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Adding short description: "Nowhere analytic, infinitely differentiable function" (Shortdesc helper) |
Asymptotic |
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The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive [[dyadic rational]] arguments.
==Asymptotic==
The rate of decay of the Fabius function for <math>x \to 0^+</math> can be expressed by an analytic function satisfying an asymptotic equivalence condition <math>f'(x) \sim 2 f(2 x)</math>. Such function is not unique; here is one example constructed using elementary functions only:
:<math>f(x)=\mathcal O\left(\exp \left(-\left(\frac{1}{2}+\frac{1}{\log 2}\right) \log x-\frac{1}{2 \log 2} \log ^2\left(-\frac{\log x}{x \log 2}\right)\right)\right),\quad x \to 0^+.</math>
==References==
*{{Citation | last1=Fabius | first1=J. | title=A probabilistic example of a nowhere analytic {{math|''C''{{hsp}}<sup>∞</sup>}}-function | mr=0197656 | year=1966 | journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete | volume=5 | issue=2 | pages=173–174 | doi=10.1007/bf00536652}}
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