# 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)

## 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.

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