Functional square root

Notation

History

• The functional square root of the exponential function was studied by Hellmuth Kneser in 1950.
• The solutions of over were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation. A particular solution is for. Babbage noted that for any given solution, its functional conjugate by an arbitrary invertible function is also a solution. In other words, the group of all invertible functions on the real line acts on the subgroup consisting of solutions to Babbage's functional equation by conjugation.

  Solutions

Examples

• is a functional square root of.
• A functional square root of the th Chebyshev polynomial,, is, which in general is not a polynomial.
• is a functional square root of.