Half-iterate

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 over $\mathbb$ (the involutions 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 ) of functions $g: \mathbb\rarr \mathbb$ 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, $g(x)=T_n(x)$, is $f(x) = \cos$, which in general is not a polynomial.
* $f(x) = x / (\sqrt + x(1-\sqrt))$ is a functional square root of $g(x)=x / (2-x)$.

: red curve">span style="color:red">red curve: blue curve">span style="color:blue">blue curve: orange curve">span style="color:orange">orange curve although this is not unique, the opposite being a solution of , too.
: lack curve above the orange curve: ashed curve
Using this extension, can be shown to be approximately equal to 0.90871.

(See.Curtright, T. L
Evolution surfaces and Schröder functional methods
. For the notation, se

.)

See also

*Iterated function
*Function composition
*Abel equation
*Schröder's equation
*Flow (mathematics)
* Superfunction
*Fractional calculus
* Half-exponential function

References

{{DEFAULTSORT:Functional Square Root
Functional analysis
Functional equations