  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.
