Wikipedia / Functional square root
Wikipedia / Functional square root
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 is a function satisfying for all .
Notation
Notations expressing that is a functional square root of are and , or rather (see 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, later providing the basis for extending tetration to non-integer heights in 2017.
The solutions of (the involutionss of the real numbers) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation. A particular solution is for . Babbage noted that for any given solution , its functional conjugate by an arbitrary invertible function 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.
Solutions
A systematic procedure to produce arbitrary functional -roots (including arbitrary real, negative, and infinitesimal relies on the solutions of Schröder's equation. Infinitely many trivial solutions exist when the domain of a root function f is allowed to be sufficiently larger than that of g.
Examples
is a functional square root of .
A functional square root of the th Chebyshev polynomial, which in general is not a polynomial.
.
[red curve]
[blue curve]
[orange curve], although this is not unique, the opposite being a solution of , too.
[black curve above the orange curve]
[dashed curve]
Using this extension, can be shown to be approximately equal to 0.90871.
(See. For the notation, see http://www.physics.miami.edu/~curtright/TheRootsOfSin.pdf .)
See also
References