152 Changes of variable in the rational functions

Consider $𝐂(s)$ and let $g(s)\in 𝐂(s)$. Then the map $\lambda \mapsto g(s)$ and $1\mapsto 1$ determines a homomorphism of fields $𝐂(\lambda )\to 𝐂(s)$ via $f(\lambda )\mapsto f(g(s)).$ Consider $a,b,c,b\in 𝐂$ such that $ad-bc\ne 0$, and write

$$\lambda =\frac{as+b}{cs+d},s=\frac{d\lambda -b}{-c\lambda +d}.$$

There is an isomorphism of fields $𝐂(\lambda )\to 𝐂(s):$ $f(\lambda )\mapsto f(\frac{as+b}{cs+d})$ with inverse $f(s)\mapsto f(\frac{d\lambda -b}{-c\lambda +d}).$ In particular, we can take

$$\lambda =\frac{1}{s+1},s=\frac{\lambda -1}{-\lambda }.$$

Let $P(\lambda )\in 𝐂[\lambda ]$. Then $P(1/(1+s))$ gives a stable rational function in $s$.