Half-exponential function

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In mathematics, a half-exponential function is a function ƒ that, if composed with itself, results in an exponential function:[1][2]

Another definition is that ƒ is half-exponential if it is non-decreasing and ƒ−1(xC) ≤ o(log x). for every C > 0.[3]

It has been proven that if a function ƒ is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then ƒ(ƒ(x)) is either subexponential or superexponential.[4][5] Thus, a Hardy L-function cannot be half-exponential.

There are infinitely many functions whose self-composition is the same exponential function as each other. In particular, for every in the open interval and for every continuous strictly increasing function g from onto , there is an extension of this function to a continuous monotonic function on the real numbers such that .[6] The function is the unique solution to the functional equation

Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.[2]

See also[edit]


  1. ^ Kneser, H. (1950). "Reelle analytische Lösungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen". Journal fur die reine und angewandte Mathematik. 187: 56–67. 
  2. ^ a b Peter Bro Miltersen; N. V. Vinodchandran; Osamu Watanabe (1999). "Super-Polynomial Versus Half-Exponential Circuit Size in the Exponential Hierarchy". Lecture Notes in Computer Science. 1627: 210–220. doi:10.1007/3-540-48686-0_21. 
  3. ^ Alexander A. Razborov; Steven Rudich (August 1997). "Natural Proofs". Journal of Computer and System Sciences. 55 (1): 24–35. doi:10.1006/jcss.1997.1494. 
  4. ^ http://mathoverflow.net/questions/45477/closed-form-functions-with-half-exponential-growth
  5. ^ "Shtetl-Optimized » Blog Archive » My Favorite Growth Rates". Scottaaronson.com. 2007-08-12. Retrieved 2014-05-20. 
  6. ^ 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