Let $\mathbb{R}$ denote the field of [real numbers](/page/Real%20Numbers), let $\mathbb{C}$ denote the field of complex numbers, and let $\mathbb{N}=\{1,2,3,\dots\}$. For the Dahlquist test equation $y'(t)=\lambda y(t)$ with $\lambda\in\mathbb{C}$ and time step $k>0$, define the stability variable $z:=k\lambda$. Applying the Backward Euler method produces the one-step recurrence for a sequence $(y_n)_{n=0}^{\infty}\subset\mathbb{C}$ given by

\begin{align*} y_{n+1}=y_n+k\lambda y_{n+1}. \end{align*}

The associated stability function is

\begin{align*} R:\mathbb{C}\setminus\{1\}\to\mathbb{C},\quad R(z)=\frac{1}{1-z}. \end{align*}

It is L-stable: it is A-stable, meaning $|R(z)|\leq 1$ for every $z\in\mathbb{C}$ with $\operatorname{Re}(z)\leq 0$, and it satisfies the stiff-decay condition

\begin{align*} \lim_{\substack{|z|\to\infty, \operatorname{Re}(z)\leq 0}} R(z)=0. \end{align*}