (w) are used to describe the convergence behavior of the continued fraction at different parts of C and for different choices of interpolation points.

Keywords

  • Significant Digit
  • Branch Point
  • Continue Fraction
  • Interpolation Point
  • Recurrence Formula

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Research supported in part by the U.S. National Science Foundation under Grant #DMS-8700498.

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References

  1. M. Abramowitz and I.E. Stegun, eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, DC, 1964).

    MATH  Google Scholar 

  2. E.T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford University Press, London, 1935).

    MATH  Google Scholar 

  3. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1965).

    MATH  Google Scholar 

  4. W.B. Gragg, Truncation error bounds for T-fractions, in: W. Cheney, ed., Approximation Theory III (Academic Press, New York, 1980) 455–460.

    Google Scholar 

  5. P. Henrici, Applied and Computational Complex Analysis, Volume 2: Special Functions, Integral Transforms, Asymptotics and Continued Fractions (Wiley, New York, 1977).

    MATH  Google Scholar 

  6. P. Henrici and P. Pluger, Truncation error estimates for Stieltjes fractions, Numer. Math. 9 (1966) 120–138.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. T.H. Jefferson, Truncation error estimates for T-fractions, SIAM J. Numer. Anal. 6 (1969) 359–364.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. W.B. Jones, Multiple point Padé tables, in: E.B. Saff and R.S. Varga, eds., Padé and Rational Approximation (Academic Press, New York, 1977) 163–171.

    CrossRef  Google Scholar 

  9. W.B. Jones, O. Njåstad and W.J. Thron, Two-point Padé expansions for a family of analytic functions, J Comp and Appl Math 9 (1983) 105–123.

    CrossRef  MATH  Google Scholar 

  10. W.B. Jones and W.J. Thron, Continued Fractions: Analytic Theory and Applications, Encyclopedia of Mathematics and its Applications 11 (Addison-Wesley, Reading, MA, 1980); (distributed by: Cambridge University Press, New York).

    Google Scholar 

  11. W.B. Jones and W.J. Thron, Continued Fractions in Numerical Analysis, Applied Numerical Mathematics 4 (1988) 143–230.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. W.J. Thron, Some properties of Continued Fractions \(1 + d_0 z + K\left( {\frac{z}{{1 + d_n z}}} \right)\), Bull. Amer. Math. Soc. 54 (1948) 206–218.

    CrossRef  MathSciNet  Google Scholar 

  13. N.J. Wyshinski, Approximations for a Family of Stieltjes Transforms Associated with the Two-Point Padé Table, M.S. Thesis, Mathematics Department, University of Colorado, Boulder, CO, 1988.

    Google Scholar 

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© 1989 Springer-Verlag

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Jones, W.B., Wyshinski, N.J. (1989). Positive T-fraction expansions for a family of special functions. In: Jacobsen, L. (eds) Analytic Theory of Continued Fractions III. Lecture Notes in Mathematics, vol 1406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096165

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  • DOI: https://doi.org/10.1007/BFb0096165

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51830-3

  • Online ISBN: 978-3-540-46820-2

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