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Multipole translation theory for the three-dimensional Laplace and Helmholtz equations. (English) Zbl 0852.31006
This paper presents a concise summary of the theory underlying the fast multipole methods (FMM) for both Laplace and Helmholtz equations in three dimensions.
First, the basic machinery needed to implement the FMM to Laplace equation is proved, taking great care to state the results in an efficient notation. Elementary proofs of three fundamental translation theorems for spherical harmonic functions due to earlier works of Carlson and Rushbroole are produced. This leads to the definition of a far-field signature function analogous to that of the Helmholtz equation.
The theory for the Helmholtz equation is developed in a new convolutional form of a translation operator, comparable with that of the translation operator for the Laplace equation. The connection with diagonal forms by means of Wigner $$3$$-$$j$$ symbols is established.
Both the exposition and notation of this paper are clear and concise.

##### MSC:
 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 35C10 Series solutions to PDEs 35C15 Integral representations of solutions to PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65R20 Numerical methods for integral equations 65N38 Boundary element methods for boundary value problems involving PDEs
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