Solving Absolute Value Complex Equations
Absolute value equations typically do not have single solutions, or even a set of solutions which can be listed. Typically, the solution describes a curve in the complex plane. To take a very simple example, the equationhas the solution given in polar form asor in cartesian form aswith
Often it is easiest to find the solution in cartesian form by substituting z=x+iy and collecting real and imaginary terms, squaring and adding them to give a real number.
Write the equation asand multiply byto give(1)
Now substitute z=x+iy.
Substitute these two expressions into (1) to obtain
Square both sides to give
Now multiply out the brackets and collect like terms.
Divide by 3 and complete the square.
This is the equation of a circle with centreand radius