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H. Cohl (Louisiana State Univ.)
In this dissertation talk we present a mathematical result that, to the best of our knowledge, has been previously undiscovered. That is, the Green's function in a variety of orthogonal coordinate systems may be expressed in terms of a single sum over the azimuthal quantum number, m, of terms involving Toroidal Harmonics. We show how this new addition theorem can be effectively applied to a variety of potential problems in gravitation, electrostatics and magnetostatics and, in particular, demonstrate how it may be used to analyze the properties of general nonaxisymmetric disk systems with and without vertical extent. Finally, we describe our numerical implementation of the addition theorem in order to determine the Newtonian potential extremely close to highly flattened mass distributions. This yields an extremely efficient technique for computing the boundary values in a general algorithm that is designed to solve the 3D Poisson equation on a cylindrical coordinate lattice.
We acknowledge support from the U.S. National Science Foundation through grant AST-9528424 and DGE-9355007, the latter of which has been issued through the NSF's Graduate Traineeships Program. This work also has been supported, in part, by grants of high-performance-computing time on NPACI facilities at SDSC and UT, Austin, and through the PET program of NAVOCEANO DoD Major Shared Resource Center in Stennis, MS.
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