Half-exponential function: Difference between revisions

Revision as of 18:54, 5 July 2021

Revision as of 20:39, 17 May 2022

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

AnotherA definition is thatfunction ''ƒ'' is upper-bounded by a half-exponential function if it is [[Monotonic function|non-decreasing]] and ''ƒ''⁻¹(''x''^''C'')&nbsp;≤&nbsp;o(log&thinsp;''x'').

for every&nbsp;''C''&thinsp;>&thinsp;0.<ref>{{cite journal |author1=Alexander A. Razborov |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|doi-access=free }}</ref>

It has been proven that ifIf a function ''ƒ'' is defined using the standard arithmetic operations, exponentials, [[logarithm]]s, and [[Real number|real]]-valued constants, then ''ƒ''(''ƒ''(''x'')) is either subexponential or superexponential.<ref>{{Cite web | url=https://mathoverflow.net/q/45477 | title=Fractional iteration - "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 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]] [[Monotonic function|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 strictly increasing function <math>f</math> on the real numbers such that <math>f(f(x))=\exp x</math>.<ref>{{cite journal