Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry
Due to the isotropy of d-dimensional hyperspherical space, one expects there to exist a spherically symmetric opposite antipodal fundamental solution for its corresponding Laplace-Beltrami operator. The R-radius hypersphere SdR with R>0, represents a Riemannian manifold with positive-constant sec...
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| Zitieren: | Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry / H.S. Cohl // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 39 назв. — англ. |
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| citation_txt | Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry / H.S. Cohl // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 39 назв. — англ. |
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| description | Due to the isotropy of d-dimensional hyperspherical space, one expects there to exist a spherically symmetric opposite antipodal fundamental solution for its corresponding Laplace-Beltrami operator. The R-radius hypersphere SdR with R>0, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric opposite antipodal fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the trigonometric sine, finite summation expressions over trigonometric functions, Gauss hypergeometric functions, and in terms of the Ferrers function of the second with degree and order given by d/2−1 and 1−d/2 respectively, with real argument x∈(−1,1).
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 108, 15 pages
Opposite Antipodal Fundamental Solution
of Laplace’s Equation in Hyperspherical Geometry
Howard S. COHL †‡
† Applied and Computational Mathematics Division, Information Technology Laboratory,
National Institute of Standards and Technology, Mission Viejo, California, 92694 USA
E-mail: howard.cohl@nist.gov
‡ Department of Mathematics, University of Auckland, 38 Princes Str., Auckland, New Zealand
Received August 18, 2011, in final form November 22, 2011; Published online November 29, 2011
https://doi.org/10.3842/SIGMA.2011.108
Abstract. Due to the isotropy of d-dimensional hyperspherical space, one expects there to
exist a spherically symmetric opposite antipodal fundamental solution for its corresponding
Laplace–Beltrami operator. The R-radius hypersphere Sd
R with R > 0, represents a Rieman-
nian manifold with positive-constant sectional curvature. We obtain a spherically symmetric
opposite antipodal fundamental solution of Laplace’s equation on this manifold in terms of
its geodesic radius. We give several matching expressions for this fundamental solution in-
cluding a definite integral over reciprocal powers of the trigonometric sine, finite summation
expressions over trigonometric functions, Gauss hypergeometric functions, and in terms of
the Ferrers function of the second with degree and order given by d/2 − 1 and 1 − d/2
respectively, with argument x ∈ (−1, 1).
Key words: hyperspherical geometry; opposite antipodal fundamental solution; Laplace’s
equation; separation of variables; Ferrers functions
2010 Mathematics Subject Classification: 35A08; 35J05; 32Q10; 31C12; 33C05
1 Introduction
We compute closed-form expressions of a spherically symmetric opposite antipodal Green’s func-
tion (opposite antipodal fundamental solution of Laplace’s equation) for a d-dimensional Rie-
mannian manifold of positive-constant sectional curvature, namely the R-radius hypersphere
with R > 0. This problem is intimately related to the solution of the Poisson equation on this
manifold and the study of spherical harmonics which play an important role in exploring col-
lective motion of many-particle systems in quantum mechanics, particularly nuclei, atoms and
molecules. In these systems, the hyperradius is constructed from appropriately mass-weighted
quadratic forms from the Cartesian coordinates of the particles. One then seeks either to identify
discrete forms of motion which occur primarily in the hyperradial coordinate, or alternatively
to construct complete basis sets on the hypersphere. This representation was introduced in
quantum mechanics by Zernike & Brinkman [39], and later invoked to greater effect in nuclear
and atomic physics, respectively, by Delves [6] and Smith [33]. The relevance of this representa-
tion to two-electron excited states of the helium atom was noted by Cooper, Fano & Prats [5];
Fock [11, 12] had previously shown that the hyperspherical representation was particularly ef-
ficient in representing the helium wave function in the vicinity of small hyperradii. There has
been a rich literature of applications ever since. Examples include Zhukov [40] (nuclear struc-
ture), Fano [10] and Lin [26] (atomic structure), and Pack & Parker [30] (molecular collisions).
A recent monograph by Berakdar [2] discusses hyperspherical harmonic methods in the general
context of highly-excited electronic systems. Useful background material relevant for the math-
ematical aspects of this paper can be found in [24, 35, 37]. Some historical references on this
topic include [19, 25, 31, 32, 38].
mailto:howard.cohl@nist.gov
https://doi.org/10.3842/SIGMA.2011.108
2 H.S. Cohl
This paper is organized as follows. In Section 2, we describe hyperspherical geometry and its
corresponding metric, global geodesic distance function, Laplace–Beltrami operator (Laplacian),
and hyperspherical global geodesic polar coordinate systems which parametrize points on this
manifold. In Section 3, for hyperspherical geometry, we show how to compute ‘radial’ harmonics
in a geodesic polar coordinate system and derive several alternative expressions for a ‘radial’
opposite antipodal fundamental solution of the Laplace’s equation on the R-radius hypersphere.
Throughout this paper we rely on the following definitions. For a1, a2, . . . ∈ C, if i, j ∈ Z and
j < i then
j∑
n=i
an = 0 and
j∏
n=i
an = 1. The set of natural numbers is given by N := {1, 2, 3, . . .},
the set N0 := {0, 1, 2, . . .} = N ∪ {0}, the set of integers is given by Z := {0,±1,±2, . . .}, the
sets of real and complex numbers are given by R and C respectively.
