I have a lengthy complex polynomial-type equation. There is a software that can solve this type of equations provided I pass it as a coupled system of two real-valued equations. Hence, I am wondering if there is an automated way or software to do this?
Note that this may seem trivial to work out manually, but for lenghty equations with complex coefficients this becomes really tedious and error prone.
Here is an example. Let $$x = a+bi, y = c+di, z = e+fi \in \mathbb{C}$$, where $a,b,c,d,e,f \in \mathbb{R}$.
For instance, the equation $xyz = s$ can be split as $$ace-bde-bcf-adf=Re(s) \\ acf-bdf+bce+ade=Im(s)$$
Already for the triple product above the split equations are not obvious. I have much more complicated one. Ideally, I need a software or an algorithm which would give me those $Re(\cdot), Im(\cdot)$.
Any ideas?
This thing is called rectangular expansion of a complex expression.
MuPAD implements this. Mathematica as well.