Precise efforts in theoretical astrophysics are needed to fully understand the mechanisms which govern the structure, stability, dynamics, formation, and evolution of differentially rotating stars. Direct computation of the physical attributes of a star can be facilitated by the use of highly compact azimuthal and separation angle Fourier formulations of the linear PDE's of mathematical physics. These mathematical reformulations are through Green's functions which represent the inhomogeneous contribution to the PDE from a point source or sink. The Green's functions for the linear PDE's of mathematical physics are of great importance in many areas of modern physics such as in diffusion and wave phenomenon. The Green's function for Laplace's equation is used in classical and quantum physics formulations of the Coulomb and Newtonian interactions. We express the Green's function formulation as sums and integrals over eigenfunction separated solutions in those specific coordinate systems which allow separable solutions to these PDE's. Those rotationally invariant coordinate systems which allow separable solutions to these linear PDE's are each commonly expressible in terms of the azimuthal angle. For these rotationally invariant coordinate systems, the Green's functions are seen to be expressible in terms of an azimuthal and separation angle Fourier basis description. This description leads to reformulations of many branches of theoretical physics and pure mathematics. The ultimate resolution of these analytical investigations will be efficient algorithmic implementations of these new schemes. We propose these methods to the 3D star community in hope that you may continue to enjoy significantly improved economical and precise boundary values for studies of analytical and numerical 3D stellar astrophysics. Numerical implementations of this approach for gravity will be presented in terms of the 3D djehuty context and describe current research investigations.

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