Isolating a complex-valued variable

[TheCanadian]

If $A, B,$ and $C$ are complex-valued variables, is there an easy way to algebraically solve for $A$ without splitting the variables into their real and imaginary parts? For example:

$$ AB^* = 20AB - A^*C + B^*C^* $$

can be isolated for $A$ but only if the real and imaginary parts are distinguished. That's fine. I am just wondering if there are any other methods to solve for $A$ in the above equation as only a function of $B, B^*, C, C^*$ (i.e. not $A^*$)

 

[mfb]

I don't think there is a useful approach that avoids splitting A into components.

 

[chiro]

Hey TheCanadian.

Apart from getting the variable separated (which is what you would need to do to get an explicit function of a variable), you are going to have to also deal with the conjugate variable A*.

You could divide by A* to remove that variable and then look at some new variable which is a function of A and A* [or some other transformation which is a function of both] so that your new variable U is such that U = f(A,A*) and then solve for your A and A* once you solve for U.

I should point out though that algebraically solving with x + iy and a single complex variable will generate the same solution so if you can do it with the two dimensional version then it would be recommended.