Thank you very much for the series approximations sheldon, but sadly the humps still occur in base \( \eta \). I'm wondering now if there is a better base to work with or if it's smarter to dump the idea of logarithmic semi-operators altogether, as they seem to be a poor extension of ackerman function to domain real.
The only interesting thing I have to report is that:
just like \( \lim_{p\to -\infty} \text{cheta}(p) = e \), \( \lim_{p\to -\infty} e\, \{p\}\, e = \ln^{\alpha -p}(2 \exp^{\alpha -p}(e)) = e \)
or that \( \lim_{p\to -\infty} \text{cheta}(p) = e\, \{p\}\, e = e \) where \( \{p\} \) is a logarithmic semi operator.
So my question was, what's the radius of convergence for the cheta series you gave me, and whats the recurrence relation so that I can produce the full \( \text{cheta}(x) \) function. I just want to test some values. for ex: if \( \text{cheta}(-1) = e + \ln(2) \) then I think we have something, but if it doesn't, oh well.
The only interesting thing I have to report is that:
just like \( \lim_{p\to -\infty} \text{cheta}(p) = e \), \( \lim_{p\to -\infty} e\, \{p\}\, e = \ln^{\alpha -p}(2 \exp^{\alpha -p}(e)) = e \)
or that \( \lim_{p\to -\infty} \text{cheta}(p) = e\, \{p\}\, e = e \) where \( \{p\} \) is a logarithmic semi operator.
So my question was, what's the radius of convergence for the cheta series you gave me, and whats the recurrence relation so that I can produce the full \( \text{cheta}(x) \) function. I just want to test some values. for ex: if \( \text{cheta}(-1) = e + \ln(2) \) then I think we have something, but if it doesn't, oh well.
