View source for Functional square root

{{Short description|Function that, applied twice, gives another function}}
{{distinguish|Root of a function}}
In [[mathematics]], a '''functional square root''' (sometimes called a '''half iterate''') is a [[Square root#In rings in general|square root]] of a [[function (mathematics)|function]] with respect to the operation of [[function composition]]. In other words, a functional square root of a function {{math| ''g''}} is a function {{math|''f''}} satisfying {{math|1=''f''(''f''(''x'')) = ''g''(''x'')}} [[for all]] {{math|''x''}}.

==Notation==  
Notations expressing that {{math|''f''}} is a functional square root of {{math|''g''}} are {{math|1=''f'' = ''g''<sub>[1/2]</sub>}} and {{math|1=''f'' = ''g''<sub>1/2</sub>}}{{citation needed|date=August 2020}}{{Dubious|talk=Notation|reason=standard notation for function iterates is as superscript|date=November 2024}}, or rather {{math|1=''f'' = ''g''<sup> 1/2</sup>}} (see [[Iterated function#Fractional_iterates_and_flows,_and_negative_iterates|Iterated Function]]), although this leaves the usual ambiguity with taking the function to that power in the multiplicative sense, just as ''f ² = f ∘ f'' can be misinterpreted as ''x ↦ f''(''x'')².

==History==
*The functional square root of the [[exponential function]] (now known as a [[half-exponential function]]) was studied by [[Hellmuth Kneser]] in 1950,<ref name="sqrtexp">{{cite journal
|author=Kneser, H. |author-link=Hellmuth Kneser
|title=Reelle analytische Lösungen der Gleichung ''&phi;''(''&phi;''(''x''))&nbsp;=&nbsp;''e''<sup>''x''</sup> und verwandter Funktionalgleichungen
|journal=[[Journal für die reine und angewandte Mathematik]]
|url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002175851
|volume=187
|year=1950
|pages=56&ndash;67|doi=10.1515/crll.1950.187.56
|s2cid=118114436
}}
</ref> later providing the basis for extending [[tetration]] to non-integer heights in 2017.{{cn|date=May 2025}}
*The solutions of {{math|1=''f''(''f''(''x'')) = ''x''}} over <math>\mathbb{R}</math> (the [[involution (mathematics)|involution]]s of the [[real number]]s) were first studied by [[Charles Babbage]] in 1815, and this equation is called Babbage's [[functional equation]].<ref>[[Jeremy Gray]] and [[Karen Parshall]] (2007) ''Episodes in the History of Modern Algebra (1800–1950)'', [[American Mathematical Society]], {{ISBN|978-0-8218-4343-7}}</ref> A particular solution is {{math|1=''f''(''x'') = (''b'' − ''x'')/(1 + ''cx'')}} for {{math|''bc'' ≠ −1}}. Babbage noted that for any given solution {{math|''f''}}, its [[Topological conjugacy|functional conjugate]] {{math|Ψ<sup>−1</sup>∘ ''f'' ∘{{space|hair}}Ψ}} by an arbitrary [[invertible function|invertible]] function {{math|Ψ}} is also a solution. In other words, the [[Group (mathematics)|group]] of all invertible functions on the real line [[Group action (mathematics)|acts]] on the subset consisting of solutions to Babbage's functional equation by [[conjugacy class|conjugation]].

==Solutions==

A systematic procedure to produce ''arbitrary'' functional {{mvar|''n''}}-roots (including arbitrary real, negative, and infinitesimal {{mvar|''n''}}) of functions <math>g: \mathbb{C}\rarr \mathbb{C}</math> relies on the solutions of [[Schröder's equation]].<ref name="schr">{{cite journal |author=Schröder, E. |author-link=Ernst Schröder (mathematician) |year=1870 |title=Ueber iterirte Functionen|journal=Mathematische Annalen |volume=3 |issue= 2|pages=296&ndash;322 | doi=10.1007/BF01443992 |s2cid=116998358 |url=http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002240998 }}</ref><ref>{{cite journal |author=Szekeres, G.|author-link=George Szekeres| year=1958|title=Regular iteration of real and complex functions |journal=Acta Mathematica |volume=100|issue=3–4 |pages=361–376 |doi= 10.1007/BF02559539 |doi-access=free }}</ref><ref>{{cite journal |author= Curtright, T.|author-link= Thomas Curtright| year= 2011|author2=Zachos, C. |author-link2=Cosmas Zachos|author3=Jin, X. |title=Approximate solutions of functional equations |journal= Journal of Physics A |volume= 44|issue= 40 |pages= 405205|doi=10.1088/1751-8113/44/40/405205|arxiv=1105.3664|bibcode=2011JPhA...44N5205C|s2cid= 119142727}}</ref> Infinitely many trivial solutions exist when the [[Domain of a function|domain]] of a root function ''f'' is allowed to be sufficiently larger than that of ''g''.

