I am trying to understand a methodology for solving implicit non-linear equations with complex variables. I would like to solve for z1 below where z2 is known. Also both z1 and z2 are complex variables.
$$ {z_1} = \frac{1}{{1 - 2{z_1}{z_2}}} + \frac{{{z_2}}}{{1 - 2{z_1}}} $$
I have the following questions
Can I discretize real and imaginary parts of z1 and solve for them separately and them combine them ? Is there a justification that it will work ?
How do I input such implicit equations into software such as R/matlab
- What do I do if I have constraints on real or imaginary parts ? Such as Imaginary portion is always positive
What you have is a two variable polynomial. In the one variable case, the solutions to a complex polynomial are zeros in $\mathbb{C}$, but in the two dimensional case they are curves in $\mathbb{C}\times\mathbb{C}$.
To visualize these curves one typically images a grid placed on $\mathbb{C}$ and imagines how the grid is deformed as you replace values of $z_1$ with points from the grid; the deformed structure is placed in the second $\mathbb{C}$.
For example, pick the unit disc. Now imagine that every point of $z_1$ was replaced with a point on the unit disc. What kind of shape would be found as you solved for the roots of $z_2$ for each of the points of $z_1$? This would give us a visualization of what this solution looks like. What does it do to this unit circle?
Of course it doesn't stop there, litterally every possible path in $\mathbb{C}$ could be asked the same question...how does it get deformed by the equation for the second variable? Every circle, line, curve, etc. The combination of ALL paths and how they trace in the second plane is the "solution to the two variable equation".
We as 3D humans can only visualize pieces of this.