Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry
For a fundamental solution of Laplace's equation on the R-radius d-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hype...
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| Cite this: | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry / H.S. Cohl, R.M. Palmer // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. |
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| author | Cohl, H.S. Palmer, R.M. |
| author_facet | Cohl, H.S. Palmer, R.M. |
| citation_txt | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry / H.S. Cohl, R.M. Palmer // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. |
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| description | For a fundamental solution of Laplace's equation on the R-radius d-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 015, 23 pages
Fourier and Gegenbauer Expansions
for a Fundamental Solution of Laplace’s Equation
in Hyperspherical Geometry?
Howard S. COHL † and Rebekah M. PALMER ‡
† Applied and Computational Mathematics Division, National Institute of Standards
and Technology, Gaithersburg, MD, 20899-8910, USA
E-mail: howard.cohl@nist.gov
URL: http://www.nist.gov/itl/math/msg/howard-s-cohl.cfm
‡ Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA
E-mail: rmaepalmer4@gmail.com
Received May 20, 2014, in final form February 09, 2015; Published online February 14, 2015
http://dx.doi.org/10.3842/SIGMA.2015.015
Abstract. For a fundamental solution of Laplace’s equation on the R-radius d-dimensional
hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three
dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental so-
lution of Laplace’s equation in hyperspherical geometry in geodesic polar coordinates. From
this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier
coefficients of a fundamental solution of Laplace’s equation on the 3-sphere. Applications
of our expansions are given, namely closed-form solutions to Poisson’s equation with uni-
form density source distributions. The Newtonian potential is obtained for the 2-disc on the
2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given
to the superintegrable Kepler–Coulomb and isotropic oscillator potentials.
Key words: fundamental solution; hypersphere; Fourier expansion; Gegenbauer expansion
2010 Mathematics Subject Classification: 31C12; 32Q10; 33C05; 33C45; 33C55; 35J05;
35A08; 42A16
Dedicated to Professor Luc Vinet,
and to his work on the 2-sphere
1 Introduction
This investigation is concerned with the study of eigenfunction expansions for a fundamental
solution of Laplace’s equation on Riemannian spaces of positive constant curvature. These
manifolds are classified as compact Riemannian, and we refer to them as d-dimensional R-
radius hyperspheres. Analysis on hyperspheres has a long history and has been studied by
luminaries such as Schrödinger in 1938 and 1940 [26, 27]. However, there is significant activity
in the literature regarding analysis on hyperspheres (e.g., [12, 18], see [22] for a topical review in
classical and quantum superintegrability). In fact, [18] makes a connection between the Askey
scheme of hypergeometric orthogonal polynomials and superintegrable systems on the 2-sphere!
In this paper, we derive expansions for a fundamental solution of Laplace’s equation on the d-
dimensional R-radius hypersphere which allow one to effectively obtain density-potential pairs
for density distributions with a large degree of rotational symmetry. Through these expan-
sions, one may further explore classical potential theory and related theories on hyperspheres.
?This paper is a contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of
Luc Vinet. The full collection is available at http://www.emis.de/journals/SIGMA/ESSA2014.html
mailto:howard.cohl@nist.gov
mailto:rmaepalmer4@gmail.com
http://dx.doi.org/10.3842/SIGMA.2015.015
http://www.emis.de/journals/SIGMA/ESSA2014.html
2 H.S. Cohl and R.M. Palmer
We have previously studied Fourier and Gegenbauer polynomial expansions for a fundamen-
tal solution of Laplace’s equation on the d-dimensional R-radius constant negative curvature
non-compact Riemannian manifold given by the hyperboloid model of hyperbolic geometry [6].
In the hyperboloid model, Fourier expansions for a fundamental solution of Laplace’s equation
were given in rotationally-invariant coordinate systems and Gegenbauer polynomial expansions
of this fundamental solution were given in standard hyperspherical coordinates. In a previous
paper [5], we have studied various special function representations for a fundamental solution of
Laplace’s equation on the compact positive constant curvature Riemannian manifold given by
the d-dimensional R-radius hypersphere. In this paper, we derive for this manifold, Fourier and
Gegenbauer expansions analogous to those presented in [6]. This exercise is motivated by the
necessity and utility of these expansions in obtaining the solution to potential theoretic problems
in this geometry.
