# Schwarz formula

A formula for a minimal surface, of the form

$$ \mathbf r ( u, v) = \mathop{\rm Re} \left \{ \mathbf r ( t) + i \int\limits [ \mathbf n , d \mathbf r ] \right \} , $$

where $ \mathbf r ( u, v) $ is the position vector of the minimal surface $ F $, $ \mathop{\rm Re} \{ \mathbf r ( t) \} $ is the position vector of an arbitrary non-closed analytic (with respect to $ t $) curve $ L $ on $ F $, and $ \mathbf n ( t) $ is the unit normal to $ F $ along $ L $( analytically dependent on $ t $). After integration, $ t $ is replaced by $ t = u+ iv $. This formula was established by H.A. Schwarz (1875); it gives an explicit solution to the Björling problem.

**How to Cite This Entry:**

Schwarz formula.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Schwarz_formula&oldid=48630