How to include sparse points in a ContourPlot if any?

I'm working with the following expressions:

$$\left(\frac{1-\tanh (X)}{\tanh (X)+1}\right)^{-k}=\pm\frac{P_2^{(k,-k)}(\tanh (X))}{P_2^{(-k,k)}(\tanh (X))}$$

where $$X\in[0,5]$$ and $$k\in\left[-\sqrt{\frac{16}{3}},\sqrt{\frac{16}{3}}\right]$$ exluding $$k\in\mathbb{Z}$$. I want to obtain an X vs. k plot, so I tried the following:

ContourPlot[{((1 - Tanh[X])/(1 + Tanh[X]))^-k ==
JacobiP[2, k, -k, Tanh[X]]/JacobiP[2, -k, k, Tanh[X]],
((1 - Tanh[X])/(1 + Tanh[X]))^-k == -(JacobiP[2, k, -k, Tanh[X]]/
JacobiP[2, -k, k, Tanh[X]])}, {X, 0, 5}, {k, -Sqrt[16/3], Sqrt[
16/3]}, MaxRecursion -> 2, PlotPoints -> 100,
PlotRange -> {{0, 5}, Full}, FrameLabel -> Automatic,
ContourStyle -> {Red, Blue},
PlotLegends -> Placed[{"+sign", "-sign"}, {0.8, 0.1}],
Exclusions ->
Flatten[{Table[k == n, {n, -4, 4, 1}],
JacobiP[2, -k, k, Tanh[X]] == 0}]]


And I got

But, since I'm reproducing a plot, I expect to obtain this

So, I dont know if ContourPlot can reach some sparse points to yield something like this

• The sparse points which you are mentioning are included in the Exclusions if I am not mistaken; see what you get for Exclusions -> Flatten[{Table[k == n, {n, 0, 4, 1}], JacobiP[2, -k, k, Tanh[X]] == 0}] Am I confusing myself?
– bmf
Jan 11, 2023 at 7:15
• @bmf The 'Exclusions' part was added to avoid $k\in\mathbb{Z}$ and possible $P_2^{(-k,k)}(\tanh (X))=0$ points. It has a 'Table' because this plot is part of a series of plots, in this case $k$ cannot be equal to $-2,-1,0,1$ and 2. Jan 11, 2023 at 7:21

You can split the plots and separately specify exclusions:

g1 = ContourPlot[{((1 - Tanh[X])/(1 + Tanh[X]))^-k == -(JacobiP[2,
k, -k, Tanh[X]]/JacobiP[2, -k, k, Tanh[X]])}, {X, 0,
5}, {k, -Sqrt[16/3], Sqrt[16/3]}, MaxRecursion -> 2,
PlotPoints -> 100, PlotRange -> {{0, 5}, Full},
FrameLabel -> Automatic, ContourStyle -> Blue,
PlotLegends -> Placed[{"-sign"}, {0.8, 0.12}],
Exclusions ->
Flatten[{k == 1, k == -1, JacobiP[2, -k, k, Tanh[X]] == 0}]]
g2 = ContourPlot[{((1 - Tanh[X])/(1 + Tanh[X]))^-k ==
JacobiP[2, k, -k, Tanh[X]]/JacobiP[2, -k, k, Tanh[X]]}, {X, 0,
5}, {k, -Sqrt[16/3], Sqrt[16/3]}, MaxRecursion -> 2,
PlotPoints -> 100, PlotRange -> {{0, 5}, Full},
FrameLabel -> Automatic, ContourStyle -> Red,
PlotLegends -> Placed[{"+sign"}, {0.8, 0.2}],
Exclusions ->
Flatten[{k == -2, k == 2, k == 0,
JacobiP[2, -k, k, Tanh[X]] == 0}]]
Show[g1, g2]