As an application and demonstration of the utility of our expansions, in Section 5.1 we
give three examples in Newtonian potential theory on the 2-sphere and 3-sphere which reduce
to elementary results in Euclidean potential theory. The density distributions we treat possess
spherical and rotational symmetry, and their potential solutions are obtained through the derived
Gegenbauer and azimuthal Fourier expansions respectively. These density distributions are
the 2-disc on the 2-sphere, the 3-ball on the 3-sphere, and the rotationally-invariant circular
curve segment on the 3-sphere. Using superintegrable potentials, one may derive a paired
density distribution assuming that the potentials satisfy Poisson’s equation. As such, one may
convolve a fundamental solution of Laplace’s equation on the hypersphere with this paired
density distribution to re-obtain the superintegrable potential. Examples of this connection to
superintegrable potentials are given in Section 5.2 for the isotropic oscillator potential and for
the Kepler–Coulomb potential on d-spheres, for d = 2, 3, . . ..
We denote the sets of real and complex numbers by R and C, respectively. Similarly, the
sets N := {1, 2, 3, . . .} and Z := {0,±1,±2, . . .} denote the natural numbers and the integers.
Furthermore, we denote the set N0 := {0, 1, 2, . . .} = N∪{0}. Throughout this paper, we adopt
the common convention that empty sums and products are zero and unity respectively. Note
that we often adopt a common notation used for fundamental solution expansions, namely if
one takes a, a′ ∈ R, then
a≶ :=
min
max
{a, a′}. (1.1)
2 Preliminaries
2.1 Coordinates
The hypersphere SdR, a smooth d-dimensional compact constant positive curvature Riemannian
manifold can be defined as Riemannian submanifold given by set of all points x = (x0, . . . , xd)
such that x2
0 + · · ·+ x2
d = R2. Standard hyperspherical coordinates are an example of geodesic
polar coordinates on this manifold 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,
(2.1)
Fourier and Gegenbauer Expansions on the Hypersphere 3
where φ ∈ [−π, π), θ2, . . . , θd ∈ [0, π], and R > 0. On SdR, the variables φ := θ1, θ := θd, are
respectively referred to as the azimuthal and radial coordinates.
It is necessary to define the geodesic distance on SdR. In any geodesic polar coordinate
system, the geodesic distance between two points d : SdR × SdR → [0, πR] is given in Vilenkin’s
polyspherical coordinates by [5, equation (2)]
d(x,x′) := R cos−1(x̂, x̂′) = R cos−1(cos θ cos θ′ + sin θ sin θ′ cos γ), (2.2)
where (·, ·) : Rd+1 × Rd+1 → R is the Euclidean inner product defined by (x,x′) := x0x
′
0 +
x1x
′
1 + · · ·+ xdx
′
d, and γ is the separation angle between the two unit vectors x̂, x̂′ ∈ Sd := Sd1.
As measured from the origin of Rd+1, these unit vectors are given by x̂ := x/R, x̂′ := x′/R. The
separation angle γ is defined through the expression cos γ := (x̂, x̂′), and is given in standard
hyperspherical coordinates as
cos γ = cos(φ− φ′)
d−2∏
i=1
sin θi sin θ′i +
d−2∑
i=1
cos θi cos θ′i
i−1∏
j=1
sin θj sin θ′j . (2.3)
2.2 Fundamental solution of Laplace’s equation on Sd
R
With u ∈ C2(Md), where Md is a d-dimensional (pseudo-)Riemannian manifold, we refer to
∆u = 0,
where ∆ is the Laplace–Beltrami operator on Md, as Laplace’s equation. We now summarize
the presentation for a fundamental solution of Laplace’s equation on SdR, given in Cohl [5,
Theorem 3.2]. Let x,x′ ∈ SdR, and let x̂ = x/R and x̂′ = x′/R be their unit vectors. We first
define Jd : (0, π)→ R as
Jd(Θ) :=
∫ π/2
Θ
dx
sind−1 x
,
and GdR : (SdR × SdR) \ {(x,x) : x ∈ SdR} → R as
GdR(x,x′) :=
Γ(d/2)
2πd/2Rd−2
Jd(Θ), (2.4)
where Θ := cos−1((x̂, x̂′)) is the geodesic distance between x̂ and x̂′. Then GdR is a fundamental
solution for −∆, where ∆ is the Laplace–Beltrami operator on SdR given by
∆f =
1
R2
[
∂2f
∂Θ2
+ (d− 1) cot Θ
∂f
∂Θ
+
1
sin2 Θ
∆Sd−1f
]
, (2.5)
where ∆Sd−1 is the corresponding Laplace–Beltrami operator on Sd−1. Moreover,
Jd(Θ) =
(d− 2)!
