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I want to plot f(x), inverse f(x) and f'(x) and then f(x)/f'(x) and inversef'(x)/inversef(x)

f[x_] := 2 a ArcTanh[(# a)/Sqrt[-1 + #^2 b]] + a Log[1 + #^2 (a^2 - b)] - 2 Log[# b + Sqrt[-1 + #^2 b]] &[x] /. {a -> 0.2, b -> 0.3};

I don't know from where I can start?

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    $\begingroup$ Your function f[x] is complex, is this feature intended? If yes your question cann#t be answered. $\endgroup$
    – Ulrich Neumann
    Dec 30 '20 at 13:32
  • $\begingroup$ Yes but I like to know inverse f(x) and its differential and their plots mostly. $\endgroup$
    – Armin Sharafi
    Dec 30 '20 at 13:35
  • $\begingroup$ Please clarify your question: You want to plot parts of a complex function: Re, Im, Abs,... $\endgroup$
    – Ulrich Neumann
    Dec 30 '20 at 13:47
  • $\begingroup$ Abs I would like to know Abs $\endgroup$
    – Armin Sharafi
    Dec 30 '20 at 13:49
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To plot Abs[f[x]]==Sqrt[f[x] Conjugate[f[x]] try

f = 2 a ArcTanh[(# a)/Sqrt[-1 + #^2 b]] + a Log[1 + #^2 (a^2 - b)] -2 Log[# b + Sqrt[-1 + #^2 b]] & /. {a -> 2/10, b -> 3/10};
Plot[Sqrt[f[x] Conjugate[f[x]]] , {x, -5, 5} , PlotRange -> All]

enter image description here

The inverse function of Abs[f[x]] is only defined in subintervalls x<-2.17172,-2.17172<x<1.94146,x>1.94146.

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  • $\begingroup$ then I have this problem that how to introduce this f for plotting inversef'(x)/inversef(x) $\endgroup$
    – Armin Sharafi
    Dec 30 '20 at 17:15
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Clear["Global`*"]

f = 2 a ArcTanh[(# a)/Sqrt[-1 + #^2 b]] + a Log[1 + #^2 (a^2 - b)] - 
     2 Log[# b + Sqrt[-1 + #^2 b]] & /. {a -> 2/10, b -> 3/10};

Plot[Abs[f[x]], {x, -5, 5},
 PlotRange -> All,
 PlotPoints -> 100,
 MaxRecursion -> 10,
 AspectRatio -> 1,
 AxesLabel -> {"x", "|f(x)|"}]

enter image description here

Use ParametricPlot to plot the inverse

ParametricPlot[{Abs[f[x]], x}, {x, -5, 5},
 PlotRange -> All,
 PlotPoints -> 100,
 MaxRecursion -> 10,
 AspectRatio -> 1,
 AxesLabel -> {"|f(x)|", "x"}]

enter image description here

Use "Log" scaling (ScalingFunctions) for the derivative

Plot[Abs[f'[x]], {x, -5, 5},
 PlotRange -> All,
 PlotPoints -> 100,
 MaxRecursion -> 10,
 AspectRatio -> 1,
 AxesLabel -> {"x", "|f'(x)|"},
 ScalingFunctions -> "Log"]

enter image description here

as well as for the ratio

Plot[Abs[f[x]/f'[x]], {x, -5, 5},
 PlotRange -> All,
 PlotPoints -> 100,
 MaxRecursion -> 10,
 AspectRatio -> 1,
 AxesLabel -> {"x", "|f(x)/f'(x)|"},
 ScalingFunctions -> "Log"]

enter image description here

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2
  • $\begingroup$ for plotting inverse f'(x) / inverse f(x) ?? $\endgroup$
    – Armin Sharafi
    Dec 31 '20 at 2:54
  • $\begingroup$ The inverses are not well-defined since they are multi-valued. You could use ParametricPlot to plot {Abs[f'[x]/f[x]], x} or {Log10[Abs[f'[x]/f[x]]], x} $\endgroup$
    – Bob Hanlon
    Dec 31 '20 at 3:06

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