View source for Löwner equation

{{TEX|done}}
A differential equation of the form

$$\frac{dw}{dt}=-w\frac{1+e^{i\alpha(t)}w}{1-e^{i\alpha(t)}w},$$

where $\alpha(t)$ is a real-valued continuous function on the interval $-\infty<t<\infty$. A generalization of the Löwner equation is the Kufarev–Löwner equation:

$$\frac{dw}{dt}=-wP(w,t),$$

where $P(w,t)$, $|w|<1$, $-\infty<t<\infty$, is a function measurable in $t$ for fixed $w$ and regular in $w$, with positive real part, normalized by the condition $P(0,t)=1$. The Löwner equation and the Kufarev–Löwner equation, which arise in the theory of univalent functions, are the basis of the [[Variation-parametric method|variation-parametric method]] of investigating extremal problems on conformal mapping.

The solution $w(t,z,\tau)$, $w(\tau,z,\tau)=z$, of the Kufarev–Löwner equation, regarded as a function of the initial value $z$, for any $t>\tau$ maps the disc $|z|<1$ conformally onto a one-sheeted simply-connected domain belonging to the disc $|w|<1$. From the formula

$$f(z)=a+b\lim_{t\to\infty}e^tw(t,z,0),$$

by a suitable choice of $P(w,t)$ in the Kufarev–Löwner equation and complex constants $a,b$ one can obtain an arbitrary regular univalent function in the disc $|z|<1$. In this way the Löwner equation generates, in particular, the conformal mappings of the disc onto domains obtained from the whole plane by making a slit along some Jordan arc (see [[#References|[1]]]–[[#References|[4]]]).

The partial differential equation

$$\frac{\partial f(z,\tau)}{\partial\tau}=z\frac{\partial f(z,\tau)}{\partial z}P(z,\tau),$$

which is satisfied by the function

$$f(z,\tau)=\lim_{t\to\infty}e^tw(t,z,\tau),$$

is also called the Kufarev–Löwner equation.

The Löwner equation was set up by K. Löwner [[#References|[1]]]; the Kufarev–Löwner equation was obtained by P.P. Kufarev (see [[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Löwner,   "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I"  ''Math. Ann.'' , '''89'''  (1923)  pp. 103–121</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.P. Kufarev,   "A theorem on solutions of a differential equation"  ''Uchen. Zap. Tomsk. Gos. Univ.'' , '''5'''  (1947)  pp. 20–21  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Pommerenke,   "Ueber die Subordination analytischer Funktionen"  ''J. Reine Angew. Math.'' , '''218'''  (1965)  pp. 159–173</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.Ya. Gutlyanskii,   "Parametric representation of univalent functions"  ''Soviet Math. Dokl.'' , '''11'''  (1970)  pp. 1273–1276  ''Dokl. Akad. Nauk SSSR'' , '''194''' :  4  (1970)  pp. 750–753</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.P. Kufarev,   "On one-parameter families of analytic functions"  ''Mat. Sb.'' , '''13'''  (1943)  pp. 87–118  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>



====Comments====
For more information see also [[Löwner method|Löwner method]].