2 Hyperspherical geometry
The Euclidean inner product for Rd+1 is given by (x,y) = x0y0 +x1y1 + · · ·+xdyd. The variety
(x,x) = x20 +x21 + · · ·+x2d = R2, for x ∈ Rd+1 and R > 0, defines the R-radius hypersphere SdR.
We denote the unit radius hypersphere by Sd := Sd1. Hyperspherical space in d-dimensions, de-
noted by SdR, is a maximally symmetric, simply connected, d-dimensional Riemannian manifold
with positive-constant sectional curvature (given by 1/R2, see for instance [24, p. 148]), whereas
Euclidean space Rd equipped with the Pythagorean norm, is a Riemannian manifold with zero
sectional curvature.
Points on the d-dimensional hypersphere SdR can be parametrized using subgroup-type coor-
dinate systems, i.e., those which correspond to a maximal subgroup chain O(d) ⊃ · · · (see for
instance [20, 22]). The isometry group of the space SdR is the orthogonal group O(d). Hyper-
spherical space SdR, can be identified with the quotient space O(d)/O(d − 1). The isometry
group O(d) acts transitively on SdR. There exist separable coordinate systems on the hyper-
sphere, analogous to parabolic coordinates in Euclidean space, which can not be constructed
using maximal subgroup chains. Polyspherical coordinates, are coordinates which correspond to
the maximal subgroup chain given by O(d) ⊃ · · · . What we will refer to as standard hyper-
spherical coordinates, correspond to the subgroup chain given by O(d) ⊃ O(d− 1) ⊃ · · · ⊃ O(2).
(For a thorough discussion of polyspherical coordinates see [37, Section IX.5].) Polyspherical
coordinates on SdR all share the property that they are described by (d+1)-variables: R ∈ [0,∞)
plus d-angles each being given by the values [0, 2π), [0, π], [−π/2, π/2] or [0, π/2] (see [20, 21]).
In our context, a useful subset of polyspherical coordinate are geodesic polar coordinates
(θ, x̂) (see for instance [29]). These coordinates, which parametrize points on SdR, have origin
at O = (R, 0, . . . , 0) ∈ Rd+1 and are given by a ‘radial’ parameter θ ∈ [0, π] which parametrizes
points along a geodesic curve emanating from O in a direction x̂ ∈ Sd−1. Geodesic polar
coordinate systems partition SdR into a family of (d − 1)-dimensional hyperspheres, each with
a ‘radius’ θ := θd ∈ (0, π), on which all possible hyperspherical coordinate systems for Sd−1 may
be used (see for instance [37]). One then must also consider the limiting case for θ = 0, π to
fill out all of SdR. Standard hyperspherical coordinates (see [23, 27]) are an example of geodesic
polar coordinates, and are given by
x0 = R cos θ,
x1 = R sin θ cos θd−1,
x2 = R sin θ sin θd−1 cos θd−2,
· · · · · · · · · · · · · · · · · · · · · · · · · · ·
xd−2 = R sin θ sin θd−1 · · · cos θ2,
xd−1 = R sin θ sin θd−1 · · · sin θ2 cosφ,
xd = R sin θ sin θd−1 · · · sin θ2 sinφ,
(1)
θi ∈ [0, π] for i ∈ {2, . . . , d}, θ = θd, and φ ∈ [0, 2π).
Opposite Antipodal Fundamental Solution of Laplace’s Equation 3
In order to study an opposite antipodal fundamental solution of Laplace’s equation on the
hypersphere, we need to describe how one computes the geodesic distance in this space. Geodesic
distances on SdR are simply given by arc lengths, angles between two arbitrary vectors, from the
origin in the ambient Euclidean space (see for instance [24, p. 82]). Any parametrization of the
hypersphere SdR, must have (x,x) = x20 + · · ·+ x2d = R2, with R > 0. The distance between two
points x,x′ ∈ SdR on the hypersphere is given by
d(x,x′) = Rγ = R cos−1
(
(x,x′)
(x,x)(x′,x′)
)
= R cos−1
(
1
R2
(x,x′)
)
. (2)
This is evident from the fact that the geodesics on SdR are great circles, i.e., intersections of SdR
with planes through the origin of the ambient Euclidean space, with constant speed parametriza-
tions.
In any geodesic polar coordinate system, the geodesic distance between two points on the
submanifold is given by
d(x,x′) = R cos−1
(
1
R2
(x,x′)
)
= R cos−1
(
cos θ cos θ′ + sin θ sin θ′ cos γ
)
, (3)
where γ is the unique separation angle given in each polyspherical coordinate system used to
parametrize points on Sd−1. For instance, the separation angle γ in standard hyperspherical
coordinates is given through
cos γ = cos(φ− φ′)
d−2∏
i=1
sin θisin θi
′ +
d−2∑
i=1
cos θicos θi
′
i−1∏
j=1
sin θjsin θj
′. (4)
Corresponding separation angle formulae for any hyperspherical coordinate system used to
parametrize points on Sd−1 can be computed using (2) and the associated formulae for the
appropriate inner-products.
