Editing Function composition (section)

==Functional powers==
{{main|Iterated function}}
If {{math|''Y'' [[subset|⊆]] ''X''}}, then <math>f:X\to Y</math> may compose with itself; this is sometimes denoted as <math> f^2</math>. That is:
{{block indent|em=1.5|text=<math> (f\circ f)(x) = f(f(x)) = f^2(x)</math>}}
{{block indent|em=1.5|text=<math> (f\circ f \circ f)(x) = f(f(f(x))) = f^3(x)</math>}}
{{block indent|em=1.5|text=<math> (f\circ f\circ f\circ f)(x) = f(f(f(f(x)))) = f^4(x)</math>}}

More generally, for any [[natural number]] {{math|''n'' ≥ 2}}, the {{mvar|n}}th '''functional [[exponentiation|power]]''' can be defined inductively by {{math|1=''f''&thinsp;<sup>''n''</sup> = ''f'' ∘ ''f''&thinsp;<sup>''n''−1</sup> = ''f''&thinsp;<sup>''n''−1</sup> ∘ ''f''}}, a notation introduced by [[Hans Heinrich Bürmann]]{{cn|date=August 2020|reason=The fact is undisputable, but for historical completeness, let's find Bürmann's original work on this and add here as a citation. It must be dated significantly before 1813 (according to Herschel in 1820 und Cajori in 1929.)}}<ref name="Herschel_1820"/><ref name="Cajori_1929"/> and [[John Frederick William Herschel]]<!-- in 1813 -->.<ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Peano_1903"/><ref name="Cajori_1929"/> Repeated composition of such a function with itself is called '''[[iterated function|function iteration]]'''.
* By convention, {{math|''f''&thinsp;<sup>0</sup>}} is defined as the identity map on {{math|''f''&thinsp;}}'s domain, {{math|id<sub>''X''</sub>}}.
* If {{math|1=''Y'' = ''X''}} and {{math|''f'': ''X'' → ''X''}} admits an [[inverse function]] {{math|''f''&thinsp;<sup>−1</sup>}}, negative functional powers {{math|''f''&thinsp;<sup>−''n''</sup>}} are defined for {{math|''n'' > 0}} as the [[additive inverse|negated]] power of the inverse function: {{math|1=''f''&thinsp;<sup>−''n''</sup> = (''f''&thinsp;<sup>−1</sup>)<sup>''n''</sup>}}.<ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Cajori_1929"/>

'''Note:''' If {{mvar|f}} takes its values in a [[ring (mathematics)|ring]] (in particular for real or complex-valued {{math|''f''&thinsp;}}), there is a risk of confusion, as {{math|''f''&thinsp;<sup>''n''</sup>}} could also stand for the {{mvar|n}}-fold product of&nbsp;{{mvar|f}}, e.g. {{math|1=''f''&thinsp;<sup>2</sup>(''x'') = ''f''(''x'') · ''f''(''x'')}}.<ref name="Cajori_1929"/> For trigonometric functions, usually the latter is meant, at least for positive exponents.<ref name="Cajori_1929"/> For example, in [[trigonometry]], this superscript notation represents standard [[exponentiation]] when used with [[trigonometric functions]]:

{{math|1=sin<sup>2</sup>(''x'') = sin(''x'') · sin(''x'')}}.

However, for negative exponents (especially &minus;1), it nevertheless usually refers to the inverse function, e.g., {{math|1=tan<sup>−1</sup> = arctan ≠ 1/tan}}.

In some cases, when, for a given function {{mvar|f}}, the equation {{math|1=''g'' ∘ ''g'' = ''f''}} has a unique solution {{mvar|g}}, that function can be defined as the [[functional square root]] of {{mvar|f}}, then written as {{math|1=''g'' = ''f''&thinsp;<sup>1/2</sup>}}.

More generally, when {{math|1=''g''<sup>''n''</sup> = ''f''}} has a unique solution for some natural number {{math|''n'' > 0}}, then {{math|''f''&thinsp;<sup>''m''/''n''</sup>}} can be defined as {{math|''g''<sup>''m''</sup>}}.

Under additional restrictions,<!---I guess, solvability of g^n = f for all n, and something like [[uniform convergence]] of f^(m/n) for m/n→r, is needed to define f^r for arbitrary r∈'''R'''. A citation is needed about that, anyway.---> this idea can be generalized so that the [[iterated function|iteration count]]  becomes a continuous parameter; in this case, such a system is called a [[flow (mathematics)|flow]], specified through solutions of [[Schröder's equation]]. Iterated functions and flows occur naturally in the study of [[fractals]] and [[dynamical systems]].

To avoid ambiguity, some mathematicians{{cn|date=August 2020|reason=Origin? Example authors?}} choose to use {{math|∘}} to denote the compositional meaning, writing {{math|''f''{{i sup|∘''n''}}(''x'')}} for the {{mvar|n}}-th iterate of the function {{math|''f''(''x'')}}, as in, for example, {{math|''f''{{i sup|∘3}}(''x'')}} meaning {{math|''f''(''f''(''f''(''x'')))}}. For the same purpose, {{math|''f''{{i sup|[''n'']}}(''x'')}} was used by [[Benjamin Peirce]]<ref name="Peirce_1852"/><ref name="Cajori_1929"/> whereas [[Alfred Pringsheim]] and [[Jules Molk]] suggested {{math|{{i sup|''n''}}''f''(''x'')}} instead.<ref name="Pringsheim-Molk_1907"/><ref name="Cajori_1929"/><ref group="nb" name="NB_Rucker"/>