= Absolute stability region
For a one-step method applied to $y'=\lambda y$, write $y_{n+1}=R(h\lambda)y_n$. Its <absolute stability region> is the set of complex $z$ where the stability <function> is defined and $|R(z)|\leq1$, using the bounded-iteration convention.
Explicit Euler has $R(z)=1+z$, giving the closed disk centered at $-1$ with radius one. For real $\lambda<0$, boundedness requires $h\leq2/|\lambda|$, while strict inequality gives decay. At the boundary $h=2/|\lambda|$, the numerical solution alternates with unchanged magnitude even though the exact solution decays. Some definitions use the strict region $|R(z)|<1$; stating the convention separates boundedness from asymptotic damping.
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