Functional square root

In mathematics, a half iterate (sometimes called a functional square root) is a square root of a function with respect to the operation of function composition. In other words, a 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. Similarly, the functional square root of the Chebyshev polynomials g(x) = Tn(x) is f(x) = cos (√n arccos(x)) , in general not a polynomial.

One notation that expresses that f is a functional square root of g is f = g½.

The functional square root of the exponential function was studied by H. Kneser in 1950.[1]

The solutions of f(f(x)) = x over the real numbers (the involutions of the reals) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation.[2]

A systematic procedure to produce arbitrary functional n-roots (including, beyond n= ½, continuous, negative, and infinitesimal n) relies on Schröder's equation.[3] [4] [5]

Example

See also

References

  1. ^ Kneser, H. (1950). "Reelle analytische Lösungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen". Journal fur die reine und angewandte Mathematik 187: 56–67. http://resolver.sub.uni-goettingen.de/purl?GDZPPN002175851. 
  2. ^ Gray, J. and Parshall, K. (2007) "Episodes in the History of Modern Algebra (1800-1950)", AMS, ISBN 978-0821843437
  3. ^ Schröder, E. (1870). "Ueber iterirte Functionen". Mathematische Annalen 3: 296–322. doi:10.1007/BF01443992. 
  4. ^ Szekeres, G. (1958). "Regular iteration of real and complex functions". Acta Mathematica 100 (3-4): 361–376. doi:10.1007/BF02559539. 
  5. ^ Curtright, T.; Zachos, C. (2011). "Approximate solutions of functional equations". Journal of Physics A 44 (40): 405205. doi:10.1088/1751-8113/44/40/405205. 
  6. ^ Curtright, T.L. Evolution surfaces and Schröder functional methods.