==Examples==
* {{math|1=''f''(''x'') = 2''x''<sup>2</sup>}} is a functional square root of {{math|1=''g''(''x'') = 8''x''<sup>4</sup>}}.
* A functional square root of the {{mvar|n}}th [[Chebyshev polynomial]], <math>g(x)=T_n(x)</math>, is <math>f(x) = \cos{(\sqrt{n}\arccos(x))}</math>, which in general is not a [[polynomial]].
* <math>f(x) = x / (\sqrt{2} + x(1-\sqrt{2}))</math> is a functional square root of <math>g(x)=x / (2-x)</math>.

[[File:Sine_iterations.svg|center|thumb|680px|
[[Iterated function|Iterates]] of the [[Sine|sine function]] (<span style="color:blue">blue</span>), in the first half-period. Half-iterate (<span style="color:orange">orange</span>), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it, and further fractional iterates up to the 1/64th iterate. The functions below sine are six integral iterates below it, starting with the second iterate (<span style="color:red">red</span>) and ending with the 64th iterate. The <span style="color:green">green</span> envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the [[Inverse function|inverse]] of sine ([[arcsin]]).]]

:{{math|sin<sub>[2]</sub>(''x'') {{=}} sin(sin(''x''))}} [<span style="color:red">red</span> curve]
:{{math|sin<sub>[1]</sub>(''x'') {{=}} sin(''x'') {{=}} rin(rin(''x''))}} [<span style="color:blue">blue</span> curve]
:{{math|sin<sub>[{{sfrac|1|2}}]</sub>(''x'') {{=}} rin(''x'') {{=}} qin(qin(''x''))}} [<span style="color:orange">orange</span> curve], although this is not unique, the opposite {{math|- rin}} being a solution of {{math|sin {{=}} rin ∘ rin}}, too. 
:{{math|sin<sub>[{{sfrac|1|4}}]</sub>(''x'') {{=}} qin(''x'')}} [black curve above the orange curve]
:{{math|sin<sub>[–1]</sub>(''x'') {{=}} arcsin(''x'')}} [dashed curve]

Using this extension, {{math|sin<sub>[{{sfrac|1|2}}]</sub>(''1'')}} can be shown to be approximately equal to 0.90871.<ref>https://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf</ref>

(See.<ref>Curtright, T.&nbsp;L. [http://www.physics.miami.edu/~curtright/Schroeder.html Evolution surfaces and Schröder functional methods] {{Webarchive|url=https://web.archive.org/web/20141030005609/http://www.physics.miami.edu/~curtright/Schroeder.html |date=2014-10-30 }}.</ref> For the notation, see [http://www.physics.miami.edu/~curtright/TheRootsOfSin.pdf] {{Webarchive|url=https://web.archive.org/web/20221205032857/http://www.physics.miami.edu/~curtright/TheRootsOfSin.pdf |date=2022-12-05 }}.)

==See also==
{{div col|colwidth=22em}}
*[[Iterated function]]
*[[Function composition]]
*[[Abel equation]]
*[[Schröder's equation]]
*[[Flow (mathematics)]]
*[[Superfunction]]
*[[Fractional calculus]]
*[[Half-exponential function]]
{{div col end}}

==References==
{{reflist}}

{{DEFAULTSORT:Functional Square Root}}
[[Category:Functional analysis]]
[[Category:Functional equations]]