Dissecting Andrew's slog solution

[jaydfox]

If we call the fixed point x, then here's a look at the coefficients a_k of Andrew's slog, divided by the real part of x^(k+1), multiplied by k (to effect the derivative), and multiplied by abs(x^2).

What fixed point?

The fixed point of the principal branch of ln(x), approximately 0.318131505+1.337235701i

With a few exceptions in the first handful of terms, the values seem to be converging on 1.0579. As it turns out, \( {\Large x}^{-1.057939991157 i} \) is equal to \( \Large{x}^{1.057939991157*{\large \left|\frac{x^{\tiny -i}}{x}\right|} \). In other words, if you start at a point very near the fixed point, then 4.44695 real iterations and -1.05794 imaginary iterations will get you to the same point.

What?

\( {\large \left(0.318131505+1.337235701 i\right)}^{-1.05793999115694 i}\ \approx\ {\large \left(0.318131505+1.337235701 i\right)}^{4.44695072006701} \)

4.44695072006701 = 1.05793999115694*(1.337235701/0.318131505)

Call the fixed point \( \chi \approx 0.318131505+1.337235701 i \). Then \( \chi^x \) and \( \chi^{-iy} \) are two complex spirals, which intersect when y is a multiple of 1.05793999115694