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f''(x)=2e^x / (1+2e^x)^2 > 0 in the leftmost x-range, and by induction for x>0. |
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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 Maclaurin series coefficients of <math>f(x-Q)</math> one by one.
achieving either one of those goals is possible.
==Application==
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