I am trying to recreate the following 2 plots: 
where the function being plotted is $|e^{2ipR}S_0^{\text{well}}(p)|$ where $|...|$ is the absolute value, $R=1$, and $S_0^{\text{well}}(p)$ is given as \begin{equation} S_0^{\text{well}}(p) = e^{-2ipR}\frac{1+ipR\cdot\text{tanc}(R\sqrt{p^2-U})}{1+ipR\cdot\text{tanc}(R\sqrt{p^2-U})} \end{equation} where $\text{tanc}(x)\equiv \tan(x)/x$ The plots are for $U = 10$ and $U = -50$ (both at $R=1$).
Here is my attempt in Mathematica:
tanc[x_] := Tan[x]/x; S0well[p_, R_, U_] := Exp[-2*I*p*R]*((1 + I*p*R*tanc[R*Sqrt[p^2 - U]])/(1 - I*p*R*tanc[R*Sqrt[p^2 - U]])); ComplexPlot[Abs[Exp[2*I*p*1]*S0well[p, 1, -50]], {p, -10 - 10*I, 10 + 10*I}] Which reproduces the following image: 
Obviously, there are a few glaring issues. The first, the color. Second, why do I have branch cuts? I want to plot the branch cuts next but by converting the $S(p)$ function to one of $S(E)$ via $E = p^2/2m$ (by setting $2m = 1$). I am also missing the "blue" colored poles corresponding to zeros, but I think I have the "red dots" which are the poles. The images and equations are originally taken from section 2.1.2 (page 36) of The Analytic S-Matrix.
Any help and/or suggestions is appreciated.
