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It is well-known that Laplace's equation
|
(3) |
admits a number of different separable solutions for the function
that are given in terms of products of known special
functions whose arguments, in turn, are given in terms a triply-orthogonal
set of curvilinear coordinates
(Bôcher 1894; §10.3 in Morse & Feshbach 1953; §3.6 in Miller
1977). These solutions can be classified as being either simply separable,
or -separable
,
where the modulation factor
(§5.1 in Morse & Feshbach 1953) is a scalar function
that has a unique specification for each coordinate system.
The simply separable coordinate systems can be geometrically characterized
by surfaces that are quadric (second-order); whereas the -separable
coordinate systems are characterized by surfaces which are cyclidic
(fourth-order). Using a Lie group theoretic approach Miller (1977)
has demonstrated that there are precisely seventeen conformally
independent separable coordinate systems that are either simply separable or
-separable for Laplace's equation. In general,
these separable Laplace systems can be broken up into three classes: a
cylindrical class (i.e., invariant under vertical translations), a
rotational class (i.e., invariant under rotations about the z-axis),
and a more general class.
Here we will focus on the rotational class of Laplace systems, that is, those
systems that correspond to the diagonalization of the z-component of the
angular momentum operator
|
(4) |
where
is the azimuthal coordinate. These
coordinate systems share the special property that their
eigenfunctions take the form
|
(5) |
where R represents the distance from the z-axis
(i.e., the cylindrical radius), z is the vertical height, and
|
(6) |
If we substitute (5) into Laplace's equation and factor
out
,
we obtain a differential equation for ,
which is the foundation of generalized axisymmetric potential theory;
written in cylindrical coordinates, for example, this key equation takes
the form,
|
(7) |
Miller (1977) also has shown that there are precisely nine conformally
independent rotational Laplace systems. Five of these coordinate systems
(cylindrical, spherical, prolate spheroidal, oblate spheroidal,
and, parabolic) are quadric and simply separable for Laplace's equation;
the remaining four (one of which is toroidal) are cyclidic and
separable for Laplace's equation. Moon & Spencer
(1961b, §IV) have tabulated ten fourth-order rotational Laplace systems.
Consulting Miller's (1977) study, we know that under any conformal symmetry
of the Laplace equation, an separable coordinate system can be
mapped to another separable coordinate system. Therefore, one
might question whether all of Moon & Spencer's (1961b) rotational Laplace
systems have unique double summation/integration Green's function
expansions, and how many such coordinate systems exist.
Discussions of the double summation/integration expressions for the
Green's function expansions (Jackson 1975, Chapter 3;
Morse & Feshbach 1953, Chapter 10) for cylindrical
(Cohl & Tohline 1999; see also §3 below) and spherical
coordinates (Cohl et al. 2000) is abundant in the literature.
Double summation/integration expressions for the Green's function
for the three remaining quadric rotational Laplace systems (oblate
spheroidal, prolate spheroidal, and, parabolic coordinates) also have
been previously presented in the literature (Hobson 1931, §§245 and 251;
Morse & Feshbach 1953, §10.3) and therefore present themselves easily
for comparison. As for the cyclidic coordinate systems, the double
summation expressions for the Green's functions in toroidal and
bispherical coordinates have been developed (cf., Morse & Feshbach 1953,
§10.3; Hobson 1931, §258; see also §6.2 below), but we have been
unable to locate the double summation/integration Green's function
expansions for the other rotational cyclidic Laplace systems. The three
remaining conformally unique cyclidic rotational Laplace systems have
separable transcendental function solutions to Laplace's
equation that are in the form of Lamé functions and Jacobi
elliptic functions. We anticipate discussing these less well known
cyclidic coordinate systems, as well as the remaining rotational
Laplace coordinate systems presented in Moon & Spencer (1961b), in
forthcoming investigations.