Geoscience Reference
In-Depth Information
Formula (1-176) shows that not only functions that are defined on the
surface of a sphere can be expanded into a series of surface spherical har-
monics. Such an expansion is even possible for rather arbitrary functions
defined on a convex surface.
A remark on terminology
The ellipsoidal-harmonic coordinates
u, ϑ
(or
β
),
λ
are the generalization of
spherical coordinates for the sole use of getting closed solutions of Laplace's
equation, in particular, for the gravity field of the reference ellipsoid in
Sect. 2.7. The brief name “ellipsoidal coordinates” frequently used for
u, β, λ
might lead to a confusion with the ellipsoidal coordinates
ϕ, λ, h
.Inthe
present topic, “ellipsoidal coordinates” will always denote “ellipsoidal ge-
ographic coordinates”, frequently also called “geodetic coordinates”, being
represented by
ϕ, λ, h
.
