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I want to draw an image of this implicit function $x^2 - y^2 = \tan ( y^x)$, but I get a lot of warning messages when I draw the image using the following method:

ContourPlot[x^2 - y^2 == Tan[y^x], {x, -10, 10}, {y, -10, 10}, 
 PlotPoints -> 50]

What can I do to create a complete and clear image of it?

enter image description here

(The drawing software used in the above figure is Grafeq.)

And the following method of chyanog can draw the outline of this image, but how to further improve its drawing speed?

With[{pow = Sign[#] Abs[#]^#2 &}, 
 ContourPlot[
  Cos[pow[y, x]] (x^2 - y^2) == Sin[pow[y, x]], {x, -2 Pi, 
   2 Pi}, {y, -2 Pi, 2 Pi}, MaxRecursion -> 1, PlotPoints -> 50]]

enter image description here

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    $\begingroup$ Everything would be much simpler if y>0. Negative y does not make much sense. $\endgroup$
    – yarchik
    Commented Mar 13, 2020 at 18:44

1 Answer 1

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The equation x^2 - y^2 == Tan[y^x] is equivalent to the series of the equations ArcTan[x^2 - y^2] == y^x + k*Pi, where k∈ Integers. Therefore,

ContourPlot[Evaluate[Table[ArcTan[x^2 - y^2] == y^x + k*Pi, {k, -2, 2}]], {x, -10,   10}, {y, -10, 10}, PlotPoints -> 50]

does the job enter image description here

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