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
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.
Iterates of the sine function, in the first half-period. Half-iterate, i.e., the sine's functional square root; the functional square root of that, the quarter-iterate above it, and further fractional iterates up to the 1/64th iterate. The functions below sine are six integral iterates below it, starting with the second iterate and ending with the 64th iterate. The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the inverse of sine.
Using this extension, can be shown to be approximately equal to 0.90871.