Analytic Continuation of the Theodorus Spiral

by Jörg Waldvogel, Seminar for Applied Mathematics,
Swiss Federal Institute of Technology ETH,
CH-8092 Zürich, Switzerland


The remarkable classical pattern of the Theodorus spiral, or square root spiral, can intuitively be supplemented by a closely related spiral asymptotic to it. A ``nice'' analytic interpolating curve of the Theodorus spiral was constructed by P.J. Davis (1993) as an infinite product satisfying the same functional equation as the discrete points. We consider the analytic continuation of the Davis solution and show that it contains the supplementing spiral as a discrete subset. We also discuss efficient evaluation algorithms as well as asymptotic expansions of the analytic functions involved.

Download a preliminary version (13 pages, work in progress) : theopaper.pdf

View the presentation, ``The Theodorus Spiral: An Exercise in Functional Equations, Summation of Series, Quadrature, and Asymptotics'',
to be given at the Swiss Numerical-Analysis Colloquium, University of Basel, April 24, 2009 (34 frames) : basel_waldvogel.pdf