One can also compute the Riemannian (volume) measure dvolg (see for instance [18, Sec-
tion 3.4]), invariant under the isometry group SO(d), of the Riemannian manifold SdR. For
instance, in standard hyperspherical coordinates (1) on SdR the volume measure is given by
dvolg = Rd sind−1 θ dθ dω := Rd sind−1 θ dθ sind−2 θd−1 · · · sin θ2 dθ1 · · · dθd−1. (5)
The distance r ∈ [0,∞) along a geodesic, measured from the origin, is given by r = θR. To show
that the above volume measure (5) reduces to the Euclidean volume measure at small distances
(see for instance [23]), we examine the limit of zero curvature, the flat-space limit. In order to
do this, we take the limit θ → 0+ and R→∞ of the volume measure (5) which produces
dvolg ∼ Rd−1 sind−1
( r
R
)
dr dω ∼ rd−1dr dω,
which is the Euclidean measure on Rd, expressed in standard Euclidean hyperspherical coordi-
nates. This measure is invariant under the Euclidean motion group E(d).
It will be useful below to express the Dirac delta distribution on SdR. The Dirac delta
distribution on the Riemannian manifold SdR with metric g is defined for an open set U ⊂ SdR
with x,x′ ∈ SdR such that∫
U
δg(x,x
′)dvolg =
{
1 if x′ ∈ U,
0 if x′ /∈ U.
(6)
For instance, using (5) and (6), in standard hyperspherical coordinates on SdR (1), we see that
the Dirac delta distribution is given by
δg(x,x
′) =
δ(θ − θ′)
Rd sind−1 θ′
δ(θ1 − θ′1) · · · δ(θd−1 − θ′d−1)
sin θ′2 · · · sind−2 θ′d−1
.
4 H.S. Cohl
2.1 The Laplace and Poisson equations on the hypersphere
Parametrizations of a submanifold embedded in Euclidean space can be given in terms of coor-
dinate systems whose coordinates are curvilinear. These are coordinates based on some trans-
formation that converts the standard Cartesian coordinates in the ambient space to a coordinate
system with the same number of coordinates as the dimension of the submanifold in which the
coordinate lines are curved.
The Laplace–Beltrami operator (Laplacian) in curvilinear coordinates ξ =
(
ξ1, . . . , ξd
)
on
a Riemannian manifold is given by
∆ =
d∑
i,j=1
1√
|g|
∂
∂ξi
(√
|g|gij ∂
∂ξj
)
, (7)
where |g| = | det(gij)|, the metric is given by
ds2 =
d∑
i,j=1
gijdξ
idξj , (8)
and
d∑
i=1
gkig
ij = δjk,
where δji ∈ {0, 1} is the Kronecker delta
δji :=
{
1 if i = j,
0 if i 6= j,
(9)
for i, j ∈ Z. The relationship between the metric tensor Gij = diag(1, . . . , 1) in the ambient
space and gij of (7) and (8) is given by
gij(ξ) =
d∑
k,l=0
Gkl
∂xk
∂ξi
∂xl
∂ξj
.
The Riemannian metric in a geodesic polar coordinate system on the submanifold SdR is
given by
ds2 = R2
(
dθ2 + sin2 θ dγ2
)
, (10)
where an appropriate expression for γ in a curvilinear coordinate system is given. If one com-
bines (1), (4), (7) and (10), then in a geodesic polar coordinate system, Laplace’s equation on SdR
is given by
∆f =
1
R2
[
∂2f
∂θ2
+ (d− 1) cot θ
∂f
∂θ
+
1
sin2 θ
∆Sd−1f
]
= 0, (11)
where ∆Sd−1 is the corresponding Laplace–Beltrami operator on Sd−1.
Consider Poisson’s equation on SdR, −∆u = ρ on a compact Riemannian manifold M with
boundary ∂M . The divergence theorem on this manifold is given by (cf. [24, p. 43])∫
M
divX dV =
∫
∂M
〈X,N〉 dṼ , (12)
Opposite Antipodal Fundamental Solution of Laplace’s Equation 5
where dV is the Riemannian volume measure on M , N is the outward unit normal to ∂M ,
and dṼ is the Riemannian volume measure of the induced metric on ∂M . If one invokes the
divergence theorem on SdR with regard to Poisson’s equation on this manifold using X = ∇u,
then since ∂SdR = ∅, one ascertains∫
Sd
R
ρdV = 0.
Hence on SdR (and on all compact manifolds without boundary), there does not exist a source
density distribution ρ, satisfying Poisson’s equation, with non-vanishing integral. In fact, a fun-
damental solution of Laplace’s equation on SdR (see Theorem 1 below), which has been pointed
out in [4, Section 5.4], is actually the solution to the Poisson equation whose inhomogeneous
source distribution is given by a point source at the origin and another with opposite sign, on
the opposite pole of the hypersphere (both modeled by Dirac delta distributions).
We define the opposite antipodal fundamental solution of Laplace’s equation AdR(x,x′), as the
solution to the following distributional partial differential equation
−∆AdR(x,x′) = δg(x,x
′)− δg(−x,x′), (13)
where g is the Riemannian metric on SdR (e.g., (10)) and δg is the Dirac delta distribution on
the manifold SdR. The total integral over the entire manifold of this source distribution vanishes.
Therefore the solution to the resulting partial differential equation must exist. The opposite
antipodal fundamental solution is the most natural fundamental solution of Laplace’s equation
on a d-dimensional R-radius hypersphere SdR because (1) it is spherically symmetric and (2)
its density distribution is composed wholly of the minimum number, two, of isolated Dirac
delta distributions. Note that many other composed density distribution may be assembled by
collecting a finite number of Dirac delta distributions, or by assembling an infinite number of
Dirac delta distributions over the manifold.
