Half-exponential function: Difference between revisions

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{{Short description|Functional square root of an exponential}}

In [[mathematics]], a '''half-exponential function''' is a [[functional square root]] of an [[exponential function]]. That is, a [[function (mathematics)|function]] <math>f</math> such that <math>f</math> [[function composition|composed]] with itself results in an exponential function:{{r|sqrtexp|miltersen}}

<math display=block>f\bigl(f(x)\bigr) = ab^x,</math>

for some constants {{nowrap|<math>a</math> and <math>b</math>.}}

[[Hellmuth Kneser]] first proposed a [[holomorphic function|holomorphic]] construction of the solution of <math>f\bigl(f(x)\bigr) = e^x</math> in 1950. It is closely related to the problem of extending [[tetration]] to non-integer values; the value of <math>{}^\frac{1}{2} a</math> can be understood as the value of <math>f\bigl(1)</math>, where <math>f\bigl(x)</math> satisfies <math>f\bigl(f(x)\bigr) = a^x</math>. Example values from Kneser's solution of <math>f\bigl(f(x)\bigr) = e^x</math> include <math>f\bigl(0) \approx 0.49856</math> and <math>f\bigl(1) \approx 1.64635</math>.

==Impossibility of a closed-form formula==

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==Construction==

For any givenAny exponential function can be written as the self-composition <math>f(f(x))</math>, there arefor infinitely many functionspossible whose self-compositionchoices isof <math>f</math>. 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>.{{r|croneu}}}} The function <math>f</math> is the unique solution to the [[functional equation]]

<math display=block> f (x) =

\begin{cases}

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[[File:Half-exponential_function.png|thumb|right|300px|Example of a half-exponential function]]

A simple example, which leads to <math>f</math> having a continuous first derivative <math>f'</math> everywhere, isand toalso takecauses
<math>A=f''\tfrac12ge 0</math> andeverywhere (i.e. <math>gf(x)=x+\tfrac12</math>, givingis concave-up,

and <math>f'(x)</math> increasing,

for all real <math>x</math>),

is to take <math>A=\tfrac12</math> and <math>g(x)=x+\tfrac12</math>, giving

<math display=block> f (x) =

\begin{cases}

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\end{cases}

</math>

Crone and Neuendorffer claim that there is no semi-exponential function f(x)

that is both (a) analytic and (b) always maps reals to reals.

The [[piecewise]] solution above achieves goal (b) but not (a).

Achieving goal (a) is possible by writing <math>e^x</math> as a Taylor

series based at a fixpoint Q (there are an infinitude of such fixpoints,

but they all are nonreal complex,

for example <math>Q=0.3181315+1.3372357i</math>), making

Q also be a fixpoint of f, that is <math>f(Q)=e^Q=Q</math>,

then computing the [[Taylor series|Maclaurin series]] coefficients of <math>f(x-Q)</math> one by one. This results in Kneser's construction mentioned above.

==Application==

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| last1 = Crone | first1 = Lawrence J.

| last2 = Neuendorffer | first2 = Arthur C.

| doi = 10.1016/0022-247X(88)90080-7 | doi-access = free

| issue = 2

| journal = Journal of Mathematical Analysis and Applications

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| editor2-last = Imai | editor2-first = Hiroshi

| editor3-last = Lee | editor3-first = D. T.

| editor4-last = Nakano | editor4-first = Shin{-}Ichiichi

| editor5-last = Tokuyama | editor5-first = Takeshi

| contribution = Super-polynomial versus half-exponential circuit size in the exponential hierarchy

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| title = Computing and Combinatorics, 5th Annual International Conference, COCOON '99, Tokyo, Japan, July 26–28, 1999, Proceedings

| volume = 1627

| year = 1999}}</ref>| isbn = 978-3-540-66200-6

}}</ref>

<ref name=razrud>{{cite journal

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| url = https://gdz.sub.uni-goettingen.de/id/PPN243919689_0187?tify%3D%7B%22pages%22%3A%5B62%5D%7D

| volume = 187

| year = 1950| doi = 10.1515/crll.1950.187.56 }}</ref>

<ref name=transseries>{{cite book

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| publisher = Springer-Verlag, Berlin

| series = Lecture Notes in Mathematics

| title = Transseries and realReal differentialDifferential algebraAlgebra

| volume = 1888

| year = 2006}}. See exercise 4.10, p. 91, according to which every such function has a comparable growth rate to an exponential or logarithmic function iterated an integer number of times, rather than the [[half-integer]] that would be required for a half-exponential function.</ref>

}}