In mathematics, a functional square root (sometimes called a half iterate) is a

square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...

of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

with respect to the operation of function composition. In other words, a functional square root of a function is a function satisfying

for all In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other w ...

Notation

Notations expressing that is a functional square root of are and .

History

*The functional square root of the

exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...

(now known as a

half-exponential function In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function f such that f composed with itself results in an exponential function: f\bigl(f(x)\bigr) = ab^x, for some constants Impossibi ...

) was studied by Hellmuth Kneser in 1950. *The solutions of over

\mathbb

(the

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...

s of the

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s) were first studied by Charles Babbage in 1815, and this equation is called Babbage's

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

. A particular solution is for . Babbage noted that for any given solution , its functional conjugate by an arbitrary

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

function is also a solution. In other words, the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

of all invertible functions on the real line

acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...

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

g: \mathbb\rarr \mathbb

relies on the solutions of

Schröder's equation Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function , find the function such that Schröder's equation is an eigenvalue equation for the composition operator that sen ...

. 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 In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...

. * is a functional square root of . Sine_iterations

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Evolution surfaces and Schröder functional methods
For the notation, se

)

References

Functional analysis Functional equations {{mathanalysis-stub

Notation

History

Solutions

Examples

See also

References