I am studying the stability of numerical methods. In this topic we take a numerical methods such as Forward Euler and we try to find the condition that make it stable or unstable.
for Forward Euler we have:
$ x_{n+1} = x_n + h f(t,x) , \qquad x' = f(x,t) = \lambda x(t) $
$ x_{n+1} = x_n + \lambda h x_n $
$ x_{n+1} = (1+\lambda h) x_n $
By induction: $x_n = (1+ \lambda h)^n x_0$
We can see that in order for the above iteration to converge when $n \rightarrow \infty$, we must have: $|(1+\lambda h)| < 1$
Now here the author of the article which I am reading sets $\lambda h = z$ and then assumes for a complex $\lambda$
$D_{Euler} = \{ z = \lambda h \in C: |z+1| < 1\}$
Which is a circle with radius 1, centered at origin of the complex plane.
My question is why do we need to consider complex numbers? It seems to me that the value of $\lambda$ can be only real. Why we need to consider $\lambda$ as a complex number?