On the hunt of singularities of isexp at 0 (islog at 1):
If sexp has a singularity on the radius \( \approx 1.4 \) then it should be visible in the conformal plot of slog as an end point of lines.
Thatswhy I did a plot of slog which I extended not via continuation/recentering but by slog(exp(x))=slog(x)+1 and slog(log(x))=slog(x)-1. Starting on the sickle re(z)>re(L) and |z|<|L| (which is contained inside the convergence disk of slog around 1 with radius |L|) where L is the primary fixed point of exp.
Indeed there is no singularity visible inside the circle with radius 1.5.
But one can see that to the right side the lines become dense which may count numerically as a singularity and thatswhy does not allow the sexp at 0 to converge for r>1.4.
If sexp has a singularity on the radius \( \approx 1.4 \) then it should be visible in the conformal plot of slog as an end point of lines.
Thatswhy I did a plot of slog which I extended not via continuation/recentering but by slog(exp(x))=slog(x)+1 and slog(log(x))=slog(x)-1. Starting on the sickle re(z)>re(L) and |z|<|L| (which is contained inside the convergence disk of slog around 1 with radius |L|) where L is the primary fixed point of exp.
Indeed there is no singularity visible inside the circle with radius 1.5.
But one can see that to the right side the lines become dense which may count numerically as a singularity and thatswhy does not allow the sexp at 0 to converge for r>1.4.