Functional square root

Functional square root

In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, the functional square root of a function "g" is a function "f" satisfying "f"("f"("x")) = "g"("x") for all "x". For example, "f"("x") = 2"x"2 is a functional square root of "g"("x") = 8"x"4.

One notation that expresses that "f" is a functional square root of "g" is "f" = "g1/2.

The functional square root of the exponential function was studied by Hellmuth Kneser in 1950.

The solutions of "f"("f"("x")) = x were first studied by Charles Babbage in 1815 and this equation is called Babbage's functional equation. [Episodes in the History of Modern Algebra (1800-1950) by Jeremy J. Gray and Karen Hunger Parshall ISBN 978-0821843437]

ee also

* Fractional calculus

References


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