New tetration method based on continuum sum and exp-series

[bo198214]

Though I'm still at a loss as to how we could use this one for computation.

Hm, I think we could create an array of coefficients \( \sigma_{n,k} \) with a first initial guess.
If we use Ansus' formula for the derivative \( g=\sigma' \) we dont need to take the exponential.
Hence we dont need to switch between phase and function space and.
We can work all the time with the coefficient array.

The algorithm would then be:
Take some initial guess for \( g_{n,k} \).
Then compute the continuum sum.
Multiplying with \( \ln(b) \).
add g'(0)/g(0)
Then take the integral.
Repeat these steps till convergent.

But my fear is that it will not converge.

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?

Hey actually I never considered graph of tetration with complex bases. I would guess that it is somewhat distorted real tetration. But I think one would need contour plots, or complex color plots, to have a bigger picture.