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I have a system of two complex equations for which I want to find the intersection ($\Gamma$).

$$p_3=Abs[\frac{\Gamma k_{32}+k_{31}}{\Gamma k_{34}+k_{33}}],p_4=Abs[\frac{\Gamma k_{42}+k_{41}}{\Gamma k_{44}+k_{43}}]$$

Where all the $k_{nm}$ are known complex number and $p_n$ are also known real numbers $0<p_n<1$. I'm trying to solve for $\Gamma$. I'm using Mathematica to assist me in those calculation.

Finding the solution graphically for this problem is quite easy. When p is plotted in function of the phase of gamma, depending on the values of $k_{nm}$, you get a circle-ish curve. If you do the same with both equations, you find two curves that will intersect at 2 points. These are the solution. I can't find any way to resolve them analytically.

For instance, with the values of k: \begin{array}{cccc} \text{k31= 0.169933 - 0.178627 I} & \text{k32= 0.244685 - 0.090219 I} & \text{k33= 1. + 0. I} & \text{k34= -0.0296144 - 0.0692198 I} \\ \end{array} \begin{array}{cccc} \text{k41= -0.1436 + 0.208844 I} & \text{k42= -0.20624 + 0.108843 I} & \text{k43= 1. + 0. I} & \text{k44= -0.0296144 - 0.0692198 I} \\ \end{array}

and $p_3=0.119056$ and $p_4=0.111597$, one of the intersecting point should be $\Gamma=0.5+0.5 i$.

$\Gamma$ contour for a fixed $p_3$ and $p_4$ On that plot, it can be seen the 2 intersecting points ( $\Gamma=0.5+0.5 i$ and $\Gamma=-0.679-0.8472 i$).

So, what should I do to find those value from the equations?

Thank you.

Julien
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    You have two equations of the form $(a \Gamma + b)(\overline a \overline\Gamma + \overline b)=p^2(c \Gamma + d)(\overline c \overline\Gamma + \overline d)$. Eliminate $\overline\Gamma$ between them and you get an equation in $\Gamma$ alone. The solutions are the intersections of the two Apollonian circles. – dxiv May 15 '23 at 17:36
  • Thank you for that insight. I should get back tomorrow with an update or a follow up question. – Julien May 15 '23 at 21:14
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    So after a bit of debugging, I can confirm that the technique suggested works perfectly. Thank you so much @dxiv! – Julien May 16 '23 at 17:01
  • what about the values for the $k_ij$? Also, it may make it easy if you put the equations in a form the majority can understand... – NoChance May 16 '23 at 17:40
  • The values of $k_{ij}$ comes from a measured physical structure. I listed their values just for an example. Please educate me, which equations are in a form that makes them hard to understand? What makes them hard to understand? Thanks – Julien May 16 '23 at 18:17

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