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 f ( 0 ) = 0 {\displaystyle f(0)=0} , the symmetry condition f ( 1 − x ) = 1 − f ( x ) {\displaystyle f(1-x)=1-f(x)} for 0 ≤ x ≤ 1 {\displaystyle 0\leq x\leq 1} , and the functional differential equation
f ′ ( x ) = 2 f ( 2 x ) {\displaystyle f'(x)=2f(2x)}
for 0 ≤ x ≤ 1 / 2 {\displaystyle 0\leq x\leq 1/2} . It follows that f ( x ) {\displaystyle f(x)} is monotone increasing for 0 ≤ x ≤ 1 {\displaystyle 0\leq x\leq 1} , with f ( 1 / 2 ) = 1 / 2 {\displaystyle f(1/2)=1/2} and f ( 1 ) = 1 {\displaystyle f(1)=1} and f ′ ( 1 − x ) = f ′ ( x ) {\displaystyle f'(1-x)=f'(x)} and f ′ ( x ) + f ′ ( 1 2 − x ) = 2 {\displaystyle f'(x)+f'({\tfrac {1}{2}}-x)=2} . All derivatives are zero at 0, i.e. f ′ ( 0 ) = f ″ ( 0 ) = f ‴ ( 0 ) = ⋯ = 0 {\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
f ^ ( z ) = ∏ m = 1 ∞ ( cos π z 2 m ) m {\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
∑ n = 1 ∞ 2 − n ξ n , {\displaystyle \sum _{n=1}^{\infty }2^{-n}\xi _{n},}
where the ξn are independent uniformly distributed random variables on the unit interval. That distribution has an expectation of 1 2 {\displaystyle {\tfrac {1}{2}}} and a variance of 1 36 {\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 + 2r) = −f(x) for 0 ≤ x ≤ 2r 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 function is closely related to the Fabius function f: u ( t ) = { f ( t + 1 ) , | t | < 1 0 , | t | ≥ 1 . {\displaystyle u(t)={\begin{cases}f(t+1),\quad |t|<1\\0,\quad |t|\geq 1\end{cases}}.} It fulfills the delay differential equation
d d t u ( t ) = 2 u ( 2 t + 1 ) − 2 u ( 2 t − 1 ) . {\displaystyle {\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 {\displaystyle f(1)=1}
f ( 1 2 ) = 1 2 {\displaystyle f({\tfrac {1}{2}})={\tfrac {1}{2}}}
f ( 1 4 ) = 5 72 {\displaystyle f({\tfrac {1}{4}})={\tfrac {5}{72}}}
f ( 1 8 ) = 1 288 {\displaystyle f({\tfrac {1}{8}})={\tfrac {1}{288}}}
f ( 1 16 ) = 143 2073600 {\displaystyle f({\tfrac {1}{16}})={\tfrac {143}{2073600}}}
f ( 1 32 ) = 19 33177600 {\displaystyle f({\tfrac {1}{32}})={\tfrac {19}{33177600}}}
f ( 1 64 ) = 1153 561842749440 {\displaystyle f({\tfrac {1}{64}})={\tfrac {1153}{561842749440}}}
f ( 1 128 ) = 583 179789679820800 {\displaystyle f({\tfrac {1}{128}})={\tfrac {583}{179789679820800}}}
with the numerators listed in OEIS: A272755 and denominators in OEIS: A272757.
Asymptotic
log f ( x ) = − log 2 x 2 log 2 + log x ⋅ log ( − log x ) log 2 − ( 1 2 + 1 + log log 2 log 2 ) log x − log 2 ( − log x ) 2 log 2 + log log 2 ⋅ log ( − log x ) log 2 + ( 6 γ 2 + 12 γ 1 − π 2 − 6 log 2 log 2 12 log 2 − 7 log 2 12 − log π 2 ) + log 2 ( − log x ) 2 log 2 ⋅ log x − log log 2 ⋅ log ( − log x ) log 2 ⋅ log x + O ( 1 log x ) {\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 x → 0 + {\displaystyle x\to 0^{+}} , where γ {\displaystyle \gamma } is Euler's constant, and γ 1 {\displaystyle \gamma _{1}} is the Stieltjes constant. Equivalently,
log f ( 2 − n ) = − n 2 log 2 2 − n log n + ( 1 + log 2 2 ) n − log 2 n 2 log 2 + ( 6 γ 2 + 12 γ 1 − π 2 12 log 2 − 7 log 2 12 − log π 2 ) − log 2 n 2 n log 2 2 + O ( 1 n ) {\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 n → ∞ {\displaystyle n\to \infty } .
References
Fabius, J. (1966), "A probabilistic example of a nowhere analytic C∞-function", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 5 (2): 173–174, doi:10.1007/bf00536652, MR 0197656, S2CID 122126180 Jessen, Børge; Wintner, Aurel (1935), "Distribution functions and the Riemann zeta function", Trans. Amer. Math. Soc., 38: 48–88, doi:10.1090/S0002-9947-1935-1501802-5, MR 1501802 Dimitrov, Youri (2006). Polynomially-divided solutions of bipartite self-differential functional equations (Thesis). Arias de Reyna, Juan (2017). "Arithmetic of the Fabius function". arXiv:1702.06487 [math.NT]. Arias de Reyna, Juan (2017). "An infinitely differentiable function with compact support: Definition and properties". arXiv:1702.05442 [math.CA]. (an English translation of the author's paper published in Spanish in 1982) Alkauskas, Giedrius (2001), Dirichlet series associated with Thue–Morse sequence, preprint. Rvachev, V. L.; Rvachev, V. A. (1979), Non-classical methods of the approximation theory in boundary value problems (in Russian), Kiev: Naukova Dumka