PaXLn corresponds to Ln in Package-X.
(*Just to load Package-X*)
PaXEvaluate[A0[1]];PaXLn uses X`Ln for numerical
evaluations
PaXLn[-4.5, 1]1.50408\, +3.14159 i
X`Ln[-4.5, 1]1.50408\, +3.14159 i
The same goes for derivatives and series expansions
D[PaXLn[x, \[Alpha]], x]\frac{1}{x}
Series[PaXLn[x, \[Alpha]], {x, 1, 2}]\log (1+i \alpha \epsilon )+(x-1)-\frac{1}{2} (x-1)^2+O\left((x-1)^3\right)