Iterated function: Difference between revisions

Revision as of 03:52, 7 January 2024

Revision as of 03:54, 7 January 2024

Line 16:

Let {{mvar|''X''}} be a set and {{math|''f'': ''X'' → ''X''}} be a [[function (mathematics)|function]].

Defining {{math| ''f'' ^''n''}} as the ''n''-th iterate of {{mvar|''f''}} (a notation introduced by [[Hans Heinrich Bürmann]]{{citation needed|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"/>), where ''n'' is a non-negative integer, by:

<math display="block">f^0 ~ \stackrel{\mathrm{def}}{=} ~ \operatorname{id}_X</math>

and

Line 23:

where {{math|id_''X''}} is the [[identity function]] on {{mvar|''X''}} and {{math|''f'' <math>\circ</math> ''g''}} denotes [[function composition]]. That is,

:{{math|(''f'' {{text| {{math| <math>\circ</math> }} }} ''g'')(''x'') {{=}} ''f'' (''g''(''x''))}}

Because the notation {{math|''f'' ^''n''}} may refer to both iteration (composition) of the function {{mvar|''f''}} or [[Exponentiation#Iterated functions|exponentiation of the function]] {{mvar|''f''}} (the latter is commonly used in [[trigonometric functions|trigonometry]]), some mathematicians{{citation needed|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'' ^[''n''](''x'')}} was used by [[Benjamin Peirce]]<ref name="Peirce_1852"/><ref name="Cajori_1929"/><ref group="nb">while {{math|''f'' ^(''n'')}} is taken for the [[Derivative#Lagrange's notation|{{math|''n''}}th derivative]]</ref> 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"/>