Half-exponential function

Summary

In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function such that composed with itself results in an exponential function:[1][2]

for some constants and .

Impossibility of a closed-form formula edit

If a function   is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then   is either subexponential or superexponential.[3] Thus, a Hardy L-function cannot be half-exponential.

Construction edit

Any exponential function can be written as the self-composition   for infinitely many possible choices of  . In particular, for every   in the open interval   and for every continuous strictly increasing function   from   onto  , there is an extension of this function to a continuous strictly increasing function   on the real numbers such that  .[4] The function   is the unique solution to the functional equation

 
 
Example of a half-exponential function

A simple example, which leads to   having a continuous first derivative everywhere, is to take   and  , giving

 

Application edit

Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.[2] A function   grows at least as quickly as some half-exponential function (its composition with itself grows exponentially) if it is non-decreasing and  , for every  .[5]

See also edit

References edit

  1. ^ Kneser, H. (1950). "Reelle analytische Lösungen der Gleichung φ(φ(x) = ex und verwandter Funktionalgleichungen". Journal für die reine und angewandte Mathematik. 187: 56–67. doi:10.1515/crll.1950.187.56. MR 0035385.
  2. ^ a b Miltersen, Peter Bro; Vinodchandran, N. V.; Watanabe, Osamu (1999). "Super-polynomial versus half-exponential circuit size in the exponential hierarchy". In Asano, Takao; Imai, Hiroshi; Lee, D. T.; Nakano, Shin-ichi; Tokuyama, Takeshi (eds.). Computing and Combinatorics, 5th Annual International Conference, COCOON '99, Tokyo, Japan, July 26–28, 1999, Proceedings. Lecture Notes in Computer Science. Vol. 1627. Springer. pp. 210–220. doi:10.1007/3-540-48686-0_21. ISBN 978-3-540-66200-6. MR 1730337.
  3. ^ van der Hoeven, J. (2006). Transseries and Real Differential Algebra. Lecture Notes in Mathematics. Vol. 1888. Springer-Verlag, Berlin. doi:10.1007/3-540-35590-1. ISBN 978-3-540-35590-8. MR 2262194. See exercise 4.10, p. 91, according to which every such function has a comparable growth rate to an exponential or logarithmic function iterated an integer number of times, rather than the half-integer that would be required for a half-exponential function.
  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 0943525.
  5. ^ Razborov, Alexander A.; Rudich, Steven (1997). "Natural proofs". Journal of Computer and System Sciences. 55 (1): 24–35. doi:10.1006/jcss.1997.1494. MR 1473047.

External links edit

  • Does the exponential function have a (compositional) square root?
  • “Closed-form” functions with half-exponential growth