01/03/2016, 07:45 PM
I made a complex plot of the third super-root. I can't seem to get it to work for complex number with negative real part, but I think it's working for positive real part, so I've attached the plot below. The branch points are where:
superroot_3(0.731531897477381 + 0.293308661285157*I) == (0.657319327367223 + 0.704370182866530*I)
this point corresponds to where \( \frac{d}{dx}x^{x^x} = 0 \), and so it's not a logarithmic singularity, it's not a singularity at all, but it looks like it creates a new branch depending on which way you travel around the branch point. These branch points are shown as black dots in the plot.
superroot_3(0.731531897477381 + 0.293308661285157*I) == (0.657319327367223 + 0.704370182866530*I)
this point corresponds to where \( \frac{d}{dx}x^{x^x} = 0 \), and so it's not a logarithmic singularity, it's not a singularity at all, but it looks like it creates a new branch depending on which way you travel around the branch point. These branch points are shown as black dots in the plot.
