Spitzer's identity for discrete random walks

Spitzer's identity describes the position of a reflected random walk over time in terms of a bivariate transform. Among its many applications in probability theory are congestion levels in queues and random walkers in physics. We present a new derivation of Spitzer's identity under the assumption that the increments of the random walk have bounded jumps to the left. This mild assumption facilitates a proof of Spitzer's identity that only uses basic properties of analytic functions and contour integration. The main novelty, believed to be of broader interest, is a reversed approach that recognizes a factored polynomial expression as the outcome of Cauchy's formula.