Half-exponential function: Difference between revisions

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|year=1950

|pages=56–67}}

</ref> <ref name="miltersen">{{cite journal |authorauthor1=Peter Bro Miltersen, |author2=N. V. Vinodchandran, |author3=Osamu Watanabe |title=Super-Polynomial Versus Half-Exponential Circuit Size in the Exponential Hierarchy |journal=Lecture Notes in Computer Science |volume=1627 |year=1999 |pages=210–220 |doi=10.1007/3-540-48686-0_21}}</ref>

: <math> f(f(x)) = ab^x. \, </math>

Another definition is that ''ƒ'' is half-exponential if it is non-decreasing and ''ƒ''⁻¹(''x''^''C'')&nbsp;≤&nbsp;o(log&nbsp;''x'').

for every&nbsp;''C''&nbsp;>&nbsp;0.<ref>{{cite journal |authorauthor1=Alexander A. Razborov and |author2=Steven Rudich |title=Natural Proofs |journal=Journal of Computer and System Sciences |volume=55 |issue=1 |date=August 1997 |pages=24–35 |doi=10.1006/jcss.1997.1494}}</ref>

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.<ref>http://mathoverflow.net/questions/45477/closed-form-functions-with-half-exponential-growth</ref> <ref>{{cite web|url=http://www.scottaaronson.com/blog/?p=263#comment-7283 |title=Shtetl-Optimized » Blog Archive » My Favorite Growth Rates |publisher=Scottaaronson.com |date=2007-08-12 |accessdate=2014-05-20}}</ref> Thus, a [[Hardy_fieldHardy field#Examples|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 <math>A</math> in the open interval <math>(0,1)</math> and for every [[continuous function|continuous]] [[strictly increasing function]] ''g'' from <math>[0,A]</math> [[surjective function|onto]] <math>[A,1]</math>, there is an extension of this function to a continuous monotonic function <math>f</math> on the real numbers such that <math>f(f(x))=\exp x</math>.<ref>{{cite journal