Green’s Functions

Abstract

Green’s functions are useful in solving the first boundary value problem (Dirichlet problem) of potential theory in itself and in the case of conformal mapping of a region onto a disk. In the latter case a relationship is needed between the conformal map and Green’s function for the region. An approximate determination of Green’s functions is an important numerical tool in solving both the Dirichlet problem for different types of regions and the related mapping problem. An integral representation of Green’s function for the disk leads to the Poisson integral. The Dirichlet problem is a special case of the Riemann—Hilbert problem which is discussed in Appendix C. Analogous to Green’s functions, the solution of the second boundary value problem (Neumann problem) of potential theory is the Neumann function which also possesses an integral representation.