Ah, now I see wherefrom the discontinuity comes. The Abel function is the logarithm (to base \( 1/\log(a) \)) of the Schroeder function and the Schroeder function crosses the line (the negative real Axis). And there the logarithm changes imaginary part from \( -\pi \) to \( \pi \) and gives this discontinuity (in the real part because we divide by another complex number).
Schroeder function \( \sigma(x) \) for \( x=-3\dots 3 \) in the complex plane:
It starts in the upper right area and evolves clockwise until it crosses the negative real axis at approximately \( x=2.1894 \).
(The peaks (which belong to \( x={^ne},n=-1,0,1 \)) are truncated due to the maple graphing routine.)
Of course this discontinuity can be removed, but in the moment I am too lazy to provide the continuous graph.
Schroeder function \( \sigma(x) \) for \( x=-3\dots 3 \) in the complex plane:
It starts in the upper right area and evolves clockwise until it crosses the negative real axis at approximately \( x=2.1894 \).
(The peaks (which belong to \( x={^ne},n=-1,0,1 \)) are truncated due to the maple graphing routine.)
Of course this discontinuity can be removed, but in the moment I am too lazy to provide the continuous graph.
