The components of the general solutions are particular solutions of the Helmholtz equation in Cartesian, cylindrical, and
spherical coordinates. (Ringgold, Inc., Portland, OR)
The three-dimensional Laplace operator in
spherical coordinates is given by [18]
For the dyadic representation of integral equations in
spherical coordinates, when all incoming waves are collected in a certain finest box, due to the discontinuities in the spherical vector components, the Gibbs phenomenon will arise if the integral operation of the spherical harmonics representation is also evaluated in
spherical coordinates [6].
Attempts to solve similar equations in
spherical coordinates in the presence of an electric dipole have been made.
Gouesbet, "T-matrix formulation and generalized Lorenz-Mie theories in
spherical coordinates," Optics Communications, vol.
where u = sin [theta] cos [phi] and v = sin [theta] sin [phi] are the usual
spherical coordinates. Eq.
The last integral using
spherical coordinates gives the estimate [N.sup.m+2([alpha]-[beta]) which tends to 0 under n [right arrow] [delta] if [beta] > [alpha] + m/2.
It is well known that the associated Legendre polynomials play an important role in the central fields when one solves the physical problems in the
spherical coordinates. However, in the case of the noncentral fields we have to introduce the universal associated Legendre polynomials [P.sup.m'.sub.l'] (x) when one studies the modified Poschl-Teller [1], the single and double ring-shaped potentials, and time-dependent potential [2-4].
Therefore, the point on the unit sphere that corresponds to our texture point (x, y) is given in
spherical coordinates as:
Spherical harmonics (SH) represent a complete set of angular functions in
spherical coordinates, where the position of a point is defined by the polar radius r and two polar and azimuthal spherical angles, [theta] and [phi], respectively.
We consider an infinitesimal mass dM of the sphere represented by its
spherical coordinates (r, [theta], [phi]), where r is the radial distance, [theta] the polar angle, and [phi] the azimuthal angle (see Figure 4).