Two new integral transforms, ideally suited for solving boundary value problems in well hydraulics, are derived from one of the Goldstein identities which generalizes a corresponding Weber identity. The two transforms are, therefore, named the Weber-Goldstein transforms. Their properties are presented. For the first, second, and third type boundary conditions, the new transforms remove the radial portion of a Laplacian in the cylindrical coordinates. They are used to straightforwardly rederive known solutions to the problems of a fully penetrating flowing well and a fully penetrating pumped well. A novel solution for a fully penetrating flowing well with infinitesimal skin situated in a leaky aquifer is also found by means of one of the new transforms. This solution is validated by comparison to a numerical solution obtained via the finite-difference method and to a quasianalytic solution obtained by numerical inversion of the corresponding solution in the Laplace domain. Based on the new solution, a flowing well test is proposed for estimating the hydraulic conductivity and specific storativity of the aquifer and the skin factor of the well. The test can also be used in a constant-head injection mode. A type-curve estimation procedure is developed and illustrated with an example. The effectiveness of the test in estimating the well skin factor and aquifer parameters depends on the availability of data on the sufficiently early well response.