Fabius function

 ( Types Of Functions ) 

In mathematics, the Fabius function is an example of an smoothness that is nowhere analytic, found by .

This function satisfies the initial condition $f(0)\; =\; 0$, the symmetry condition $f(1-x)\; =\; 1\; -\; f(x)$ for , and the functional differential equation

$f\text{'}(x)\; =\; 2\; f(2\; x)$

for . It follows that $f(x)$ is monotone increasing for , with $f(1/2)=1/2$ and $f(1)=1$ and $f\text{'}(1-x)=f\text{'}(x)$ and .

It was also written down as the Fourier transform of

$\backslash hat\{f\}(z)\; =\; \backslash prod\_\{m=1\}^\backslash infty\; \backslash left(\backslash cos\backslash frac\{\backslash pi\; z\}\{2^m\}\backslash right)^m$

by .

The Fabius function is defined on the unit interval, and is given by the cumulative distribution function of

$\backslash sum\_\{n=1\}^\backslash infty2^\{-n\}\backslash xi\_n,$

where the are independent uniformly distributed on the unit interval. That distribution has an expectation of $\backslash tfrac\{1\}\{2\}$ and a variance of .

There is a unique extension of to the real numbers that satisfies the same differential equation for all x. This extension can be defined by for , for , and for with a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the Thue–Morse sequence.

The Rvachëv up function is closely related to the Fabius function : It fulfills the delay differential equation $$\backslash frac\{d\}\{dt\}u(t)=2u(2t+1)-2u(2t-1).$$ (See Delay differential equation for another example.)

---

Values The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive dyadic rational arguments. For example:

• $f(1)=1$
• $f(\backslash tfrac1\{2\})\; =\backslash tfrac\{1\}\{2\}$
• $f(\backslash tfrac1\{4\})\; =\backslash tfrac\{5\}\{72\}$
• $f(\backslash tfrac1\{8\})\; =\backslash tfrac\{1\}\{288\}$
• $f(\backslash tfrac1\{16\})\; =\backslash tfrac\{143\}\{2073600\}$
• $f(\backslash tfrac1\{32\})\; =\backslash tfrac\{19\}\{33177600\}$
• $f(\backslash tfrac1\{64\})\; =\backslash tfrac\{1153\}\{561842749440\}$
• $f(\backslash tfrac1\{128\})\; =\backslash tfrac\{583\}\{179789679820800\}$

with the numerators listed in and denominators in .

---

Asymptotic

for , where $\backslash gamma$ is Euler's constant, and $\backslash gamma\_1$ is the Stieltjes constant. Equivalently,

$\backslash log\; f\backslash !\backslash left(2^\{-n\}\backslash right)=-\backslash frac\{n^2\backslash log2\}2-n\backslash log\; n+\backslash left(1+\backslash frac\{\backslash log2\}2\backslash right)n\; -\backslash frac\{\backslash log^2n\}\{2\backslash log2\}+\backslash left(\backslash frac\{6\backslash gamma\; ^2+12\backslash gamma\_1-\backslash pi^2\}\{12\backslash log\; 2\}-\backslash frac\{7\backslash log\; 2\}\{12\}-\backslash frac\{\backslash log\backslash pi\}2\backslash right)-\backslash frac\{\backslash log^2n\}\{2n\backslash log^22\}+O\backslash !\backslash left(\backslash frac1n\backslash right)$

for .

• (an English translation of the author's paper published in Spanish in 1982)
• , target="_blank" rel="nofollow"> preprint.

Post Comment