3 An opposite antipodal Green’s function on the hypersphere
3.1 Harmonics in geodesic polar coordinates
The harmonics (solutions to Laplace’s equation) in a geodesic polar coordinate system are given
in terms of a ‘radial’ solution (‘radial’ harmonics) multiplied by the angular solution (angular
harmonics).
Using polyspherical coordinates on Sd−1, one can compute the normalized hyperspherical
harmonics in this space by solving the Laplace equation using separation of variables. This
results in a general procedure which, for instance, is given explicitly in [20, 21]. These angu-
lar harmonics are given as general expressions involving trigonometric functions, Gegenbauer
polynomials and Jacobi polynomials. The angular harmonics are eigenfunctions of the Laplace–
Beltrami operator on Sd−1 which satisfy the following eigenvalue problem (see for instance [34,
equation (12.4) and Corollary 2 to Theorem 10.5])
∆Sd−1Y K
l (x̂) = −l(l + d− 2)Y K
l (x̂), (14)
where x̂ ∈ Sd−1, Y K
l (x̂) are normalized angular hyperspherical harmonics, l ∈ N0 is the angular
momentum quantum number, and K stands for the set of (d− 2)-quantum numbers identifying
degenerate harmonics for each l and d. The degeneracy
(2l + d− 2)
(d− 3 + l)!
l!(d− 2)!
6 H.S. Cohl
(see [37, equation (9.2.11)]), tells you how many linearly independent solutions exist for a par-
ticular l value and dimension d. The angular hyperspherical harmonics are normalized such
that ∫
Sd−1
Y K
l (x̂)Y K′
l′ (x̂)dω = δl
′
l δ
K′
K ,
where dω is the Riemannian (volume) measure on Sd−1, which is invariant under the isometry
group SO(d) (cf. (5)), and for x + iy = z ∈ C, z = x − iy, represents complex conjugation.
The angular solutions (hyperspherical harmonics) are well-known (see [37, Chapter IX] and
[9, Chapter 11]). The generalized Kronecker delta symbol δK
′
K (cf. (9)) is defined such that it
equals 1 if all of the (d−2)-quantum numbers identifying degenerate harmonics for each l and d
coincide, and equals zero otherwise.
We now focus on the ‘radial’ solutions (harmonics) on SdR. These satisfy the following ordinary
differential equation (cf. (11) and (14))
d2u
dθ2
+ (d− 1) cot θ
du
dθ
− l(l + d− 2)
sin2 θ
u = 0. (15)
Four solutions of this ordinary differential equation ud,l1,±, u
d,l
2,± : (−1, 1)→ C are given by
ud,l1,±(cos θ) :=
1
(sin θ)d/2−1
P
±(d/2−1+l)
d/2−1 (± cos θ),
and
ud,l2,±(cos θ) :=
1
(sin θ)d/2−1
Q
±(d/2−1+l)
d/2−1 (± cos θ), (16)
where Pµν ,Q
µ
ν : (−1, 1) → C are Ferrers functions of the first and second kind. The Ferrers
functions of the first and second kind (see [28, Chapter 14]) can be defined respectively in terms
of a sum over two Gauss hypergeometric functions, for all ν, µ ∈ C such that ν + µ 6∈ −N,
Pµν (x) :=
2µ+1
√
π
sin
[π
2
(ν + µ)
] Γ
(
ν+µ+2
2
)
Γ
(
ν−µ+1
2
)x(1− x2)−µ/22F1
(
1− ν − µ
2
,
ν − µ+ 2
2
;
3
2
;x2
)
+
2µ√
π
cos
[π
2
(ν + µ)
] Γ
(
ν+µ+1
2
)
Γ
(
ν−µ+2
2
)(1− x2)−µ/22F1
(
−ν − µ
2
,
ν − µ+ 1
2
;
1
2
;x2
)
(cf. [28, equation (14.3.11)]), and
Qµν (x) :=
√
π2µ cos
[π
2
(ν + µ)
] Γ
(
ν+µ+2
2
)
Γ
(
ν−µ+1
2
)x(1− x2)−µ/22F1
(
1− ν − µ
2
,
ν − µ+ 2
2
;
3
2
;x2
)
−
√
π2µ−1 sin
[π
2
(ν + µ)
] Γ
(
ν+µ+1
2
)
Γ
(
ν−µ+2
2
)(1− x2)−µ/22F1
(
−ν − µ
2
,
ν − µ+ 1
2
;
1
2
;x2
)
(17)
(cf. [28, equation (14.3.12)]). The Gauss hypergeometric function 2F1 : C × C × (C \ −N0) ×
C \ [1,∞)→ C, can be defined in terms of the infinite series
2F1(a, b; c; z) :=
∞∑
n=0
(a)n(b)n
(c)nn!
zn
Opposite Antipodal Fundamental Solution of Laplace’s Equation 7
(see [28, equation (15.2.1)]), and elsewhere in z by analytic continuation. On the unit circle
|z| = 1, the Gauss hypergeometric series converges absolutely if Re(c−a−b) ∈ (0,∞), converges
conditionally if z 6= 1 and Re(c − a − b) ∈ (−1, 0], and diverges if Re(c − a − b) ∈ (−∞,−1].
For z ∈ C and n ∈ N0, the Pochhammer symbol (z)n (also referred to as the rising factorial) is
defined as (cf. [28, equation (5.2.4)])
(z)n :=
n∏
i=1
(z + i− 1).
