Positive T-fraction expansions for a family of special functions

  • William B. Jones
  • Nancy J. Wyshinski
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1406)


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.


Significant Digit Branch Point Continue Fraction Interpolation Point Recurrence Formula 
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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • William B. Jones
    • 1
  • Nancy J. Wyshinski
    • 2
  1. 1.Department of MathematicsUniversity of Colorado at BoulderBoulderUSA
  2. 2.Department of MathematicsUniversity of Colorado at DenverDenverUSA

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