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 over (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 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 .

[<span style="color:red">red</span> curve]

[<span style="color:blue">blue</span> curve]

[<span style="color:orange">orange</span> 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

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

References