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The spiral of Theodorus. (English) Zbl 1077.39002

The notation of the spiral of Theodorus is referred to the representation by P. J. Davis [Spirals from Theodorus to chaos, A. K. Peters, Wellesley, MA (1993; Zbl 0940.00002)]. The underlying so-called Theodorus function \(T(x)\) is a particular solution of the difference equation \[ f(x+ 1)= \Biggl(1+{i\over\sqrt{n+1}}\Biggr)\cdot f(x)\;(- 1< x< \infty),\;f(0)= 1.\tag{\(*\)} \] The discrete points of this spiral are considered, which can be presented in the Gaussian plane recursively by \(z_0= 1\) and \(z_{n+1}= (1+ {i\over\sqrt{n+1}})\cdot z_n\) for \(n= 0,1,\dots\). There are infinitely many possibilities for connecting these points by a continuous curve.
The question is: How can one characterize such a spiral-like curve through the “Theodorus points”? The author gives a characterisation of \(T(x)\) among the various solutions of the difference equation \((*)\) by way of monotonicity criteria.
Finally, two further interesting examples of solutions are given.

MSC:

39A10 Additive difference equations

Citations:

Zbl 0940.00002
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