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Function that, applied twice, gives another function From Wikipedia, the free encyclopedia

In mathematics, a **functional square root** (sometimes called a **half iterate**) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function *g* is a function *f* satisfying *f*(*f*(*x*)) = *g*(*x*) for all *x*.

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Notations expressing that *f* is a functional square root of *g* are *f* = *g*_{[1/2]} and *f* = *g*_{1/2}.^{[citation needed]}

- The functional square root of the exponential function (now known as a half-exponential function) was studied by Hellmuth Kneser in 1950.
^{[1]} - The solutions of
*f*(*f*(*x*)) =*x*over (the involutions of the real numbers) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation.^{[2]}A particular solution is*f*(*x*) = (*b*−*x*)/(1 +*cx*) for*bc*≠ −1. Babbage noted that for any given solution*f*, its functional conjugate Ψ^{−1}∘*f*∘ Ψ by an arbitrary invertible function Ψ is also a solution. In other words, the group of all invertible functions on the real line acts on the subset consisting of solutions to Babbage's functional equation by conjugation.

A systematic procedure to produce *arbitrary* functional *n*-roots (including arbitrary real, negative, and infinitesimal *n*) of functions relies on the solutions of Schröder's equation.^{[3]}^{[4]}^{[5]} Infinitely many trivial solutions exist when the domain of a root function *f* is allowed to be sufficiently larger than that of *g*.

*f*(*x*) = 2*x*^{2}is a functional square root of*g*(*x*) = 8*x*^{4}.- A functional square root of the nth Chebyshev polynomial, , is , which in general is not a polynomial.
- is a functional square root of .

- sin
_{[2]}(*x*) = sin(sin(*x*)) [red curve] - sin
_{[1]}(*x*) = sin(*x*) = rin(rin(*x*)) [blue curve] - sin
_{[.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/2]}(*x*) = rin(*x*) = qin(qin(*x*)) [orange curve] - sin
_{[1/4]}(*x*) = qin(*x*) [black curve above the orange curve] - sin
_{[–1]}(*x*) = arcsin(*x*) [dashed curve]

(See.^{[6]} For the notation, see Archived 2022-12-05 at the Wayback Machine.)

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