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Subject Item

dbr:Half-exponential_function

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Half-exponential function

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In mathematics, a half-exponential function is a functional square root of an exponential function, that is, a function ƒ that, if composed with itself, results in an exponential function: Another definition is that ƒ is half-exponential if it is non-decreasing and ƒ−1(xC) ≤ o(log x). for every C > 0. It has been proven that if a function ƒ is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then ƒ(ƒ(x)) is either subexponential or superexponential. Thus, a Hardy L-function cannot be half-exponential.

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dbc:Analysis_of_algorithms dbc:Computational_complexity_theory

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dbo:abstract

In mathematics, a half-exponential function is a functional square root of an exponential function, that is, a function ƒ that, if composed with itself, results in an exponential function: Another definition is that ƒ is half-exponential if it is non-decreasing and ƒ−1(xC) ≤ o(log x). for every C > 0. It has been proven that if a function ƒ is defined using the standard arithmetic operations, exponentials, logarithms, and real-valued constants, then ƒ(ƒ(x)) is either subexponential or superexponential. Thus, a Hardy L-function cannot be half-exponential. There are infinitely many functions whose self-composition is the same exponential function as each other. In particular, for every in the open interval and for every continuous strictly increasing function g from onto , there is an extension of this function to a continuous strictly increasing function on the real numbers such that . The function is the unique solution to the functional equation A simple example, which leads to ƒ having a continuous first derivative everywhere, is to take and , giving Half-exponential functions are used in computational complexity theory for growth rates "intermediate" between polynomial and exponential.(See also: Iterated function, Schröder's equation, Functional square root, and Abel equation)

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