User:Michael Hardy/Cyclic function

Sometimes a deleted article is near the borderline between that which should be deleted and that which should be kept, and looks as if some day it could evolve into an article worth keeping. Just in case this one is such an instance, I've put a copy of it here. One concern is that even if the term "cyclic function" cannot be found in authoritative secondary sources, the topic rather than that particular name of the concept might still be treated in the literature.

Wikipedia:Articles_for_deletion/Cyclic_function.

Cyclic function

Further information: functional notation

In mathematics, a cyclic function f is a function that when iterated some finite number of times yields the identity function, thus:

$f(f(f(\cdots (f(((\cdots (x)\cdots )=x\,$

One can express this as

$f^{n}(x)=(\,\underbrace {f\circ \cdots \circ f} _{n{\text{ iterations}}}\,)(x)=x.$

for all values of x in the domain of f. The number of iterations needed is the order of cyclicity, so that if n iterations are needed then one says that f is cyclic of order n. A cyclic function of order 2 is called an involution.

Cyclic functions can be used in solving problems by substituting a function for its cyclic pair.

References

• Cyclic Functions at AoPS
• Merriam Webster
• Ruscyzk, Richard (2010). Intermediate Algebra. AoPS Incorporated.