Abstract
This chapter focuses on the elementary solutions of the 3D Helmholtz equation in spherical coordinates. The major advantage of using spherical coordinates is that they provide a countable basis of functions orthogonal on a sphere that can be used for the representation of any other solution of the Helmholtz equation. Some of the properties of special functions constituting such solutions are described and two basic types of solutions, regular and singular are realized. Several differentiation theorems, which enable to represent derivatives of the elementary solutions via the elementary solutions of different degree and order are proved. In addition two different types of solutions, multipoles and plane waves, which can be expressed through the elementary spherical basis functions for the Helmholtz equation are also considered. Finally, the process through which convergent infinite series over the spherical basis functions, appear from the far field, and local expansions of an arbitrary solution of the Helmholtz equation are considered.
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