mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 . E ...

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 mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...

. 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, a ...

(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 Hellmuth Kneser (16 April 1898 – 23 August 1973) was a Baltic German mathematician, who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition for 3-manifold ...

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 real ...

s) were first studied by

Charles Babbage Charles Babbage (; 26 December 1791 – 18 October 1871) was an English polymath. A mathematician, philosopher, inventor and mechanical engineer, Babbage originated the concept of a digital programmable computer. Babbage is considered ...

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 Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the chang ...

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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

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 The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...

, , 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 exa ...

. * is a functional square root of . Sine_iterations

: red_curve.html" ;"title="span style="color:red">red curve">span style="color:red">red curve: blue_curve.html" ;"title="span style="color:blue">blue curve">span style="color:blue">blue curve: orange_curve.html" ;"title="span style="color:orange">orange curve">span style="color:orange">orange curve:

lack curve above the orange curve Lack may refer to: Places * Lack, County Fermanagh, a townland in Northern Ireland * Lack, Poland * Łąck, Poland * Lack Township, Juniata County, Pennsylvania, US Other uses * Lack (surname) * Lack (manque), a term in Lacan's psychoanalyti ...

: ashed curve (See.Curtright, T. L
Evolution surfaces and Schröder functional methods
For the notation, se

)

References

Functional analysis Functional equations {{mathanalysis-stub

Notation

History

Solutions

Examples

See also

References