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.

Example: Solve

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

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