Functional square root
In mathematics, a functional square root 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, 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 ↦ f².History
- The functional square root of the 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 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 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.