  It has been known for some time that the Green's function of a planar domain
can be defined in terms of the exit time of Brownian motion, and this
definition has been extended to stopping times more general than exit times. In
this paper, we extend the notion of conformal invariance of Green's function to
analytic functions which are not injective, and use this extension to calculate
the Green's function for a stopping time defined by the winding of Brownian
motion. These considerations lead to a new proof of the Riemann mapping
theorem. We also show how this invariance can be used to deduce several
identities, including the standard infinite product representations of several
trigonometric functions.
