I have this function: $$\frac{1}{16} \left(\sinh (\pi x) \left(64 \left(x^2-4\right) \cosh \left(\frac{2 \pi x}{3}\right) \cos (y)+\left(x^2+4\right)^2+256 x \sinh \left(\frac{2 \pi x}{3}\right) \sin (y)\right)+\left(x^2-12\right)^2 \sinh \left(\frac{7 \pi x}{3}\right)-2 \left(x^2+4\right)^2 \sinh \left(\frac{5 \pi x}{3}\right)\right)+2 \left(x^2-4\right) \sinh \left(\frac{\pi x}{3}\right)$$ I use ContourPlot
f[x_, y_] :=
2 (-4 + x^2) Sinh[( π x)/3] +
1/16 (((4 + x^2)^2 + 64 (-4 + x^2) Cos[y] Cosh[(2 π x)/3] +
256 x Sin[y] Sinh[(2 π x)/3]) Sinh[ π x] -
2 (4 + x^2)^2 Sinh[(5 π x)/
3] + (-12 + x^2)^2 Sinh[(7 π x)/3]);
ContourPlot[f[x, y] == 0, {x, 3.42, 3.5}, {y, 0.5, 1.5},
PlotPoints -> 100]
and this is the result
Then, when I diminish the ranges,
ContourPlot[
f[x, y] == 0, {x, 3.4657283, 3.4657285}, {y, 1.046788, 1.046793},
PlotPoints -> 500]
I get this plot
Now, how can I make sure whether the two curves cross each other or not?