  In this paper, we find a holomorphic Darboux chart around any immersed
noncompact holomorphic Legendrian curve in a complex contact manifold
$(X,\xi)$. By using such a chart, we show that every holomorphic Legendrian
immersion $R\to X$ from an open Riemann surface can be approximated on
relatively compact subsets by holomorphic Legendrian embeddings, and every
holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann
surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such
that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian
embedding. We also establish a contact neighborhood theorem for isotropic Stein
submanifolds, and we find a holomorphic Darboux chart around any contractible
isotropic Stein submanifolds in an arbitrary complex contact manifold.
