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 (the involutions of the real numbers) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation.Jeremy Gray and Karen Parshall (2007) Episodes in the History of Modern Algebra (1800–1950), American Mathematical Society, 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 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, , is , which in general is not a polynomial. is a functional square root of .
[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.Curtright, T. L. Evolution surfaces and Schröder functional methods . For the notation, see .)
See also
Abel equation Schröder's equation Flow (mathematics) Superfunction Fractional calculus
References