# How can I plot this transcedental equation?

How can I plot $\kappa(\epsilon_{dd},\lambda)$ this transcendental equation? $$3\kappa^2 \epsilon_{dd}\left[\left(\frac{\lambda^2}{2}+1\right)\frac{f_s(\kappa)}{1-\kappa^2}-1\right]+(\epsilon_{dd}-1)(\kappa^2-\lambda^2)=0$$ where $\lambda=1,2,3,4$ and $$f_s(\kappa)=\frac{1+2\kappa^2}{1-\kappa^2}-\frac{3\kappa^2 artanh \sqrt{1-\kappa^2} }{(1-\kappa^2)^{3/2}}.$$

My original problem is not that, but it's similar. If you help me with this, maybe I can solve mine.

Here are the codes of equations:

fs[kappa_] := (1 +2 kappa^2)/(1 - kappa^2) - (3 kappa^2 ArcTanh[
Sqrt[1 - kappa^2]])/(1 - kappa^2)^(3/2)

3 kappa^2 edd (((lambda^2/2) -1 ) fs[kappa]/(1 - kappa^2) -
1) + (edd - 1) (kappa^2 - lambda^2) == 0

• Mathematica code would increase your chance to get help... – Ulrich Neumann Dec 29 '17 at 12:08
• I really do not know how to work with transcendental equations. I tried to use a SOLVE, but it did not work. – Dinesh Shankar Dec 29 '17 at 12:17
• If you make the code of your equations available, answering and helping would be easier ... – Ulrich Neumann Dec 29 '17 at 12:21
• Sorry, my mistake. I already fixed it. – Dinesh Shankar Dec 29 '17 at 14:12
• you have a typo in your fs expression. @anderstood answer is correct and produces the plot in the paper if you fix that. – george2079 Dec 29 '17 at 15:43

Use ContourPlot.

fs[kappa_] := (1 + 2 kappa^2)/(1 -
kappa^2) - (3 kappa^2 ArcTanh[Sqrt[1 - kappa^2]])/(1 -
kappa^2)^(3/2)

zero[kappa_, edd_, lambda_] =
3 kappa edd (((lambda^2/2) + 1) fs[kappa]/(1 - kappa^2) -
1) + (edd - 1) (kappa^2 - lambda^2);

Show[{ContourPlot[
Evaluate@
Table[zero[kappa, edd, lambda] == 0, {lambda, 0, 2, 1/3}], {edd,
0, 1.8}, {kappa, 0, 2}, FrameLabel -> Automatic,
AspectRatio -> 6/10],
ContourPlot[edd + 1, {edd, 0, 1.8}, {kappa, 0, 2},
FrameLabel -> Automatic, AspectRatio -> 6/10,
RegionFunction -> Function[{x, y, z}, 1 < x < 2],
ContourStyle -> {Directive[Lighter[Red, 0.8], Dashed]},
Contours -> 100, ContourShading -> None],
ContourPlot[edd + 1, {edd, 0, 1.8}, {kappa, 0, 2},
FrameLabel -> Automatic, AspectRatio -> 6/10,
RegionFunction -> Function[{x, y, z}, (x - 2)^2 + (y)^2 < 1],
ContourStyle -> {Directive[Lighter[Blue, 0.7], Dashed]},
Contours -> 100, ContourShading -> None]},
Epilog -> {Text[unstable, {1.4, 0.5}], Text[metastable, {1.4, 1.5}],
Text[stable, {0.6, 1.8}]}]

• @MariuszIwaniuk Thanks for your edit! – anderstood Dec 29 '17 at 17:42

You consider four equations[lamda=1,2,3,4] in epsdd and kappa. It is very easy to solve for epsdd=f[kappa;lamda]. The four solutions can be plotted for different lamda with

ParametricPlot[{f[kappa;lamda],kappa},{kappa,...}]

if you know the kappa-range!

solution(with MMA-code offered) and the corrected formulas:

fs[kappa_] := (1 + 2 kappa^2)/(1 -kappa^2) - (3 kappa^2 ArcTanh[Sqrt[1 - kappa^2]])/(1 -kappa^2)^(3/2)
gl = 3 kappa edd (((lambda^2/2) + 1) fs[kappa]/(1 - kappa^2) -1) + (edd - 1) (kappa^2 - lambda^2) == 0

ergedd =  Solve[gl, edd][[1]] (* implicit soulution *)
(* {edd -> (kappa^2 - lambda^2)/(kappa^2 - lambda^2 +
3 kappa (-1 + ((1 + lambda^2/2) ((1 + 2 kappa^2)/(1 - kappa^2) -
(3 kappa^2 ArcTanh[Sqrt[1 - kappa^2]])/(1 - kappa^2)^(3/2)))/(1-kappa^2)))} *)

Show[Table[
ParametricPlot[  {edd /. ergedd, kappa} , {kappa, 0, lambda},
PlotStyle -> RGBColor[lambda/4, 0, 1 - lambda/4]]  , {lambda, 1,
4}]
, AspectRatio -> 1 ,PlotRange->{0,4}]

the result is as expected:

• I want $\kappa(\epsilon_{dd})$. – Dinesh Shankar Dec 29 '17 at 12:18
• You wanted to plot the solution, ParametricPlot does it! Looking for an explicit solution of your equations is a much harder effort. Perhaps you can approximate the implicit solution epsd=f[kappa]... – Ulrich Neumann Dec 29 '17 at 12:26
• I just modified the equation. However, the code does not compile. – Dinesh Shankar Dec 29 '17 at 14:04
• this also produces the paper plot if you get the expressions correct. – george2079 Dec 29 '17 at 15:57
• could you show your graph? – Dinesh Shankar Dec 29 '17 at 20:01