  In this work, we study the inverse spectral problems for the Sturm-Liouville
operators on [0,1] with complex coefficients and a discontinuity at
$x=a\in(0,1)$. Assume that the potential on (a,1) and some parameters in the
discontinuity and boundary conditions are given. We recover the potential on
(0,a) and the other parameters from the eigenvalues. This is the so-called
partial inverse problem. The local solvability and stability of the partial
inverse problems are obtained for $a\in(0,1)$, in which the error caused by the
given partial potential is considered. As a by-product, we also obtain two new
uniqueness theorems for the partial inverse problem.
