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Yes, every holomorphic vector bundle of any rank is trivial on the punctured disk $\dot{\Delta}$ . Indeed, since $\dot{\Delta}$ is a Stein manifold ( like any non-compact Riemann surface ! ) the Oka meta-principle (here a Theorem of Grauert ) says that the classification of holomorphic vector bundles on that manifold is the same as that of topological vector bundles. Since the punctured disk is homotopically equivalent to a circle , all topological complex vector bundles are trivial. (Do not confuse with the Möbius vector bundle, which is a non-trivial real vector bundle!)