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) = 2x2 is a functional square root of g(x) = 8x4.
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.[1]
See also
References
- ^ Episodes in the History of Modern Algebra (1800-1950) by Jeremy J. Gray and Karen Hunger Parshall ISBN 978-0821843437
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