Schwarz equation

The non-linear ordinary differential equation of the third order

$$\frac{z'''}{z'}-\frac32\left(\frac{z''}{z'}\right)^2=2I(t).\label{1}\tag{1}$$

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

$$z(t)=\frac{x_1(t)}{x_2(t)}\label{3}\tag{3}$$

satisfies the Schwarz equation \eqref{1}, where

$$I(t)=q(t)-\frac14p^2(t)-\frac12p'(t)$$

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.

References

[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)

Comments

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

References

[a1]	E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1969)
[a2]	Z. Nehari, "Conformal mapping" , Dover, reprint (1975) pp. Chapt. 7, §7