= Solution
Apply the numerical method to the scalar test equation $y'=\lambda y$, with $z=h\lambda$. Its <linear stability domain> consists of those $z\in\mathbb C$ for which the discrete solution stays bounded as the step count tends to infinity. For a one-step method with <stability function> $R$, this is
$$
\boxed{\mathcal S=\{z:|R(z)|\leq1\}.}
$$
If asymptotic decay is required, use $|R(z)|<1$ instead; this only changes the boundary conventions in part (b). A method is <A-stable> if its stability domain contains the left half-plane, so every decaying linear test equation is stable for every positive step size. For a multistep method, the corresponding definition uses the <root condition for a multistep method> for every characteristic root, including simplicity of unit-modulus roots.
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