02/07/2020, 09:39 AM

Dear friends -

I've contributed in a thread in MSE https://math.stackexchange.com/questions...olve-xxx-1 on the question for finding of solutions for z^z^z = -1 . The question is of 2015 and because of lots of chaotic numerical difficulties I left the question with finding three roots using the log(log(z^z^z))=log(log(-1)), exposed in a Newton-fractal.

This year I came back to the question, partially overcame the numerical chaos in Pari/GP and have now a discussion worth to mention it also here.

A finding is, that the separation of the complex values z^z^z+1 into real and imaginary components gives families of continuous curves of zero values, and the lines occur with some periodicity.

Because roots of the full complex values of the function are only there where both components are zero, we find the roots on a discrete lattice-style set of surely infinite points.

See the overlay of the contour-plots of real and imaginary components in the neighbourhood of a known root at about 5.277+11.641i . The roots are on the intersections of the white curves.

I've contributed in a thread in MSE https://math.stackexchange.com/questions...olve-xxx-1 on the question for finding of solutions for z^z^z = -1 . The question is of 2015 and because of lots of chaotic numerical difficulties I left the question with finding three roots using the log(log(z^z^z))=log(log(-1)), exposed in a Newton-fractal.

This year I came back to the question, partially overcame the numerical chaos in Pari/GP and have now a discussion worth to mention it also here.

A finding is, that the separation of the complex values z^z^z+1 into real and imaginary components gives families of continuous curves of zero values, and the lines occur with some periodicity.

Because roots of the full complex values of the function are only there where both components are zero, we find the roots on a discrete lattice-style set of surely infinite points.

See the overlay of the contour-plots of real and imaginary components in the neighbourhood of a known root at about 5.277+11.641i . The roots are on the intersections of the white curves.

Gottfried Helms, Kassel