Half-exponential function: Difference between revisions

Revision as of 01:12, 26 March 2021

Revision as of 18:54, 5 July 2021

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Another definition is that ''ƒ'' is half-exponential if it is [[Monotonic function|non-decreasing]] and ''ƒ''⁻¹(''x''^''C'') ≤ o(log ''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 if 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.

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| title = Functional powers near a fixed point

| volume = 132

| year = 1988| doi-access = free
}}</ref> The function <math>f</math> is the unique solution to the [[functional equation]]

:<math> f (x) =