Half-exponential function: Difference between revisions

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In [[mathematics]], a '''half-exponential function''' is a [[functional square root]] of an [[exponential function]], that is, a [[function (mathematics)|function]] ''&fnof;'' that, if [[function composition|composed]] with itself, results in an exponential function:<ref name="sqrtexp">{{cite journal

|author=Kneser, H. |authorlink=Hellmuth Kneser

|title=Reelle analytische Lösungen der Gleichung ''&phi;''(''&phi;''(''x''))&nbsp;=&nbsp;''e''^''x'' und verwandter Funktionalgleichungen

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</ref><ref name="miltersen">{{cite book |author1=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|isbn=978-3-540-66200-6 |citeseerx=10.1.1.16.2908 }}</ref>

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

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

for every&nbsp;''C''&nbspthinsp;>&nbspthinsp;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}}</ref>

It has been proven that if a function ''ƒ'' is defined using the standard arithmetic operations, exponentials, logarithms[[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]] function ''g'' from <math>[0,A]</math> [[surjective function|onto]] <math>[A,1]</math>, there is an extension of this function to a continuous monotonicstrictly increasing function <math>f</math> on the real numbers such that <math>f(f(x))=\exp x</math>.<ref>{{cite journal

| last1 = Crone | first1 = Lawrence J.

| last2 = Neuendorffer | first2 = Arthur C.