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Under what constraints on the parameters a,b, and c does the transcendental equation

$$x+a+be^{-x}+ce^{-kx}=0$$ ($k$ is a constant) have ALL its roots with negative real parts? Alternatively, any good references to this type of problem are most welcome.

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    $\begingroup$ This might be more appropriate for math.SE. It also doesn't look like complex analysis to me. Is $k>0$? I would start by reducing the four-dimensional space to a three-dimensional one with the substitution $x\rightarrow x-a$, which brings the equation into the form $x+pe^{-x}+qe^{-kx}=0$. $\endgroup$
    – user21349
    Commented Aug 4, 2012 at 23:13

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Could you explain a little bit how you arrived at your result? I'm assuming you employed Rouch\'{e}'s theorem or something along those lines.

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One sufficient condition is that $k > 0$ and $\text{Re}(a) - |b| - |c| > 0$.

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