Functional square root

<h2 id="notation">Notation</h2>
Notations expressing that f is a functional square root of g are f = g[1/2] and f = g1/2[dubious – discuss], or rather f = g 1/2 (see <a href="/facts/Iterated_function/Vzsx64An">Iterated Function</a>), 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.

<h2 id="history">History</h2>
<ul><li>The functional square root of the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> (now known as a <a href="/facts/Half-exponential_function/82llf04t">half-exponential function</a>) was studied by <a href="/facts/Hellmuth_Kneser/IhDlnYET">Hellmuth Kneser</a> in 1950,<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> later providing the basis for extending <a href="/facts/Tetration/5rKeVVYv">tetration</a> to non-integer heights in 2017.</li>
<li>The solutions of f(f(x)) = x over 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 (the <a href="/facts/Involution_(mathematics)/kKxYrUVa">involutions</a> of the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>) were first studied by <a href="/facts/Charles_Babbage/T3jklQcd">Charles Babbage</a> in 1815, and this equation is called Babbage's <a href="/facts/Functional_equation/4ovU4CAt">functional equation</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> A particular solution is f(x) = (b − x)/(1 + cx) for bc ≠ −1. Babbage noted that for any given solution f, its <a href="/facts/Topological_conjugacy/vQvYmcIA">functional conjugate</a> Ψ−1∘ f ∘ Ψ by an arbitrary <a href="/facts/Invertible_function/Tg5nnXlu">invertible</a> function Ψ is also a solution. In other words, the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of all invertible functions on the real line <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">acts</a> on the subset consisting of solutions to Babbage's functional equation by <a href="/facts/Conjugacy_class/AFuPIFNW">conjugation</a>.</li></ul>
<h2 id="solutions">Solutions</h2>
A systematic procedure to produce arbitrary functional n-roots (including arbitrary real, negative, and infinitesimal n) of functions 
 
 
 
 g
 :
 
 C
 
 →
 
 C
 
 
 
 {\displaystyle g:\mathbb {C} \rightarrow \mathbb {C} }
 
 relies on the solutions of <a href="/facts/Schr%25C3%25B6der%2527s_equation/jWWTbrra">Schröder's equation</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> Infinitely many trivial solutions exist when the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of a root function f is allowed to be sufficiently larger than that of g.

<h2 id="examples">Examples</h2>
<ul><li>f(x) = 2x2 is a functional square root of g(x) = 8x4.</li>
<li>A functional square root of the nth <a href="/facts/Chebyshev_polynomial/HoBINqgC">Chebyshev polynomial</a>, 
 
 
 
 g
 (
 x
 )
 =
 
 T
 
 n
 
 
 (
 x
 )
 
 
 {\displaystyle g(x)=T_{n}(x)}
 
, is 
 
 
 
 f
 (
 x
 )
 =
 cos
 ⁡
 
 (
 
 
 n
 
 
 arccos
 ⁡
 (
 x
 )
 )
 
 
 
 {\displaystyle f(x)=\cos {({\sqrt {n}}\arccos(x))}}
 
, which in general is not a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a>.</li>
<li>
 
 
 
 f
 (
 x
 )
 =
 x
 
 /
 
 (
 
 
 2
 
 
 +
 x
 (
 1
 −
 
 
 2
 
 
 )
 )
 
 
 {\displaystyle f(x)=x/({\sqrt {2}}+x(1-{\sqrt {2}}))}
 
 is a functional square root of 
 
 
 
 g
 (
 x
 )
 =
 x
 
 /
 
 (
 2
 −
 x
 )
 
 
 {\displaystyle g(x)=x/(2-x)}
 
.</li></ul>

sin[2](x) = sin(sin(x)) [red curve]
sin[1](x) = sin(x) = rin(rin(x)) [blue curve]
sin[⁠1/2⁠](x) = rin(x) = qin(qin(x)) [orange curve], although this is not unique, the opposite - rin being a solution of sin = rin ∘ rin, too.
sin[⁠1/4⁠](x) = qin(x) [black curve above the orange curve]
sin[–1](x) = arcsin(x) [dashed curve]
Using this extension, sin[⁠1/2⁠](1) can be shown to be approximately equal to 0.90871.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
(See.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> For the notation, see <a href="http://www.physics.miami.edu/~curtright/TheRootsOfSin.pdf">[1]</a> <a href="https://web.archive.org/web/20221205032857/http://www.physics.miami.edu/~curtright/TheRootsOfSin.pdf">Archived</a> 2022-12-05 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>.)

<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Iterated_function/Vzsx64An">Iterated function</a></li>
<li><a href="/facts/Function_composition/r5Pvnmth">Function composition</a></li>
<li><a href="/facts/Abel_equation/VR16bsDB">Abel equation</a></li>
<li><a href="/facts/Schr%25C3%25B6der%2527s_equation/jWWTbrra">Schröder's equation</a></li>
<li><a href="/facts/Flow_(mathematics)/CrmJPfeV">Flow (mathematics)</a></li>
<li><a href="/facts/Superfunction/WLMaT1vv">Superfunction</a></li>
<li><a href="/facts/Fractional_calculus/PxN1pMcj">Fractional calculus</a></li>
<li><a href="/facts/Half-exponential_function/82llf04t">Half-exponential function</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Kneser, H. (1950). "Reelle analytische Lösungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen". Journal für die reine und angewandte Mathematik. 187: 56–67. doi:10.1515/crll.1950.187.56. S2CID 118114436. <a href="/wiki/Hellmuth_Kneser" target="_blank">/wiki/Hellmuth_Kneser</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Jeremy Gray and Karen Parshall (2007) Episodes in the History of Modern Algebra (1800–1950), American Mathematical Society, ISBN 978-0-8218-4343-7 <a href="/wiki/Jeremy_Gray" target="_blank">/wiki/Jeremy_Gray</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Schröder, E. (1870). "Ueber iterirte Functionen". Mathematische Annalen. 3 (2): 296–322. doi:10.1007/BF01443992. S2CID 116998358. <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" target="_blank">/wiki/Ernst_Schr%C3%B6der_(mathematician)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Szekeres, G. (1958). "Regular iteration of real and complex functions". Acta Mathematica. 100 (3–4): 361–376. doi:10.1007/BF02559539. <a href="/wiki/George_Szekeres" target="_blank">/wiki/George_Szekeres</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Curtright, T.; Zachos, C.; Jin, X. (2011). "Approximate solutions of functional equations". Journal of Physics A. 44 (40): 405205. arXiv:1105.3664. Bibcode:2011JPhA...44N5205C. doi:10.1088/1751-8113/44/40/405205. S2CID 119142727. <a href="/wiki/Thomas_Curtright" target="_blank">/wiki/Thomas_Curtright</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">https://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf <a href="https://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf" target="_blank">https://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Curtright, T. L. Evolution surfaces and Schröder functional methods Archived 2014-10-30 at the Wayback Machine. <a href="http://www.physics.miami.edu/~curtright/Schroeder.html" target="_blank">http://www.physics.miami.edu/~curtright/Schroeder.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Functional square root open-in-new

Functional square root