(01/10/2016, 06:46 PM)sheldonison Wrote: How fast does the superroot3 grow on the negative real axis? Something funny happens somewhere near -35.83, maybe because the imaginary part goes to zero. Is this another singularity, or an exponential/logarithmic branch problem?
z = -0.368593375973251 + 8.24287825516783 E-7*I
z^z^z = -35.83
z = -0.36859401116538
z^z^z = -35.830698526398
I think I know what is going on here. Firstly, it appears that if we use the above branch cuts, then the imag(superroot_3(z)) would cross 0 around z = -36.
Secondly, the reason why this creates a discontinuity is that this corresponds to a branch cut of superpower_3 (also known as z^z^z):
You can kind of think of traveling along the negative real axis in superroot_3 is like traveling the cyan (light blue) region in superpower_3. The dot in the plot above is approximately where the value of superpower_3 == -64, and the point would be the value of superroot_3(-64) if it was continuous, but you won't get this value on the main branch of superpower_3, because by traveling along the negative real axis of superroot_3, you've crossed a branch cut of superpower_3. In order to make that region continuous, then you would have to choose a branch of superpower_3 that is continuous with the upper-left quadrant for the "green region" above, and choose a branch of superpower_3 that is continuous with lower-left quadrant for the "blue region" above. So in order to calculate the roots of z^z^z, we need a way to choose branches of it...
