Content deleted Content added
m →References: Added 1 dois to journal cites using AWB (10094) |
Add two graphs of the Fabius function, and mention a few more properties Tag: nowiki added |
||
Line 1:
[[Image:Graph of the Fabius function between 0 and 1.png|thumb|Graph of the Fabius function on the interval <nowiki>[0,1]</nowiki>.]]
[[Image:Graph of the Fabius function.png|thumb|Extension of the function on the nonnegative real axis.]]
In mathematics, the '''Fabius function''' is an example of an infinitely differentiable function that is nowhere [[analytic function|analytic]], found by {{harvtxt|Fabius|1966}}.
Line 6 ⟶ 9:
where the ''ξ''<sub>''n''</sub> are [[independence (probability)|independent]] [[uniform distribution (continuous)|uniformly distributed]] [[random variable]]s on the [[unit interval]].
This function satisfies the functional equation ''f''′(''x'')=2''f''(2''x'') (where ''f''′ denotes the derivative of ''f'') for 0≤''x''≤1. There is a unique extension of ''f'' to the nonnegative real numbers which satisfies the same equation: it can be defined by ''f''(''x''+1) = 1−''f''(''x'') for 0≤''x''≤1 and ''f''(''x''+2<sup>''r''</sup>) = −''f''(''x'') for 0≤''x''≤2<sup>''r''</sup> with ''r''≥1 integer; it is strongly related to the [[Thue–Morse sequence]].
==References==
| |||