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In mathematics, a functional square root (sometimes called a half iterate) is a square root of a
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(See.Curtright, T. L
Evolution surfaces and Schröder functional methods
For the notation, se
)
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 .
Notation
Notations expressing that is a functional square root of are and .History
*The functional square root of theexponential 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) 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-manifo ...
in 1950.
*The solutions of over (the involutions of the real numbers) 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. 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
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 change ...
.
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 thedomain
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
* ...
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 thChebyshev 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 Chebys ...
, , is , which in general is not a polynomial.
* is a functional square root of .
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(See.Curtright, T. LEvolution surfaces and Schröder functional methods
For the notation, se
)
See also
*Iterated function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function ...
* Function composition
*Abel equation
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
:f(h(x)) = h(x + 1)
or
:\alpha(f(x)) = \alpha(x)+1.
The forms are equivalent when is invertible. or control the iteration of .
Equivalence
The ...
* Schröder's equation
* Flow (mathematics)
*Superfunction
In mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function ''f'' and for some variable ''x'', the superfunction could be defined by the expression
:S(z;x) ...
* Fractional calculus
* Half-exponential function
References
Functional analysis Functional equations {{mathanalysis-stub