Fabius function

In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966).

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

${\displaystyle f'(x)=2f(2x)}$

for ⁠${\displaystyle 0\leq x\leq 1/2}$⁠. It follows that ${\displaystyle f(x)}$ is monotone increasing for ⁠${\displaystyle 0\leq x\leq 1}$⁠, with ${\displaystyle f(1/2)=1/2}$ and ${\displaystyle f(1)=1}$ and ${\displaystyle f'(1-x)=f'(x)}$ and ⁠${\displaystyle f'(x)+f'({\tfrac {1}{2}}-x)=2}$⁠. All derivatives are zero at 0, i.e. ${\displaystyle f'(0)=f''(0)=f'''(0)=\cdots =0}$, and are also all zero at all positive integers.

It was also written down as the Fourier transform of

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

by Børge Jessen and Aurel Wintner (1935).

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

${\displaystyle \sum _{n=1}^{\infty }2^{-n}\xi _{n},}$

where the ξ_n are independent uniformly distributed random variables on the unit interval. That distribution has an expectation of ${\displaystyle {\tfrac {1}{2}}}$ and a variance of ⁠${\displaystyle {\tfrac {1}{36}}}$⁠.

There is a unique extension of f to the real numbers that satisfies the same differential equation for all x. This extension can be defined by f(x) = 0 for x ≤ 0, f(x + 1) = 1 − f(x) for 0 ≤ x ≤ 1, and f(x + 2^r) = −f(x) for 0 ≤ x ≤ 2^r with r 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 function¹ is closely related to the Fabius function f: $${\displaystyle u(t)={\begin{cases}f(t+1),\quad |t|<1\\0,\quad |t|\geq 1\end{cases}}.}$$ It fulfills the delay differential equation² $${\displaystyle {\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).}$$ (See Delay differential equation for another example.)

The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive dyadic rational arguments. For example:³⁴

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

with the numerators listed in OEIS: A272755 and denominators in OEIS: A272757.

${\displaystyle {\begin{aligned}\log f(x)&=-{\frac {\log ^{2}x}{2\log 2}}+{\frac {\log x\cdot \log(-\log x)}{\log 2}}-\left({\frac {1}{2}}+{\frac {1+\log \log 2}{\log 2}}\right)\log x-{\frac {\log ^{2}(-\log x)}{2\log 2}}+{\frac {\log \log 2\cdot \log(-\log x)}{\log 2}}\\&+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}-6\log ^{2}\log 2}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)+{\frac {\log ^{2}(-\log x)}{2\log 2\cdot \log x}}-{\frac {\log \log 2\cdot \log(-\log x)}{\log 2\cdot \log x}}+O\!\left({\frac {1}{\log x}}\right)\end{aligned}}}$

for ⁠${\displaystyle x\to 0^{+}}$⁠, where ${\displaystyle \gamma }$ is Euler's constant, and ${\displaystyle \gamma _{1}}$ is the Stieltjes constant. Equivalently,

${\displaystyle \log f\!\left(2^{-n}\right)=-{\frac {n^{2}\log 2}{2}}-n\log n+\left(1+{\frac {\log 2}{2}}\right)n-{\frac {\log ^{2}n}{2\log 2}}+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)-{\frac {\log ^{2}n}{2n\log ^{2}2}}+O\!\left({\frac {1}{n}}\right)}$

for ⁠${\displaystyle n\to \infty }$⁠.