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I am trying to plot the graph between P and x. My teacher wants me to normalize this graph by introducing a constant $k$ to get $P/k^2$ on the y-axis and $xk$ on the x-axis. I did that by doing a short cut, by manually seeing the max value of P and plotted the graph by doing $P/0.0035$ which was the maximum value of $P$ from graph. But how can I do it efficiently by using mathematical commands? My code is below:

gamma = 1/Sqrt[1 - D[r (-1 + 1/v) + ArcTanh[r], r]^-2]

Then I plotted $p-x$ in the limits, ${x,-1,1}$ as follows: Definition:

power = 1/(6 Pi) D[gamma, r]^2 

Plot

pow = Plot[{power/0.0035 /. v -> 1/2, power/0.0035 /. v -> 1/3, 
   power/0.0035 /. v -> 1/4}, {r, -1, 1}, 
  AxesLabel -> {Style["x\[Kappa]", Black, FontFamily -> "Times", 
     FontSize -> 15], Style["P/\[Kappa]^2", Black, FontFamily -> "Times", FontSize -> 15]}, 
  LabelStyle -> {FontSize -> 12, FontFamily -> "Times", Black, Bold}, 
  PlotStyle -> {{Red, Thickness[0.005]}, {Purple, 
     Thickness[0.005]}, {Yellow, Thickness[0.005]}}, 
  PlotLegends -> 
   Placed[LineLegend[{Directive[Thickness[0.1], Red], 
      Directive[Thickness[0.1], Purple], 
      Directive[Thickness[0.1], Yellow]}, {Style["s=1/2", 11, Bold], 
      Style["s=1/3", 11, Bold], Style["s=1/4", 11, Bold]}, 
     LegendMarkerSize -> {{21, 15}}], {0.6, 0.6}]]

If I introduce a $k^2$ quantity, my x-axis should also be scaled. How can I completely normalize this graph with both axes calling through the quantity $k$ using the mathematica commands? I have mathematica 9.0.

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    $\begingroup$ k = Sqrt[MaxValue[{power, 1/4 <= v <= 1/2, -1 <= r <= 1}, {r, v}]] which evaluates to (2 Sqrt[2/π])/27 The approximate value of k^2 is 0.00349311 $\endgroup$
    – Bob Hanlon
    Nov 5, 2023 at 0:53

1 Answer 1

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You may use "Maximize" to get the r value of the maximum. With this you get the maximal value for P: pmax:

gamma = 1/Sqrt[1 - D[r (-1 + 1/v) + ArcTanh[r], r]^-2];
power = 1/(6 Pi) D[gamma, r]^2;
pmax = power /. v -> 1/2 /. 
   Maximize[{power /. v -> 1/2, -1 <= r <= 1}, r][[2]];
pow = Plot[{power/pmax /. v -> 1/2, power/pmax /. v -> 1/3, 
   power/pmax /. v -> 1/4}, {r, -1, 1}, 
  AxesLabel -> {Style["x\[Kappa]", Black, FontFamily -> "Times", 
     FontSize -> 15], 
    Style["P/\[Kappa]^2", Black, FontFamily -> "Times", 
     FontSize -> 15]}, 
  LabelStyle -> {FontSize -> 12, FontFamily -> "Times", Black, Bold}, 
  PlotStyle -> {{Red, Thickness[0.005]}, {Purple, 
     Thickness[0.005]}, {Yellow, Thickness[0.005]}}, 
  PlotLegends -> 
   Placed[LineLegend[{Directive[Thickness[0.1], Red], 
      Directive[Thickness[0.1], Purple], 
      Directive[Thickness[0.1], Yellow]}, {Style["s=1/2", 11, Bold], 
      Style["s=1/3", 11, Bold], Style["s=1/4", 11, Bold]}, 
     LegendMarkerSize -> {{21, 15}}], {0.6, 0.6}]]

enter image description here

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  • $\begingroup$ Thanks dude, that's helpful. $\endgroup$
    – Jpmg
    Nov 8, 2023 at 0:12

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