andydude Wrote:......................................
(About the inflection zeros, ... or ... flexion !? I am pertubeted by the French language. GFR comment.)
It looks as though this function is very well-behaved for \( b > e \), but is quite slow to converge for lower bases. So my guess is that GFR's conjecture that f(2)=0 is probably not true.
Two bases for which this function seems to converge quickly are \( (e, \pi) \), so I have included numerical data for these. I have been meaning to implement my own version of Jay's accelerated natural slog, but until I do that I can only use very small approximations.
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(About finding a possible "mother" differential equation. GFR comment.)
@GFR
Finding such a differential equation would be amazing! But maybe we should model it after the differential equation of exponentiation:
\( \frac{d}{dt}(x^y) = x^{y-1}\left(y \frac{dx}{dt} + x \ln(x) \frac{dy}{dt}\right) \)although I think your idea has a better chance of working.
Andrew Robbins
Sorry for the flexion/inflection business. What a pity, but, unfortunately, I think you are right!
Thank you for your second observation. Perhaps somebody might have some new good ideas about that. My ... animal instinct suggests that the solution of the problem of extending tetration to the reals includes this fantomatic equation, together with the implementatiomn of a continuous iteration of the exponential function. Maybe, they are two aspects of the same problem.
GFR
