Since we're dealing with half-iterates of \( \text{sexp}_\eta(z) \), does this tell us anything about pentation? If \( \text{pent}_\eta(z) \) is pentation base \( \eta \),
if \( \text{sexp}_\eta^{\frac{1}{2}}(z) = \text{pent}_\eta(\text{pent}_\eta^{-1}(z) + \frac{1}{2}) \) holds up,
I would assume since \( \text{sexp}_\eta^{\frac{1}{2}}(z) \) isn't analytic about e, this says \( \text{pent}_\eta(z) \) and \( \text{pent}_\eta^{-1}(z) \) aren't analytic about e also right? I'm not sure, but the composition of two analytic functions is always an analytic function right? so if \( \text{sexp}_\eta^{\frac{1}{2}}(z) \) isn't analytic about e, they probably aren't. I guess this glueing together has consequences for all hyperoperators base \( \eta \), because if \( \text{pent}_\eta(z) \) isn't analytic about e \( \text{pent}_\eta^{\frac{1}{2}}(z) \) isn't analytic about e, so the same is true about hexation, etc etc..
Still though, it's very interesting that you can already develop half-iterates of tetration. Very fascinating. I guess using some sort of complicated recursive formula, I'm pretty sure you can theoretically evaluate any value for all hyperoperators base \( \eta \). That's pretty incredible. At least, using the cheta function method.
With this knowledge, I know it's possible for me to extend logarithmic semi-operators base \( \eta \) to domain C. This would give a complex extension of the Ackerman function.
This thread was all very beautiful. Thank you for posting it.
if \( \text{sexp}_\eta^{\frac{1}{2}}(z) = \text{pent}_\eta(\text{pent}_\eta^{-1}(z) + \frac{1}{2}) \) holds up,
I would assume since \( \text{sexp}_\eta^{\frac{1}{2}}(z) \) isn't analytic about e, this says \( \text{pent}_\eta(z) \) and \( \text{pent}_\eta^{-1}(z) \) aren't analytic about e also right? I'm not sure, but the composition of two analytic functions is always an analytic function right? so if \( \text{sexp}_\eta^{\frac{1}{2}}(z) \) isn't analytic about e, they probably aren't. I guess this glueing together has consequences for all hyperoperators base \( \eta \), because if \( \text{pent}_\eta(z) \) isn't analytic about e \( \text{pent}_\eta^{\frac{1}{2}}(z) \) isn't analytic about e, so the same is true about hexation, etc etc..
Still though, it's very interesting that you can already develop half-iterates of tetration. Very fascinating. I guess using some sort of complicated recursive formula, I'm pretty sure you can theoretically evaluate any value for all hyperoperators base \( \eta \). That's pretty incredible. At least, using the cheta function method.
With this knowledge, I know it's possible for me to extend logarithmic semi-operators base \( \eta \) to domain C. This would give a complex extension of the Ackerman function.
This thread was all very beautiful. Thank you for posting it.
