  We consider a problem relating to magnetic confinement devices known as
stellarators. Plasma is confined by magnetic fields generated by
current-carrying coils, and here we investigate how closely to the plasma they
need to be positioned. Current-carrying coils are represented as singularities
within the magnetic field and therefore this problem can be modelled
mathematically as finding how far we can harmonically extend a vector field
from the boundary of a domain.
  For this paper we consider two-dimensional domains with real analytic
boundary, and prove that a harmonic extension exists if and only if the
boundary data satisfies a combined compatibility and regularity condition. Our
method of proof uses a generalisation of a result of Hadamard on the Cauchy
problem for the Laplacian. We then provide a lower bound on how far we can
harmonically extend the vector field from the boundary via the
Cauchy--Kovalevskaya Theorem.
