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  "body": "\n## Definition\n\nLet $U \\subseteq \\C$ be a [[domain|097KNJFJ7V]], and let $f \\colon U \\to \\C$ be [[holomorphic|0DG0Z72CQD]] with $f'(z) \\ne 0$ for all $z \\in U$.\nThe *Schwarzian derivative* of $f$ is the holomorphic function $S(f) \\colon U \\to \\C$ given by\n$$S(f) = \\left(\\frac{f''}{f'}\\right)' - \\frac{1}{2}\\left(\\frac{f''}{f'}\\right)^2 = \\frac{f'''}{f'} - \\frac{3}{2}\\left(\\frac{f''}{f'}\\right)^2.$$\n\n## Remarks\n\nOther notations are $\\set{f, z}$ and $S_f$.\nThe two expressions agree, since $(f''/f')' = f'''/f' - (f''/f')^2$.\nExample: for $f(z) = e^z$ on $\\C$, $f'' / f' = 1$, so $S(f) = 0 - \\frac{1}{2} = -\\frac{1}{2}$.\nFor a [[Möbius transformation|0B346DDAM5]] $T(z) = (az + b)/(cz + d)$ with $ad - bc \\ne 0$, one gets $S(T) = 0$; conversely $S(f) = 0$ on a domain only for restrictions of Möbius maps.\nNon-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 $0$.\nThe 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 $T$.\nThe definition extends to locally injective meromorphic functions, with $S(f)$ holomorphic, since $S(1/f) = S(f)$.\n",
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