Functional square root

Template:Short description Template:Distinguish Template:MOS 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 Template:Math is a function Template:Math satisfying Template:Math for all Template:Math.

Notation

Notations expressing that Template:Math is a functional square root of Template:Math are Template:Math and Template:MathTemplate:Citation neededTemplate:Dubious, or rather Template:Math (see Iterated function#Fractional_iterates_and_flows,_and_negative_iterates), 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.
• The solutions of Template:Math over $ℝ$ (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 Template:Math for Template:Math. Babbage noted that for any given solution Template:Math, its functional conjugate Template:Math by an arbitrary invertible function Template:Math 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 Template:Mvar-roots (including arbitrary real, negative, and infinitesimal Template:Mvar) of functions $g:ℂ\to ℂ$ 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

• Template:Math is a functional square root of Template:Math.
• A functional square root of the Template:Mvarth Chebyshev polynomial, $g(x)=T_n(x)$, is $f(x)=\cos (\sqrt{n}\arccos (x))$, which in general is not a polynomial.
• $f(x)=x/(\sqrt{2}+x(1-\sqrt{2}))$ is a functional square root of $g(x)=x/(2-x)$.

Iterates of the sine function (blue), in the first half-period. Half-iterate (orange), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) 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 (red) 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 (arcsin).

Template:Math [red curve]

Template:Math [blue curve]

Template:Math [orange curve], although this is not unique, the opposite Template:Math being a solution of Template:Math, too.

Template:Math [black curve above the orange curve]

Template:Math [dashed curve]

(See. For the notation, see [1] Template:Webarchive.)

See also

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