The Pochhammer symbol is expressible in terms of a quotient of gamma functions as [28, equa-
tion (5.2.5)]
(z)n =
Γ(z + n)
Γ(z)
,
for all z ∈ C\−N0. The gamma function Γ: C\−N0 → C (see [28, Chapter 5]) is an important
combinatoric function and is ubiquitous in special function theory. It is naturally defined over
the right-half complex plane through Euler’s integral (see [28, equation (5.2.1)])
Γ(z) :=
∫ ∞
0
tz−1e−tdt,
Re z > 0, and elsewhere by analytic continuation. The Euler reflection formula allows one to
obtain values of the gamma function in the left-half complex plane [28, equation (5.5.3)], namely
Γ(z)Γ(1− z) =
π
sinπz
,
0 < Re z < 1, for Re z = 0, z 6= 0, and then for z shifted by integers using the following
recurrence relation (see [28, equation (5.5.1)])
Γ(z + 1) = zΓ(z).
An important formula which the gamma function satisfies is the duplication formula [28, equa-
tion (5.5.5)]
Γ(2z) =
22z−1√
π
Γ(z)Γ
(
z +
1
2
)
, (18)
provided 2z 6∈ −N0.
Due to the fact that the space SdR is homogeneous with respect to its isometry group, the
orthogonal group O(d), and therefore an isotropic manifold, we expect that there exist a fun-
damental solution on this space with spherically symmetric dependence. We specifically expect
these solutions to be given in terms of the Ferrers function of the second kind with argument
given by cos θ. The Ferrers function of the second kind naturally fits our requirements because
it is singular at θ = 0, whereas the Ferrers function of the first kind, with the same argument, is
regular at θ = 0. We require there to exist a singularity at the origin of an opposite antipodal
fundamental solution of Laplace’s equation on SdR, since it is a manifold and must behave locally
like a Euclidean fundamental solution of Laplace’s equation which also has a singularity at the
origin.
8 H.S. Cohl
3.2 Opposite antipodal fundamental solution of −∆u = 0 on Sd
R
In computing an opposite antipodal fundamental solution of the Laplacian on SdR, we must
solve (13) where g is the Riemannian metric on SdR (e.g., (10)) and δg is the Dirac delta dis-
tribution on the manifold SdR (e.g., (6)). In general since we can add any harmonic function
to an opposite antipodal fundamental solution of Laplace’s equation and still have an opposite
antipodal fundamental solution, we will use this freedom to make our fundamental solution as
simple as possible. It is reasonable to expect that there exists a particular spherically symmetric
fundamental solution AdR on the hypersphere with pure ‘radial’, θ := d(x̂, x̂′) (e.g., (3)), and
constant angular dependence due to the influence of the point-like nature of the Dirac delta
distribution in (13). For a spherically symmetric solution to the Laplace equation, the corre-
sponding ∆Sd−1 term in (11) vanishes since only the l = 0 term survives in (15). In other words
we expect there to exist an opposite antipodal fundamental solution of Laplace’s equation on SdR
such that AdR(x,x′) = f(θ) (cf. (3)), where R is a parameter of this fundamental solution.
We will prove that on the R-radius hypersphere SdR, an opposite antipodal Green’s function
for the Laplace operator (fundamental solution of Laplace’s equation) can be given as follows.
Theorem 1. Let d ∈ {2, 3, . . .}. Define Id : (0, π)→ R as
Id(θ) :=
∫ π/2
θ
dx
sind−1 x
,
x,x′ ∈ SdR, and AdR :
(
SdR × SdR
)
\
{
(x,x) : x ∈ SdR
}
→ R defined such that
AdR(x,x′) :=
Γ(d/2)
2πd/2Rd−2
Id(θ),
where θ := cos−1 ([x̂, x̂′]) is the geodesic distance between x̂ and x̂′ on the unit radius hyper-
sphere Sd, with x̂ = x/R, x̂′ = x′/R, then AdR is an opposite antipodal fundamental solution
for −∆ where ∆ is the Laplace–Beltrami operator on SdR. Moreover,
Id(θ) =
(d− 3)!!
(d− 2)!!
[
log cot
θ
2
+ cos θ
d/2−1∑
k=1
(2k − 2)!!
(2k − 1)!!
1
sin2k θ
]
if d even,
(
d− 3
2
)
!
(d−1)/2∑
k=1
cot2k−1 θ
(2k − 1)(k − 1)!((d− 2k − 1)/2)!
,
or
(d− 3)!!
(d− 2)!!
cos θ
(d−1)/2∑
k=1
(2k − 3)!!
(2k − 2)!!
1
sin2k−1 θ
,
if d odd,
=
cos θ 2F1
(
1
2
,
d
2
;
3
2
; cos2 θ
)
,
cos θ
sind−2 θ
2F1
(
1,
3− d
2
;
3
2
; cos2 θ
)
,
(d− 2)!
Γ (d/2) 2d/2−1
1
(sin θ)d/2−1
Q
1−d/2
d/2−1(cos θ).
