New tetration method based on continuum sum and exp-series

[mike3]

Mike, take a look here: http://math.eretrandre.org/tetrationforu...hp?tid=442

Already did.

[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.

[mike3]

Numerical algorithm: The problem is, what should the lower bound on the integral be? And how do you find \( \frac{g'(0)}{g(0)} \)?

Graph of tetration at the complex base: So did the graph I gave surprise you? Or what?

[tommy1729]

Numerical algorithm: The problem is, what should the lower bound on the integral be? And how do you find \( \frac{g'(0)}{g(0)} \)?

that partly is why i was , and im , skeptical.

tommy1729

[mike3]

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.

I just finished the color graph of the complex tetration \( ^z (2.33 + 1.28i) \) using the same method as to generate the the graph on the real axis. Dang this took a long time to compute -- I'm gonna need to revise my program...

   

The scale runs from -10 to +10 on both axes (real = horizontal, imag = vertical). The white regions should have more detail in them, but due to overflow only limited detail is available.

What do you think?

[andydude]

Wow.

The first thing that I see is that its not symmetric across the real axis, but that would probably be a side-effect of the base not being real.

[andydude]

Also, if you'd like my python tool to add lines, here it is: