= Solution
Apply a one-step numerical method to the <scalar> test <ordinary differential equation>
$$
y'=\lambda y.
$$
If one step has the form
$$
y_{n+1}=R(z)y_n,
\qquad z=h\lambda,
$$
then $R$ is the <stability function> and the <linear stability domain> is
$$
\boxed{\mathcal S=\{z\in\mathbb C:|R(z)|\leq1\}}.
$$
The method is <A-stable> when
$$
\boxed{\{z:\operatorname{Re}z\leq0\}\subseteq\mathcal S},
$$
so it does not amplify any mode that the exact <differential equation> does not amplify.
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