engleski

Generalized Fresnel integrals and fractal properties of related spirals

We obtain a new asymptotic expansion of generalized Fresnel integrals $x(t)=\int_0^t\cos q(s)\, ds$ for large $t$, where $q(s)\sim s^p$ when $s\to\infty$, 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.

generalized Fresnel integrals; generalized Euler or Cornu spiral; chirp; box dimension; Minkowski content

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