In Von Neumann's stability analysis of Finite Difference Equations, the Euler's formula is used to describe a perturbation. Now the finite difference equation is stable if this perturbation does not grow in time.
Now, my question really is: how do I plot the perturbation (Equation 1) three-dimensions(?)/Im vs Real dimensions?
$$\epsilon(x,t) = \sum_{m=1}^{M} e^{at} e^{ik_m x} \ldots \ldots \text{(1)}$$
I realize that the the Euler's formula $e^{ik_m x}$ describes a circle. Clearly the multiplicative amplitude factor $e^{at}$ either "grows" this circle or decreases it whether or not $e^{at}$ is greater than 1 or not.
I would like to describe that as a Manipulate
or a ListAnimate
in Mathematica but I don't understand which function to use.
So far I have understood that PolarPlot
and ParametricPlot
draw this circle in 2D.
Parametric Plot example
ParametricPlot[G {Re[Exp[I x]], Im[Exp[I x]]}, {x, 0, 2 \[Pi]},
PlotRange -> {{-1, 1}, {-1, 1}}]
Polar Plot example
PolarPlot[{Re[Exp[I x]], Im[Exp[I x]]}, {x, 0, Pi}]