# Plot FDE Amplification Factor in Complex Plane

I am reviewing stability of finite difference methods. I am using the von Neumann method to study a linear PDE and an explicit finite difference scheme. The goal is to develop an expression for an amplification factor $$G$$.

I've read that if $$G$$ is a complex function it can be plotted in the complex plane to more easily determine where the solution of the finite difference equation will be stable. For example, consider if $$G$$ was

$$G = (1-2d) + 2d\cos\theta$$

where $$d$$ is determined from my grid and time step sizes.

What is the template or formulas used to re-write this in terms of a circle (or some other shape, whatever it is) in the complex plane? Once I know that, I can plot it and hopefully tell where the stable region is.

My reference indicates this example of $$G$$ is an oscillation centered at $$(1-2d,0i)$$ with an amplitude of $$2d$$. It isn't clear to me how that was determined or how you could use that to plot it.