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Difference between revisions of "Schwarz equation"

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The non-linear ordinary differential equation of the third order
 
The non-linear ordinary differential equation of the third order
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083510/s0835101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$\frac{z'''}{z'}-\frac32\left(\frac{z''}{z'}\right)^2=2I(t).\tag{1}$$
  
Its left-hand side is called the Schwarzian derivative of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083510/s0835102.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083510/s0835103.png" />. H.A. Schwarz applied this equation in his studies [[#References|[1]]].
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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 [[#References|[1]]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083510/s0835104.png" /> is a [[Fundamental system of solutions|fundamental system of solutions]] of the second-order linear differential equation
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If $x_1(t),x_2(t)$ is a [[Fundamental system of solutions|fundamental system of solutions]] of the second-order linear differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083510/s0835105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$x''+p(t)x'+q(t)x=0,\quad p\in C^1,\quad q\in C,\tag{2}$$
  
then on any interval where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083510/s0835106.png" />, the function
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then on any interval where $x_2(t)\neq0$, the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083510/s0835107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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$$z(t)=\frac{x_1(t)}{x_2(t)}\tag{3}$$
  
satisfies the Schwarz equation (1), where
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satisfies the Schwarz equation \ref{1}, where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083510/s0835108.png" /></td> </tr></table>
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$$I(t)=q(t)-\frac14p^2(t)-\frac12p'(t)$$
  
is the so-called invariant of the linear equation (2). Conversely, any solution of the Schwarz equation (1) can be presented in the form (3), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083510/s0835109.png" /> are linearly independent solutions of (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.
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is the so-called invariant of the linear equation \ref{2}. Conversely, any solution of the Schwarz equation \ref{1} can be presented in the form \ref{3}, where $x_1(t),x_2(t)$ are linearly independent solutions of \ref{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====
 
====References====

Revision as of 11:51, 10 August 2014

The non-linear ordinary differential equation of the third order

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

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

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

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

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

satisfies the Schwarz equation \ref{1}, where

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

is the so-called invariant of the linear equation \ref{2}. Conversely, any solution of the Schwarz equation \ref{1} can be presented in the form \ref{3}, where $x_1(t),x_2(t)$ are linearly independent solutions of \ref{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
How to Cite This Entry:
Schwarz equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarz_equation&oldid=32796
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article