In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces. It is named after the German mathematician Hermann Schwarz.

The Schwarzian derivative of a holomorphic function f of one complex variable z is defined by

$${\displaystyle (Sf)(z)=\left({\frac {f''(z)}{f'(z)}}\right)'-{\frac {1}{2}}\left({\frac {f''(z)}{f'(z)}}\right)^{2}={\frac {f'''(z)}{f'(z)}}-{\frac {3}{2}}\left({\frac {f''(z)}{f'(z)}}\right)^{2}.}$$

The same formula also defines the Schwarzian derivative of a C³ function of one real variable. The alternative notation

$${\displaystyle \{f,z\}=(Sf)(z)}$$

is frequently used.

The Schwarzian derivative of any Möbius transformation

$${\displaystyle g(z)={\frac {az+b}{cz+d}}}$$