Half-exponential function

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In mathematics, a half-exponential function is a function ƒ so that if ƒ is composed with itself the result is exponential:[1]

 f(f(x)) = ab^x. \,

Another definition is that ƒ is half-exponential if it is non-decreasing and ƒ−1(xC) ≤ o(log x). for every C > 0.[2]

It has been proven that every function ƒ composed of basic arithmetic operations, exponentials, and logarithms, then ƒ(ƒ(x)) is either subexponential or superexponential:[3] half-exponential functions are not expressible in terms of elementary functions.

There are infinitely many functions whose self-composition is the same exponential function as each other. In particular, for every A in the open interval (0,1) and for every continuous strictly increasing function g from [0,A] onto [A,1], there is an extension of this function to a continuous monotonic function f on the real numbers such that f(f(x))=\exp x.[4] In particular,

 f (x) =
g (x) & \mbox{if } x \in [0,A], \\
\exp (g^{-1} (x)) & \mbox{if } x \in (A,1], \\
\exp (f ( \ln (x))) & \mbox{if } x \in (1,\infty), \\
\ln (f ( \exp (x))) & \mbox{if } x \in (-\infty,0). \\

Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.[1]

See also[edit]


  1. ^ a b Peter Bro Miltersen, N. V. Vinodchandran, Osamu Watanabe. "Super-Polynomial Versus Half-Exponential Circuit Size in the Exponential Hierarchy". Lecture Notes in Computer Science Volume 1627, 1999, pp 210–220.
  2. ^ Alexander A. Razborov and Steven Rudich, "Natural Proofs", ournal of Computer and System Sciences, Volume 55, Issue 1, August 1997, pages 24–35
  3. ^ "Shtetl-Optimized » Blog Archive » My Favorite Growth Rates". Scottaaronson.com. 2007-08-12. Retrieved 2014-05-20. 
  4. ^ Crone, Lawrence J.; Neuendorffer, Arthur C. (1988), Functional powers near a fixed point, Journal of Mathematical Analysis and Applications 132 (2): 520–529, doi:10.1016/0022-247X(88)90080-7, MR 943525 .

External links[edit]

  1. http://mathoverflow.net/questions/12081/does-the-exponential-function-have-a-square-root
  2. http://mathoverflow.net/questions/45477/closed-form-functions-with-half-exponential-growth