Schwarzian derivative and second-order linear ODEs

Statement

Let $D\subseteq\mathbb C$ be simply connected, let $q$ be holomorphic on $D$ , and let $y_1,y_2$ be linearly independent solutions of

y''+\frac12q\,y=0.

Then the meromorphic map

f=\frac{y_1}{y_2}:D\longrightarrow\widehat{\mathbb C}

is locally univalent and has Schwarzian derivative

S(f)=q.

Basis independence

Replacing $(y_1,y_2)$ by another basis of the two-dimensional solution space postcomposes $f$ by a Möbius transformation. The Schwarzian chain rule therefore makes $S(f)$ independent of the chosen solution basis.

Converse

Locally, every locally univalent meromorphic function $f$ arises as such a ratio with $q=S(f)$ . One may take

y_2=(f')^{-1/2}, \qquad y_1=f(f')^{-1/2}

after choosing a local square root. This correspondence explains why solutions of the Schwarzian equation $S(f)=q$ are unique up to Möbius postcomposition.

References

Zeev Nehari, Conformal Mapping, Dover, 1975. Relevant: Chapter VI.
R. C. Gunning, Lectures on Riemann Surfaces, Princeton University Press, 1966. Relevant: projective connections and second-order differential equations.