On toroidal Green’s functions

Green’s functions are valuable analytical tools for solving a myriad of boundary-value problems in mathematical physics. Here, Green’s functions of the Laplacian and biharmonic operators are derived for a three-dimensional toroidal domain. In some sense, the former result may be regarded as “standard,” but the latter is most certainly not. It is shown that both functions can be constructed to have zero value on a specified toroidal surface with a circular cross section. Additionally, the Green’s function of the biharmonic operator may be chosen to have the property that its normal derivative also vanishes there. A “torsional” Green’s function is derived for each operator which is useful in solving some boundary-value problems involving axisymmetric vector equations. Using this approach, the magnetic vector potential of a wire loop is computed as a simple example.