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the example f is concave-up for all real x as may be seen by, in the leftmost c range, induction. |
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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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<math>f''\ge 0</math> everywhere (i.e. <math>f(x)</math> is concave-up,
and <math>f'(x)</math> increasing,
for all real <math>x
is to take <math>A=\tfrac12</math> and <math>g(x)=x+\tfrac12</math>, giving
<math display=block> f (x) =
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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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