I’m reading the wolfram page on complex functions and noticed that one can actually plot the values of these functions on the axis (cf. image attached). How can I do this in mathematics, say for $sech(z)^{-1}$? I.e., how to replicate those 4 figures?
Those and the associated notebook were generated under an earlier version of Matheamtica and are poorly drawn and confussing. Consider the inverse hyperbolic secant function. If you look under the "Details" of ArcSech, (type in ArcSech, hover over it, press "I") it states the "default" branch cuts run $(-\infty,0$ and $(1,\infty)$. So that if you try and plot explicitly across those domains, sometimes with the built-in plotting functions, the plot will show a jagged, seemingly-smooth (and incorrect) transition across the branch cuts when in fact there is suppose to be a sharp discontinuity across the cut. So to correct this in this particular case, we break up the plot into 2 parts: one from $-\pi$ to 0 and the other from 0 to $\pi$. Then join them:
f[z_] := ArcSech[z];
pLow =
ParametricPlot3D[{Re[z], Im[z], Im[f[z]]} /.
z -> r Exp[I t], {r, 0, 3}, {t, -Pi, 0}];
pHigh =
ParametricPlot3D[{Re[z], Im[z], Im[f[z]]} /. z -> r Exp[I t], {r, 0,
3}, {t, 0, Pi}];
Show[{pLow, pHigh}, BoxRatios -> {1, 1, 1}, PlotRange -> All]
The syntax {Re[z], Im[z], Im[f[z]]} /.z -> r Exp[I t] means parameterize the real part of z, imag part of z and imag part of f[z] given by $z=re^{it}$ over the indicated range. To plot the real part of f(z) just switch to Re[f[z]] or absolute value of f[z] by Abs[f[z]]
Branch Cutin the upper right corner click:Download Wolfram Notebook. $\endgroup$