Γ(d/2)2d/2−1
1
sind/2−1 Θ
Q
1−d/2
d/2−1(cos Θ), (2.6)
where Q−νν : (−1, 1)→ C is a Ferrers function of the second kind defined as [5, equation (30)]
Q−νν (x) :=
√
πx(1− x2)ν/2
2νΓ
(
ν + 1
2
) 2F1
(
1
2 , ν + 1
3
2
;x2
)
,
where ν ∈ C \ {−1
2 ,−
3
2 ,−
5
2 , . . .} and 2F1 is the Gauss hypergeometric function (see [24, Chap-
ter 15]). On Euclidean space Rd, a fundamental solution for −∆, the Newtonian potential,
N d : Rd ×Rd \ {(x,x) : x ∈ Rd} is given as follows (see for example [9, p. 75])
N d(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.
(2.7)
4 H.S. Cohl and R.M. Palmer
2.3 Harmonics on the d-dimensional hypersphere
The angular harmonics are eigenfunctions of the Laplace–Beltrami operator on Sd−1 which
satisfy the following eigenvalue problem:
∆Sd−1Y K
l (x̂) = −l(l + d− 2)Y K
l (x̂), (2.8)
where x̂ ∈ Sd−1, Y K
l (x̂) are normalized hyperspherical harmonics, l ∈ N0 is the angular mo-
mentum quantum number, and K stands for the set of (d − 2)-quantum numbers identifying
degenerate harmonics for each l [6].
The Riemannian (volume) measure d volg of the Riemannian manifold SdR with Riemannian
metric g in standard hyperspherical coordinates (2.1) is given by
d volg = Rd sind−1 θdθdω := Rd sind−1 θ sind−2 θd−1 · · · sin θ2dθdθd−1 · · · dθ2dφ. (2.9)
In computing a fundamental solution of Laplace’s equation on SdR, we know that
−∆GdR(x,x′) = δg(x,x
′), (2.10)
where δg(x,x
′) is the Dirac delta distribution on the diagonal manifold SdR × SdR. This is the
Schwartz kernel of the identity map (thought of as a scalar distribution.) The Dirac delta
distribution on the Riemannian manifold SdR with metric g for an open set U ⊂ SdR with
x,x′ ∈ SdR satisfies∫
U
δg(x,x
′)d volg =
{
1 if x′ ∈ U,
0 if x′ /∈ U.
(2.11)
Using (2.9) and (2.11) in standard hyperspherical coordinates (2.1) on SdR, the Dirac delta
distribution is given by
δg(x,x
′) =
δ(θ − θ′)δ(φ− φ′)δ(θ2 − θ′2) · · · δ(θd−2 − θ′d−2)
Rd sind−1 θ′ sin θ′2 · · · sind−2 θ′d−1
. (2.12)
Due to the fact that the space SdR is homogeneous with respect to its isometry group
SO(d+ 1), it is therefore an isotropic manifold. Therefore there exists a fundamental solu-
tion of the Laplace–Beltrami operator with purely radial, spherically symmetric dependence.
This will be important when we perform our analysis in Section 4. The radial part of Laplace’s
equation on SdR satisfies the following differential equation
d2u
dθ2
+ (d− 1) cot θ
du
dθ
− l(l + d− 2)
sin2 θ
u = 0.
The above differential equation is a (transformed) associated Legendre differential equation and
originates from separating out the radial part of (2.5). Four solutions of this ordinary differential
equation ud,l1±, u
d,l
2± : (−1, 1)→ C are given by [6]
ud,l1±(cos θ) :=
1
sind/2−1 θ
P
±(d/2−1+l)
d/2−1 (cos θ), (2.13)
ud,l2±(cos θ) :=
1
sind/2−1 θ
Q
±(d/2−1+l)
d/2−1 (cos θ), (2.14)
where Pµν ,Q
µ
ν : (−1, 1)→ C are Ferrers functions of the first and second kind, respectively [24,
Section 14.3(i)].
