Generalized Fresnel integrals and fractal properties of related spirals

摘要

We obtain a new asymptotic expansion of generalized Fresnel integrals x(t)=∫0tcosq(s)ds for large t, where q(s)∼sp when s→∞, and p>1. The terms of the expansion are defined via a simple iterative algorithm. Using this we show that the box dimension of the related q-clothoid, also called the generalized Euler or Cornu spiral, is equal to d=2p/(2p-1). Furthermore, this curve is Minkowski measurable, and we compute its d-dimensional Minkowski content. We also find oscillatory dimension of Fresnel integrals by studying the corresponding chirps.