The Jordan decomposition theorem for square matrices with complex coefficients is most commonly proved by means of algebraic methods. Every good theorem has several proofs which give different insights and generalise into different directions. The aim of this talk is to present an approach using complex analysis. We derive the Jordan decomposition theorem from the Laurent expansions of the resolvent about the eigenvalues of the matrix. This talk is presented by Alex Campbell and is based on joint work with Daniel Daners.