Hmm. How could you derive an iterating formula from this similar to the iterating formula for the original continuum sum formula? That is, one like \( \mathrm{tet}_b(x) = <\mathrm{something\ involving\ tet_b(x)}> \), but that doesn't also require you to explicitly take the derivative w.r.t the base? Also, with base e we get a zero denominator since \( \ln(\ln(e)) = 0 \). So what happens there? Or do you mean \( \ln(z)^2 \) (i.e. multiplicative power, not compositional power, of logarithm) by "\( \ln^2(z) \)", in which case there is no problem?
But this is really a cool result, since by Mueller's formula we can sum the continuum sums, and the reciprocal of tetration's derivative for real bases greater than \( \eta = e^{1/e} \) converges to zero very very quickly, so quickly (tetrationally quickly
) that for any conceivable approximation level we need only a single-digit number of terms in the Mueller formula. Thus if an iterating formula can be found, it may be the fastest and most efficient method yet for the computation of the tetrational function, way better than all the other methods proposed so far (natural iteration, Carleman matrix iteration, Cauchy integral, etc.), perhaps good enough to enable the computation of tetration to extremely huge precision.
But this is really a cool result, since by Mueller's formula we can sum the continuum sums, and the reciprocal of tetration's derivative for real bases greater than \( \eta = e^{1/e} \) converges to zero very very quickly, so quickly (tetrationally quickly
) that for any conceivable approximation level we need only a single-digit number of terms in the Mueller formula. Thus if an iterating formula can be found, it may be the fastest and most efficient method yet for the computation of the tetrational function, way better than all the other methods proposed so far (natural iteration, Carleman matrix iteration, Cauchy integral, etc.), perhaps good enough to enable the computation of tetration to extremely huge precision.
