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Final ansatz

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I'm currently looking to solve an equation of the general form: [itex] \sqrt{x^2-y^2}+\sqrt{\epsilon x^2-y^2} = \beta[/itex]. I'm interested in solving this equation for [itex]x[/itex] assuming [itex]y>0[/itex], [itex]\epsilon>1[/itex] and [itex]\beta \in \mathbb{C}[/itex]. By squaring the equation twice I can find four potential solutions of the form:

[tex]x = (-1)^n \sqrt{ \frac{\beta^2}{(1-\epsilon)^2}\Big[1+\epsilon+(-1)^m \frac{2}{\beta}\sqrt{\epsilon y^2(2-\epsilon)+\epsilon\beta^2-y^2}\Big]} \ \ \ \ \ \ \ \mathrm{for}\ \{n,m\}\in \{1,2\}.[/tex]

Now, I have tested these solutions numerically for parameters roughly in the range [itex]y\in ]1.67, 2[ [/itex] and with [itex] \epsilon \in ]1.01, 4[ [/itex]. Generally, I seem to be getting proper solutions if [itex]|\beta|[/itex] is "large" - but if I set e.g. [itex]y = 1.9[/itex], [itex]\epsilon = 2[/itex] and [itex] \beta = 0.02 + i 0.01[/itex] then the solutions are wrong.

I'm consequently quite convinced that [itex] \sqrt{x^2-y^2}+\sqrt{\epsilon x^2-y^2} = \beta[/itex] only has solutions for certain parameters choices - what I want to find out is; can I analytically express when the equation has a solution? I.e. when is the solution domain of the equation empty?

I'll look forward to reading your replies!