Proof. Since a spherically symmetric choice for an opposite antipodal fundamental solution
satisfies Laplace’s equation everywhere except at θ ∈ {0, π}, we may first set g = f ′ in (11) and
solve the first-order ordinary differential equation
g′ + (d− 1) cos θ g = 0,
Opposite Antipodal Fundamental Solution of Laplace’s Equation 9
which is integrable and clearly has the general solution
g(θ) =
df
dθ
= c0(sin θ)
1−d, (19)
where c0 ∈ R is a constant. Now we integrate (19) to obtain an opposite antipodal fundamental
solution for the Laplacian on SdR
AdR(x,x′) = c0Id(θ) + c1, (20)
where Id : (0, π)→ R is defined as
Id(θ) :=
∫ π/2
θ
dx
sind−1 x
, (21)
and c0, c1 ∈ R are constants (independent of θ) which depend on d and R. Notice that we can
add any harmonic function to (20) and still have an opposite antipodal fundamental solution of
the Laplacian since an opposite antipodal fundamental solution of the Laplacian must satisfy∫
Sd
R
(−∆ϕ)(x′)AdR(x,x′) dvol′g = ϕ(x),
for all ϕ ∈ S
(
SdR
)
, where S is the space of test functions, and dvol′g is the Riemannian (volume)
measure on SdR in the primed coordinates. Notice that our fundamental solution of Laplace’s
equation on the hypersphere (20), has the property that it tends towards +∞ as θ → 0+ and
tends towards −∞ as θ → π−. Therefore our fundamental solution attains all real values.
As an aside, by the definition therein (see [16, 17]), SdR is a parabolic manifold. Since the
hypersphere SdR is bi-hemispheric, we expect that an opposite antipodal fundamental solution
of Laplace’s equation on the hypersphere should vanish at θ = π/2. It is therefore convenient
to set c1 = 0 leaving us with
AdR(x,x′) = c0Id(θ). (22)
In Euclidean space Rd, a Green’s function for Laplace’s equation (fundamental solution for
the Laplacian) is well-known and is given by the following expression (see [13, p. 94], [14, p. 17],
[3, p. 211], [7, p. 6]). Let d ∈ N. Define
Gd(x,x′) =
Γ(d/2)
2πd/2(d− 2)
‖x− x′‖2−d if d = 1 or d ≥ 3,
1
2π
log ‖x− x′‖−1 if d = 2,
(23)
then Gd is a fundamental solution for −∆ in Euclidean space Rd, where −∆ is the positive
Laplace operator on Rd. Note that most authors only present the above theorem for the case
d ≥ 2 but it is easily-verified to also be valid for the case d = 1 as well.
The hypersphere SdR, being a manifold, must behave locally like Euclidean space Rd. There-
fore for small θ we have eθ ' 1 + θ and e−θ ' 1− θ and in that limiting regime
Id(θ) ≈
∫ 1
θ
dx
xd−1
'
− log θ if d = 2,
1
θd−2
if d ≥ 3,
which has exactly the same singularity as a Euclidean fundamental solution. Therefore the
proportionality constant c0 is obtained by matching locally to a Euclidean fundamental solution
AdR = c0Id ' Gd, (24)
in a small neighborhood of the singularity at x = x′, as the curvature vanishes, i.e., R→∞.
10 H.S. Cohl
We have shown how to compute an opposite antipodal fundamental solution of the Laplace–
Beltrami operator on the hypersphere in terms of an improper integral (21). We now prove
several equivalent finite summation expressions for Id(θ). We wish to compute the antiderivative
Im : (0, π)→ R, which is defined to within a constant in x as
Im(x) :=
∫
dx
sinm x
,
where m ∈ N. This antiderivative satisfies the following recurrence relation
Im(x) = − cosx
(m− 1) sinm−1 x
+
(m− 2)
(m− 1)
Im−2(x), (25)
which follows from the identity
1
sinm x
=
1
sinm−2 x
+
cosx
sinm x
cosx,
and integration by parts. The antiderivative Im(x) naturally breaks into two separate classes,
namely∫
dx
sin2n+1 x
= −(2n− 1)!!
(2n)!!
[
log cot
x
2
+ cosx
n∑
k=1
(2k − 2)!!
(2k − 1)!!
1
sin2k x
]
+ C, (26)
and
∫
dx
sin2n x
=
−(2n− 2)!!
(2n− 1)!!
cosx
n∑
k=1
(2k − 3)!!
(2k − 2)!!
1
sin2k−1 x
+ C, or
−(n− 1)!
n∑
k=1
cot2k−1 x
(2k − 1)(k − 1)!(n− k)!
+ C,
(27)
where C is a constant. The double factorial (·)!! : {−1, 0, 1, . . .} → N is defined by
n!! :=
n · (n− 2) · · · 2 if n even ≥ 2,
n · (n− 2) · · · 1 if n odd ≥ 1,
1 if n ∈ {−1, 0}.
Note that (2n)!! = 2nn! for n ∈ N0. The finite summation formulae for Im(x) all follow trivially
by induction using (25) and the binomial expansion (cf. [28, equation (1.2.2)])
(1 + cos2 x)n = n!
n∑
k=0
cot2k x
k!(n− k)!
.
The formulae (26) and (27) are essentially equivalent to [15, equations (2.515.1) and (2.515.2)],
except [15, equation (2.515.2)] is in error with the factor 28k being replaced with 2k. This is
also verified in the original citing reference [36]. By applying the limits of integration from the
definition of Id(θ) in (21) to (26) and (27) we obtain the following finite summation expressions
Id(θ) =
(d− 3)!!
(d− 2)!!
log cot
θ
2
+ cos θ
d/2−1∑
k=1
(2k − 2)!!
(2k − 1)!!
1
sin2k θ
if d even,
(
d− 3
2
)
!
