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Let {{mvar|''X''}} be a set and {{math|''f'': ''X'' → ''X''}} be a [[function (mathematics)|function]].
Defining {{math| ''f'' <sup>''n''</sup>}} 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.)}} 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
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