Fourier and Gegenbauer Expansions on the Hypersphere 5
2.4 The flat-space limit from hyperspherical to Euclidean geometry
Because each point on a manifold is locally Euclidean, one expects that quantities defined in hy-
perspherical geometry should reduce to their flat-space counterparts in the flat-space limit. The
flat-space limit is a procedure which allows one to compare mathematical quantities defined on
on d-dimensional spaces of constant curvature (such as hyperspherical and hyperbolic geometry)
with corresponding quantities defined on d-dimensional Euclidean space. In hyperspherical geo-
metry, this method has previously been used in many different contexts [2, 13, 14, 15]. The limi-
ting case in 3-dimensions from the hypersphere to Euclidean space, are treated in all Helmholtz
separable coordinate systems in [25]. These limiting processes are often termed (Inönü–Wigner)
contractions. For contractions of Helmholtz subgroup-type separable coordinate systems from
SdR → Rd, see [17].
To obtain the corresponding Euclidean limit for a quantity expressed in a specific coordinate
system on the curved space, one performs an asymptotic expansion as R→∞ of that quantity
with coordinates approaching zero. Take for instance the example given just below [5, equa-
tion (5)]. In standard hyperspherical coordinates (2.1) on SdR, the Riemannian volume measure
is given by (2.9). The flat-space limit of the Riemannian volume measure on SdR is applied by
taking R → ∞ with the geodesic radial distance r = RΘ is fixed as measured from the origin
θ = 0. In the flat-space limit
Rd sind−1 θdθ ∼ Rd sind−1
( r
R
)
R−1dr ∼ Rd
( r
R
)d−1
R−1dr = rd−1dr,
which is exactly the radial volume element in standard hyperspherical coordinates on Rd. This
example demonstrates how the flat-space limit can be used (and has been used extensively)
to show how Laplace–Beltrami separable coordinate systems on SdR (such as (2.1)) reduce to
Laplace separable coordinate systems (such as polar or standard hyperspherical coordinates)
on Rd.
Another interesting example of the flat-space limit which we will use in Section 5.1, is Hopf
coordinates (sometimes referred to as cylindrical coordinates) on S3
R. Using Hopf coordinates,
points on S3
R are parametrized using
x0 = R cosϑ cosφ1, x1 = R cosϑ sinφ1,
x2 = R sinϑ cosφ2, x3 = R sinϑ sinφ2, (2.15)
where ϑ ∈ [0, π2 ], and φ1, φ2 ∈ [−π, π). Note that (x,x) = R2. In the flat-space limit we fix
r = Rϑ, z = Rφ1 and let R→∞. In the flat-space limit
x ∼
(
R cos
( r
R
)
cos
( z
R
)
, R cos
( r
R
)
sin
( z
R
)
, R sin
( r
R
)
cosφ2, R sin
( r
R
)
sinφ2
)
∼
(
R,R
( z
R
)
, R
( r
R
)
cosφ2, R
( r
R
)
sinφ2
)
∼ (R, z, r cosφ2, r sinφ2),
with the first coordinate approaching infinity and the last three coordinates being circular
cylindrical coordinates on R3.
3 Fourier expansions for a fundamental solution
of Laplace’s equation
A fundamental solution of Laplace’s equation on the unit radius hypersphere gd : (Sd × Sd) \
{(x,x) : x ∈ Sd} → R, is given, apart from a multiplicative constant, as (2.4)
gd(x̂, x̂′) := Jd(d(x̂, x̂′)) =
2πd/2Rd−2
Γ(d/2)
GdR(x,x′). (3.1)
6 H.S. Cohl and R.M. Palmer
Note that the d in gd (as in Gd and Sd) is an index for g representing the dimension of the
manifold and not some power of g. In standard hyperspherical coordinates (2.1), we would like
to perform an azimuthal Fourier expansion for a fundamental solution of Laplace’s equation on
the hypersphere; more precisely, we would like
gd(x̂, x̂′) =
∞∑
m=0
cos(m(φ− φ′))Gd/2−1
m (θ, θ′, θ2, θ
′
2, . . . , θd−1, θ
′
d−1), (3.2)
where G
d/2−1
m : [0, π]2d−2 → R which are given by
Gd/2−1
m (θ, θ′, θ2, θ
′
2, . . . , θd−1, θ
′
d−1) :=
εm
π
∫ π
0
gd(x̂, x̂′) cos(m(φ− φ′))d(φ− φ′), (3.3)
where εm is the Neumann factor defined as εm := 2 − δ0
m. Using (2.6), (3.1), we may write
gd(x̂, x̂′) as
gd(x̂, x̂′) =
(d− 2)!
