A generalized form of the Poisson Integral Formula is presented and applied to obtain a new class of two-dimensional, incompressible, potential flowfield solutions for free and confined jets. Two basic configurations are considered that demonstrate the procedure, a jet issuing from a straight wall or flowing from a semi-infinite, constant area duct. In either case the jet can be free or confined, with or without uniform external flow. A discretized form of the Poisson Integral Formula is also presented so that arbitrary boundary value distributions (i.e., boundary conditions) can be handled, including those with functional and/or slope discontinuities. The procedure makes use of conformal mapping and complex variable theory so that jet flowfield determination is reduced to solution of a boundary value problem on a simple domain (i.e., upper half-plane). The entire velocity field is described analytically; stream function and velocity potential contour maps are readily constructed. These solutions can serve as the zero-order inviscid outer solution (replacing the usual assumption of uniform or quiescent outer flow) in a matched asymptotic expansion process whereby viscous effects are introduced near the jet axis by means of a boundary-layer-type analysis. A number of example solutions are presented, most without any form of singularity in the flowfield as a result of having the freedom to arbitrarily select boundary conditions. Solutions are also presented having separation and recirculation regions adjacent to sharp jet exit corners. Several additional examples of the generalized Poisson Integral Formula applied to other types of problems are also described, including periodic boundary distributions and distributions having discontinuities.