Out of curiosity I'm playing around with the concept of involutory functions. An involutory function (involution) is a function whose composition with itself is the identity function (i.e. ${f \circ f} = id$). In other words, involutions are functions that are their own inverses (i.e. $f = f^{-1}$).

Applying this concept to univariate real polynomials; it seems that all involutory polynomials have degree one(!) and belong to one of these two families:

1. $id$: $id(x) = x$ (the identity function, trivially involutory)
2. $p_1$: $p_1(x) = {-x + c}$, for some $c \in \mathbb{R}$

One well-known generalization of polynomials are infinite series, so that raises the question: Is there some pretty classification of involutory real infinite series, too?

Clarification in response to a comment: this is about "analytic functions which are involutory within some part of their region of convergence", but ideally from an infinite series perspective. If that perspective doesn't make sense, an explanation as to why it doesn't make sense would be nice.

posted 9 months ago

CC BY-SA 4.0

9mo ago

nsajko‭

16 reputation 1 0 3 5