Functional Powers
If then may compose with itself; this is sometimes denoted . Thus:
Repeated composition of a function with itself is called function iteration.
The functional powers for natural follow immediately.
- By convention, the identity map on the domain of .
- If admits an inverse function, negative functional powers are defined as the opposite power of the inverse function, .
Note: If f takes its values in a ring (in particular for real or complex-valued f ), there is a risk of confusion, as f n could also stand for the n-fold product of f, e.g. f 2(x) = f(x) · f(x).
(For trigonometric functions, usually the latter is meant, at least for positive exponents. For example, in trigonometry, this superscript notation represents standard exponentiation when used with trigonometric functions: sin2(x) = sin(x) · sin(x). However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., tan−1 = arctan (≠ 1/tan).
In some cases, an expression for f in g(x) = f r(x) can be derived from the rule for g given non-integer values of r. This is called fractional iteration. For instance, a half iterate of a function f is a function g satisfying g(g(x)) = f(x). Another example would be that where f is the successor function, f r(x) = x + r. This idea can be generalized so that the iteration count becomes a continuous parameter; in this case, such a system is called a flow.
Iterated functions and flows occur naturally in the study of fractals and dynamical systems.
Read more about this topic: Function Composition
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