Schwarz equation

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The non-linear ordinary differential equation of the third order


Its left-hand side is called the Schwarzian derivative of the function $z(t)$ and is denoted by $\{z,t\}$. H.A. Schwarz applied this equation in his studies [1].

If $x_1(t),x_2(t)$ is a fundamental system of solutions of the second-order linear differential equation

$$x''+p(t)x'+q(t)x=0,\quad p\in C^1,\quad q\in C,\label{2}\tag{2}$$

then on any interval where $x_2(t)\neq0$, the function


satisfies the Schwarz equation \eqref{1}, where


is the so-called invariant of the linear equation \eqref{2}. Conversely, any solution of the Schwarz equation \eqref{1} can be presented in the form \eqref{3}, where $x_1(t),x_2(t)$ are linearly independent solutions of \eqref{2}. Solutions of a Schwarz equation with a rational right-hand side in the complex plane are closely connected with the problem of describing the functions that conformally map the upper half-plane into a polygon bounded by a finite number of segments of straight lines and arcs of circles.


[1] H.A. Schwarz, "Ueber diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elements darstellt (nebst zwei Figurtafeln)" J. Reine Angew. Math. , 75 (1873) pp. 292–335
[2] V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)


For the relation with conformal mapping see [a2] and Christoffel–Schwarz formula.


[a1] E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1969)
[a2] Z. Nehari, "Conformal mapping" , Dover, reprint (1975) pp. Chapt. 7, §7
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Schwarz equation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article