New tetration method based on continuum sum and exp-series

[mike3]

I'm not sure if this method is going to work. Take what happens when \( n = 0 \)...

Oh thats my fault! I was too sloppy with the case \( n=0 \) the correct formula is (I changed that in my original post too):
\( \sigma(z)=\lambda+\sum_{n=1}^{\infty}\sum_{k=-\infty}^{\infty} \sigma_{n,k} e^{(\kappa n + 2\pi i k) z} \)

Ah! Now we can take the continuum sum! Though I'm still at a loss as to how we could use this one for computation. I posted a thread in the Computation forum to discuss the construction of the numerical algorithm. Perhaps it will be better than the periodic approximation approach (get more accuracy, do more bases, etc.). Want to discuss that there?

Also, what about my question about the graph? Did you have any prior idea what the graph of tetration for a complex base would look like?