(d−1)/2∑
k=1
cot2k−1 θ
(2k − 1)(k − 1)!((d− 2k − 1)/2)!
,
or
(d− 3)!!
(d− 2)!!
cos θ
(d−1)/2∑
k=1
(2k − 3)!!
(2k − 2)!!
1
sin2k−1 θ
,
if d odd.
(28)
Opposite Antipodal Fundamental Solution of Laplace’s Equation 11
Moreover, the antiderivative (indefinite integral) can be given in terms of the Gauss hyper-
geometric function as∫
dθ
sind−1 θ
= − cos θ 2F1
(
1
2
,
d
2
;
3
2
; cos2 θ
)
+ C, (29)
where C ∈ R. This is verified as follows. By using
d
dz
2F1(a, b; c; z) =
ab
c
2F1(a+ 1, b+ 1; c+ 1; z)
(see [28, equation (15.5.1)]), and the chain rule, we can show that
− d
dθ
cos θ 2F1
(
1
2
,
d
2
;
3
2
; cos2 θ
)
= sin θ
[
2F1
(
1
2
,
d
2
;
3
2
; cos2 θ
)
+
d
3
cos2 θ 2F1
(
3
2
,
d+ 2
2
;
5
2
; cos2 θ
)]
.
The second hypergeometric function can be simplified using Gauss’ relations for contiguous
hypergeometric functions, namely
z 2F1(a+ 1, b+ 1; c+ 1; z) =
c
a− b
[
2F1(a, b+ 1; c; z)− 2F1(a+ 1, b; c; z)
]
(see [8, p. 58]), and
2F1(a, b+ 1; c; z) =
b− a
b
2F1(a, b; c; z) +
a
b
2F1(a+ 1, b; c; z)
(see [28, equation (15.5.12)]). By applying these formulae, the term with the hypergeometric
function cancels leaving only a term which is proportional to a binomial through
2F1(a, b; b; z) = (1− z)−a
(see [28, equation (15.4.6)]), which reduces to 1/ sind−1 θ. By applying the limits of integration
from the definition of Id(θ) in (21) to (29), we obtain the following Gauss hypergeometric
representation
Id(θ) = cos θ 2F1
(
1
2
,
d
2
;
3
2
; cos2 θ
)
. (30)
Using (30), we can write another expression for Id(θ). Applying Euler’s transformation
2F1(a, b; c; z) = (1− z)c−a−b 2F1 (c− a, c− b; c; z)
(see [1, equation (2.2.7)]), to (30) produces
Id(θ) =
cos θ
sind−2 θ
2F1
(
1,
3− d
2
;
3
2
; cos2 θ
)
.
Our derivation for an opposite antipodal fundamental solution of Laplace’s equation on the
R-radius hypersphere in terms of Ferrers functions of the second kind is as follows. If we let
ν + µ = 0 in the definition of the Ferrers function of the second kind Qµν : (−1, 1)→ C (17), we
derive
Q−νν (x) =
√
π
2ν
x
(
1− x2
)ν/2
Γ
(
ν + 1
2
) 2F1
(
1
2
, ν + 1;
3
2
;x2
)
,
12 H.S. Cohl
for all ν ∈ C. If we let ν = d/2− 1 and substitute x = cos θ, then we have
Q
1−d/2
d/2−1(cos θ) =
√
π
2d/2−1
cos θ sind/2−1 θ
Γ
(
d−1
2
) 2F1
(
1
2
,
d
2
;
3
2
; cos2 θ
)
. (31)
Using the duplication formula for gamma functions (18), then through (31) we have
Id(θ) =
(d− 2)!
Γ(d/2)2d/2−1
1
sind/2−1 θ
Q
1−d/2
d/2−1(cos θ). (32)
We have therefore verified that the harmonics computed in Section 3.1, namely ud,02,+(cos θ) (16),
give an alternate form for an opposite antipodal fundamental solution of the Laplacian on the
hypersphere.
The constant c0 in an opposite antipodal fundamental solution for the Laplace operator on
the hypersphere SdR (22) is computed by locally matching up, through (24), to the singularity
of an opposite antipodal fundamental solution for the Laplace operator in Euclidean space (23).
The coefficient c0 depends on d and R. For d ≥ 3 we take the asymptotic expansion for c0Id(θ) as
θ → 0+, and match this to a fundamental solution of Laplace’s equation for Euclidean space (23).
This yields
c0 =
Γ (d/2)
2πd/2
. (33)
For d = 2 we take the asymptotic expansion for
c0I2(θ) = −c0 log tan
θ
2
' c0 log ‖x− x′‖−1,
as θ → 0+, and match this to G2(x,x′) = (2π)−1 log ‖x − x′‖−1, therefore c0 = (2π)−1. This
exactly matches (33) for d = 2. The R dependence of c0 originates from (21), where x and θ
represents geodesic distances (cf. (3)). The distance r ∈ [0,∞) along a geodesic, as measured
from the origin of SdR, is given by r = θR. To show that an opposite antipodal fundamental solu-
tion (22) reduces to the Euclidean fundamental solution at small distances (see for instance [23]),
we examine the flat-space limit of zero curvature. In order to do this, we take the limit θ → 0+
and R → ∞ of (21) with the substitution x = r/R which produces a factor of Rd−2. So an
opposite antipodal fundamental solution of Laplace’s equation on the Riemannian manifold SdR
is given by
AdR(x,x′) :=
Γ (d/2)
2πd/2Rd−2
Id (θ).
