# ContourPlot U^2+W^2=V^2 -> V(U), where U, V solutions of equation from special functions [closed]

CountourPlot W(U) I have this following code of it:

eps1 = 2;
eps2 = 1.5;
n = 0;
\[Delta] = 1/2 (1 - eps2/eps1);

ContourPlot[
Evaluate[{(D[BesselJ[n, U],
U]/(BesselJ[n, U] * U) + (1 - 2*\[Delta])/W  *
D[BesselK[n, W], W]/(BesselK[n, W] ))*
(D[BesselJ[n, U], U]/(BesselJ[n, U] * U) +
1/W  * D[BesselK[n, W], W]/(BesselK[n, W] )) ==  0}], {U, 1,
10}, {W, 1, 9}, WorkingPrecision -> 10, FrameLabel -> Automatic,
PlotLegends -> Automatic, PlotPoints -> 150]


but I need to plot as following V(U), where V=Sqrt(U^2+W^2)

If I put W = Sqrt[V^2 - U^2] in eq like this:

ContourPlot[
Evaluate[{(D[BesselJ[n, U],
U]/(BesselJ[n, U] * U) + (1 - 2*\[Delta])/Sqrt[V^2 - U^2]  *
D[BesselK[n, Sqrt[V^2 - U^2]],
Sqrt[V^2 - U^2]]/(BesselK[n, Sqrt[V^2 - U^2]] ))*
(D[BesselJ[n, U], U]/(BesselJ[n, U] * U) +
1/Sqrt[V^2 - U^2]  *
D[BesselK[n, Sqrt[V^2 - U^2]],
Sqrt[V^2 - U^2]]/(BesselK[n, Sqrt[V^2 - U^2]] )) ==
0}], {U, 1, 10}, {V, 1, 15}, WorkingPrecision -> 10,
FrameLabel -> Automatic, PlotLegends -> Automatic, PlotPoints -> 150]


I'll get this picture, where is nothing... I actually don't understand, how to do it. Pls help

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• Use ParametricPlot? Can you expand on your problem a bit more and add some context? Jun 21 at 21:07
• Just substitute in the expression that you are plotting that $W=\pm\sqrt{V^2-U^2}$ taking care the sign. Jun 21 at 21:22
• @yarchik, this way doesn't help Jun 21 at 22:19
• D[BesselK[n, Sqrt[V^2 - U^2]], Sqrt[V^2 - U^2]] is incorrect code, as Mathematica states whey it is executed. Instead, try D[BesselK[n, z], z] /. z -> Sqrt[V^2 - U^2]. Jun 21 at 22:34

Use ContourPlot3D for U^2+W^2==V^2 and f[U,V] as MeshFunction sine the plot is the intersection of U^2+W^2==V^2 and f[U,V]==0.

Clear[f];
eps1 = 2;
eps2 = 1.5;
n = 0;
δ = 1/2 (1 - eps2/eps1);
f[U_, W_] = (D[BesselJ[n, U], U]/(BesselJ[n, U]*U) + (1 - 2*δ)/
W*D[BesselK[n, W], W]/(BesselK[n, W]))*(D[BesselJ[n, U],
U]/(BesselJ[n, U]*U) + 1/W*D[BesselK[n, W], W]/(BesselK[n, W]));
ContourPlot3D[U^2 + W^2 == V^2, {U, 0, 10}, {V, -10, 10}, {W, 0, 10},
MeshFunctions -> Function[{U, V, W}, f[U, W] // Evaluate],
Mesh -> {{0}}, MeshStyle -> Red, ContourStyle -> None,
ViewProjection -> "Orthographic", ViewPoint -> {0, 0, 1},
BoundaryStyle -> None, AxesLabel -> {"U", "V", None},
Ticks -> {Automatic, Automatic, None}, PlotPoints -> 150,
MaxRecursion -> 4]


The same as ( We use  {V, U, 10} in order to set V>U)

ContourPlot[
f[U, W] == 0 /. W -> Sqrt[V^2 - U^2] // Evaluate, {U, 0, 10}, {V, U,
10}, PlotPoints -> 150, MaxRecursion -> 4]