# Löwner equation

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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 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 ).

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 ; the Kufarev–Löwner equation was obtained by P.P. Kufarev (see ).

How to Cite This Entry:
Löwner equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%B6wner_equation&oldid=33195
This article was adapted from an original article by V.Ya. Gutlyanskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article