This completes the proof. �
Apart from the well-known historical results in two and three dimensions, the closed form
expressions for an opposite antipodal fundamental solution of Laplace’s equation on the R-radius
hypersphere given by Theorem 1 in Section 3.2 appear to be new. Furthermore, the Ferrers
function representations in Section 3.1 for the radial harmonics on the R-radius hypersphere do
not appear to be have previously appeared in the literature.
3.3 Some examples for Id(θ) and Q
1−d/2
d/2−1(cos θ)
We would now like to express the integral Id(cos θ) for d ∈ {2, . . . , 7} in terms of trigonometric
and/or logarithmic functions. The integral Id can be computed using elementary methods
through its definition (21). In d = 2 we have
I2(θ) =
∫ π/2
θ
dx
sinx
=
1
2
log
cos θ + 1
cos θ − 1
= log cot
θ
2
,
Opposite Antipodal Fundamental Solution of Laplace’s Equation 13
and in d = 3 we have
I3(θ) =
∫ π/2
θ
dx
sin2 x
= cot θ.
In d ∈ {4, 5, 6, 7} we have
I4(θ) =
1
2
log cot
θ
2
+
cos θ
2 sin2 θ
,
I5(θ) = cot θ +
1
3
cot3 θ,
I6(θ) =
3
8
log cot
θ
2
+
3 cos θ
8 sin2 θ
+
cos θ
4 sin2 θ
, and
I7(θ) = cot θ +
2
3
cot3 θ +
1
5
cot5 θ.
Equivalently, some relevant Ferrers functions of the second kind Q
1−d/2
d/2−1(cos θ) for d ∈ {2, . . . , 7}
are (cf. (28) and (32))
Q0(cos θ) = log cot
θ
2
,
1
(sin θ)1/2
Q
−1/2
1/2 (cos θ) =
√
π
2
cot θ,
1
sin θ
Q−11 (cos θ) =
1
2
log cot
θ
2
+
cos θ
2 sin2 θ
,
1
(sin θ)3/2
Q
−3/2
3/2 (cos θ) =
1
2
√
π
2
(
cot θ +
1
3
cot3 θ
)
,
1
(sin θ)2
Q−22 (cos θ) =
1
8
log cot
θ
2
+
cos θ
8 sin2 θ
+
cos θ
12 sin4 θ
, and
1
(sin θ)5/2
Q
−5/2
5/2 (cos θ) =
1
8
√
π
2
(
cot θ +
2
3
cot3 θ +
1
5
cot5 θ
)
.
Acknowledgments
Much thanks to Ernie Kalnins, Willard Miller Jr., George Pogosyan, and Charles Clark for
valuable discussions. Much thanks as well to Richard Chapling for his comments in [4], in re-
ference to an original version of the present paper and its implications. I would like to express
my sincere gratitude to the anonymous referees and an editor at SIGMA whose helpful com-
ments improved this paper. This work was partly conducted while H.S. Cohl was a National
Research Council Research Postdoctoral Associate in the Information Technology Laboratory
at the National Institute of Standards and Technology, Gaithersburg, Maryland, USA.
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1 Introduction
2 Hyperspherical geometry
2.1 The Laplace and Poisson equations on the hypersphere
3 An opposite antipodal Green's function on the hypersphere
3.1 Harmonics in geodesic polar coordinates
3.2 Opposite antipodal fundamental solution of -u=0 on SRd
3.3 Some examples for Id() and Qd/2-11-d/2(cos)
References
|
| id | nasplib_isofts_kiev_ua-123456789-148085 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:26:24Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cohl, H.S. 2019-02-16T20:48:17Z 2019-02-16T20:48:17Z 2011 Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry / H.S. Cohl // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35A08; 35J05; 32Q10; 31C12; 33C05 DOI: https://doi.org/10.3842/SIGMA.2011.108 https://nasplib.isofts.kiev.ua/handle/123456789/148085 Due to the isotropy of d-dimensional hyperspherical space, one expects there to exist a spherically symmetric opposite antipodal fundamental solution for its corresponding Laplace-Beltrami operator. The R-radius hypersphere SdR with R>0, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric opposite antipodal fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the trigonometric sine, finite summation expressions over trigonometric functions, Gauss hypergeometric functions, and in terms of the Ferrers function of the second with degree and order given by d/2−1 and 1−d/2 respectively, with real argument x∈(−1,1). Much thanks to Ernie Kalnins, Willard Miller Jr., George Pogosyan, and Charles Clark for
 valuable discussions. Much thanks as well to Richard Chapling for his comments in [4], in reference to an original version of the present paper and its implications. I would like to express my sincere gratitude to the anonymous referees and an editor at SIGMA whose helpful comments improved this paper. This work was partly conducted while H.S. Cohl was a National Research Council Research Postdoctoral Associate in the Information Technology Laboratory at the National Institute of Standards and Technology, Gaithersburg, Maryland, USA. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry Article published earlier |
| spellingShingle | Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry Cohl, H.S. |
| title | Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_full | Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_fullStr | Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_full_unstemmed | Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_short | Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_sort | opposite antipodal fundamental solution of laplace's equation in hyperspherical geometry |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148085 |
| work_keys_str_mv | AT cohlhs oppositeantipodalfundamentalsolutionoflaplacesequationinhypersphericalgeometry |