  Let $u$ be a harmonic function in a $C^1$-Dini domain, such that $u$ vanishes
on an open set of the boundary. We show that near every point in the open set,
$u$ can be written uniquely as the sum of a non-trivial homogeneous harmonic
polynomial and an error term of higher degree (depending on the Dini
parameter). In particular, this implies that $u$ has a unique tangent function
at every such point, and that the convergence rate to the tangent function can
be estimated. We also study the relationship of tangent functions at nearby
points in a special case.
