Schwarzian derivative

Remarks

Other notations are $\set{f, z}$ and $S_f$. The two expressions agree, since $(f''/f')' = f'''/f' - (f''/f')^2$. Example: for $f(z) = e^z$ on $\C$, $f'' / f' = 1$, so $S(f) = 0 - \frac{1}{2} = -\frac{1}{2}$. For a Möbius transformation $T(z) = (az + b)/(cz + d)$ with $ad - bc \ne 0$, one gets $S(T) = 0$; conversely $S(f) = 0$ on a domain only for restrictions of Möbius maps. Non-example: $f(z) = z^2$ on $\C$ has $f'(0) = 0$, so it is excluded; indeed $S(f)(z) = -\frac{3}{2z^2}$ on $\C \setminus \set{0}$ has a pole at $0$. The chain rule is $S(f \circ g) = (S(f) \circ g)\,(g')^2 + S(g)$, so $S(T \circ f) = S(f)$ for every Möbius $T$. The definition extends to locally injective meromorphic functions, with $S(f)$ holomorphic, since $S(1/f) = S(f)$.