  A method is given of deriving the distribution of planar Brownian motion
evaluated at certain stopping times using analytic functions. This method
relies upon a generalization of the standard conformal invariance of harmonic
measure. A number of examples are given, including several in which the
stopping time in question is not the exit time of a domain. It is also shown
how appropriate choices of domains and stopping times can lead to new proofs of
identities, including Euler's Basel sum and a generalization of Leibniz's
formula for $\pi$.
