Half-exponential function: Difference between revisions

Revision as of 00:47, 18 May 2022

Revision as of 00:48, 18 May 2022

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==Construction==

There are infinitely many functions whose self-composition is the same exponential function as each other. In particular, for every <math>A</math> in the [[open interval]] <math>(0,1)</math> and for every [[continuous function|continuous]] [[Monotonic function|strictly increasing]] function <math>g</math> from <math>[0,A]</math> [[surjective function|onto]] <math>[A,1]</math>, there is an extension of this function to a continuous strictly increasing function <math>f</math> on the real numbers such that {{nowrap|<math>f\bigl(f(x)\bigr)=\exp x</math>.<ref>{{cite journal

| last1 = Crone | first1 = Lawrence J.

| last2 = Neuendorffer | first2 = Arthur C.

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\begin{cases}

g (x) & \mbox{if } x \in [0,A], \\

\exp (g^{-1} (x)) & \mbox{if } x \in (A,1], \\

\exp (f ( \ln (x))) & \mbox{if } x \in (1,\infty), \\

\ln (f ( \exp (x))) & \mbox{if } x \in (-\infty,0). \\

\end{cases}

</math>