# Remarks on a Tetration extension

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### I.N. Galidakis

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Aug 9, 2007, 2:56:09 PM8/9/07
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Many people have remarked over the years that my solution to the tetration
extension x^^y is only piecewise differentiable with respect to y. At the time I
constructed the solution, I didn't even bother to examine differentiability.
Yesterday I saw that Lambert's W function explains the discrepancy for this
particular construct.

The differentiability of the particular construct basically depends on the base
x. There seem to exist solutions based on this construction which are
arbitrarily many times differentiable (which I just found), but the only C^{oo}
solution (based on this construction) is e^^y. (i.e. when the base is e).

I have added a relevant section (differentiability) to the corresponding
article, which shows the conditions which force x=e.

http://ioannis.virtualcomposer2000.com/math/exponents4.html#diff

Therefore at least one (true) C^{oo} solution exists. I say "at least one",
because from all the solutions I have seen so far, none actually exhibits a true
C^{oo} solution. At least as far as I understand the rest of the solutions
presented.
--
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/

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