The Functional Equation of the Square Root Spiral

Abstract

The square roots of the positive integers can be placed on a well known square root spiral. In order to characterize it, polar coordinates are introduced with θ = g(r). Then we find the general solution to the functional equation

$g\left( {\sqrt {r^2 + 1} } \right) = g(r) + \arctan \left( {\frac{1} {r}} \right) $

where g(1) = 0 and g(r) is monotone increasing for r > 0. The resulting curve θ = g(r) gives a continuous square root spiral