Schwarzian derivative ($S[u]=\{u,x\}$)

For a thrice differentiable function with $u^{\prime }\ne 0$, the Schwarzian derivative isIt vanishes exactly for fractional linear functions $u(x)=(ax+b)/(cx+d)$ on intervals where the denominator does not vanish. In a two-dimensional conformal field theory, it is the inhomogeneous term in the finite conformal transformation of a stress-energy tensor with nonzero central charge.

For $u_j=u^{(j)}$, the $n$th prolongations of the translation, dilation, and special projective generators acting on the independent variable are

For $V_1={\partial }_x$, $V_2=x{\partial }_x$, and $V_3=x^2{\partial }_x$, direct application of their third prolongations givesThus the Schwarzian is invariant under translation and has weight minus two under dilation; the third identity is its infinitesimal projective transformation law.