Γ
(
d
2
)
2d/2−1(sin d(x̂, x̂′))d/2−1
Q
1−d/2
d/2−1(cos d(x̂, x̂′)). (3.4)
Through (2.2), (3.3), (3.4) in standard hyperspherical coordinates (2.1), the azimuthal Fourier
coefficients are given by
Gd/2−1
m (θ, θ′, θ2, θ
′
2, . . . , θd−1, θ
′
d−1) =
εm(d− 2)!
πΓ(d2)2d/2−1
∫ π
0
Q
1−d/2
d/2−1(Bd cosψ +Ad) cos(mψ)(√
1− (Bd cosψ +Ad)2
)d/2−1
dψ,
where ψ := φ− φ′ and Ad, Bd : [0, π]2d−2 → R are defined through (2.2), (2.3) as
Ad(θ, θ
′, θ2, θ
′
2, . . . , θd−1, θ
′
d−1) := cos θ cos θ′ + sin θ sin θ′
d−1∑
i=2
cos θi cos θ′i
i−1∏
j=1
sin θj sin θ′j ,
Bd(θ, θ
′, θ2, θ
′
2, . . . , θd−1, θ
′
d−1) := sin θ sin θ′
d−1∏
i=2
sin θi sin θ′i.
3.1 Fourier expansion for a fundamental solution of Laplace’s equation
on the 2-sphere
On the unit sphere S2, a fundamental solution of Laplace’s equation is given by g2 := 2πG2
(cf. (3.1)), where [5, Theorem 3.2]
g2(x̂, x̂′) = log cot
d(x̂, x̂′)
2
=
1
2
log
1 + cos d(x̂, x̂′)
1− cos d(x̂, x̂′)
. (3.5)
Note that, as in (3.1), the 2 in g2 is an index representing the dimension of the manifold and not
some power of g. We can further deconstruct g2 in standard hyperspherical coordinates (2.1).
Using (2.2), (2.3) to obtain
cos d(x̂, x̂′) = cos θ cos θ′ + sin θ sin θ′ cos(φ− φ′),
from (3.5) one obtains
g2(x̂, x̂′) =
1
2
log
1 + cos θ cos θ′ + sin θ sin θ′ cos(φ− φ′)
1− cos θ cos θ′ − sin θ sin θ′ cos(φ− φ′)
. (3.6)
Fourier and Gegenbauer Expansions on the Hypersphere 7
Theorem 3.1. Let φ, φ′ ∈ [−π, π). Then, the azimuthal Fourier expansion for a fundamental
solution of Laplace’s equation on the unit sphere S2 expressed in standard spherical coordi-
nates (2.1) is given by
g2(x̂, x̂′) =
1
2
log
1 + cos θ>
1− cos θ>
+
∞∑
n=1
cos[n(φ− φ′)]
n
(
1− cos θ<
1 + cos θ<
)n/2
×
[(
1 + cos θ>
1− cos θ>
)n/2
− (−1)n
(
1− cos θ>
1 + cos θ>
)n/2]
.
Note that we are using the notation (1.1).
Proof. Replacing ψ := φ− φ′ in (3.6) and rearranging the logarithms yields
g2(x̂, x̂′) =
1
2
[
log
1 + cos θ cos θ′
1− cos θ cos θ′
+ log(1 + z+ cosψ)− log(1− z− cosψ)
]
, (3.7)
where
z± :=
sin θ sin θ′
1± cos θ cos θ′
. (3.8)
Note that that z± ∈ (−1, 1) for θ, θ′ ∈ [0, π]. For x ∈ (−1, 1), we have the Maclaurin series
log(1± x) = −
∞∑
n=1
(∓1)nxn
n
.
Therefore, away from the singularity at x = x′, we have λ± : (−1, 1) × (−π, π) → (−∞, log 2)
defined by
λ±(z±, ψ) := log(1± z± cosψ) = −
∞∑
k=1
(∓1)kzk± cosk(ψ)
k
. (3.9)
We can expand powers of cosine using the following trigonometric identity:
cosk ψ =
1
2k
k∑
n=0
(
k
n
)
cos[(2n− k)ψ],
which is the standard expansion for powers [10, p. 52] when using Chebyshev polynomials of
the first kind Tm(cosψ) = cos(mψ). Inserting the above expression into (3.9), we obtain the
double-summation expression
λ±(z±, ψ) = −
∞∑
k=1
k∑
n=0
(∓1)kzk±
2kk
(
k
n
)
cos[(2n− k)ψ]. (3.10)
Our current goal is to identify closed-form expressions for the azimuthal Fourier cosine
coefficients cn′ of the sum given by
λ±(z±, ψ) =
∞∑
n′=0
cn′(z±) cos(n′ψ).
