Generalized Weierstrass function

Richardson, in his 1926 investigation of turbulence, observed that the velocity field of the atmospheric wind is so erratic that it probably cannot be described by an analytic function [79]. He suggested a Weierstrass function as a candidate to represent the velocity field, since the function is continuous everywhere, but nowhere differentiable, properties he observed in the wind-field data. Here we investigate a generalization of the Weierstrass function in order to simplify some of the discussion ... 

In general, minimal surfaces display self-intersections. The most usual cases are surfaces that intersect themselves everywhere, and the "surface" wraps onto itself repeatedly, eventually densely filling the embedding space. We are only interested in translationally (or orientationally) periodic minimal surfaces, which are free of self-intersections (thereby generating a bicontinuous geometry) or periodic surfaces with limited self-intersections. Elucidation of these cases of interest requires judicious choice of the complex function R(a>) in the Weierstrass equations (1.18). 

In the second half of the nineteenth century, the theories of real numbers and sets were created (Weierstrass, Dedekind, Cantor [17]). These allow one to give a general and rigorous mathematical definition of a function. 

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