# Schwarz formula

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