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LucasBrown (talk | contribs) Changing short description from "Functional square root of the exponential function" to "Functional square root of an exponential" |
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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,
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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.
==Application==
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