We accomplish this by performing a double-index replacement as in [6, equation (41)] to (3.10).
The double sum over k, n is rearranged into two separate double sums with different indices k′, n′,
8 H.S. Cohl and R.M. Palmer
first for k ≤ 2n and second for k ≥ 2n. If k ≤ 2n, let k′ = k − n, n′ = 2n − k which implies
k = 2k′ + n′ and n = n′ + k′. If k ≥ 2n, let k′ = n and n′ = k − 2n, which implies k = 2k′ + n′
and n = k′. Note that the identity
(
n
k
)
=
(
n
n−k
)
implies
(
2n′+k′
n′+k′
)
=
(
2k′+n′
k′
)
. After halving the
the double counting when k = 2n and replacing k′ 7→ k and n′ 7→ n, we obtain
λ±(z±, ψ) = −
∞∑
k=1
z2k
±
22k(2k)
(
2k
k
)
− 2
∞∑
n=1
(∓1)n cos(nψ)
∞∑
k=0
z2k+n
±
22k+n(2k + n)
(
2k + n
k
)
.
If we define I± : (−1, 1)→ (− log 2, 0) and Jn,± : (−1, 1)→ (0, (∓2)n/n) such that
λ±(z±, ψ) = I±(z±) +
∞∑
n=0
Jn,±(z±) cos(nψ),
then these terms are given by [6, Section 4.1]
I±(z±) := −
∞∑
k=1
z2k
±
22k(2k)
(
2k
k
)
= − log 2 + log
(
1 +
√
1− z±
)
,
Jn,±(z±) := −2(∓1)n
∞∑
k=0
z2k+n
±
22k+n(2k + n)
(
2k + n
k
)
=
2(∓1)n
n
1−
√
1− z2
±
1 +
√
1− z2
±
n/2
.
Finally, from (3.8), we have
λ± = − log 2 + log
(1 + cos θ<)(1± cos θ>)
1± cos θ cos θ′
−
∞∑
n=1
2(∓1)n
n
cos(nψ)
(
(1− cos θ<)(1∓ cos θ>)
(1 + cos θ<)(1± cos θ>)
)n/2
.
Utilizing this expression in (3.7) completes the proof. �
Note, as shown in [6, equation (39)], that (using the notation (1.1))
n2(x,x′) = log ‖x− x′‖−1 = − log r> +
∞∑
n=1
cos[n(φ− φ′)]
n
(
r<
r>
)n
, (3.11)
where n2 := 2πN 2 is defined using (2.7), normalized as in (3.1). It is straightforward to check
that g2(x̂, x̂′)→ n2(x,x′) as θ, θ′ → 0+ in the flat-space limit R→∞.
3.2 Fourier expansion for a fundamental solution of Laplace’s equation
on the 3-sphere
On the hypersphere S3
R, a fundamental solution of Laplace’s equation g3 := 4πRG3
R (cf. (3.1))
is given by [5, Theorem 3.2]
g3(x̂, x̂′) = cot(d(x̂, x̂′)) =
cos(d(x̂, x̂′))√
1− cos(d(x̂, x̂′))2
, (3.12)
Fourier and Gegenbauer Expansions on the Hypersphere 9
where (2.2), (2.3) still hold. As indicated in (3.1), the 3 in g3 above, should be interpreted as
an index for g representing the dimension of the hypersphere, and not some power of g. If we
define A : [0, π]4 → [−1, 1] and B : [0, π]4 → [0, 1] respectively as
A(θ, θ′, θ2, θ
′
2) := cos θ cos θ′ + sin θ sin θ′ cos θ2 cos θ′2, (3.13)
B(θ, θ′, θ2, θ
′
2) := sin θ sin θ′ sin θ2 sin θ′2, (3.14)
then the azimuthal Fourier coefficients G
1/2
m : [0, π]4 → R of g3 are expressed through
g3(x̂, x̂′) =
∞∑
m=0
G1/2
m (θ, θ′, θ2, θ
′
2) cos(mψ), (3.15)
where ψ := φ − φ′. By application of orthogonality for Chebyshev polynomials of the first
kind [24, Table 18.3.1] to (3.15), using (2.2) and (3.12), we produce an integral representation
for the azimuthal Fourier coefficients
G1/2
m (θ, θ′, θ2, θ
′
2) =
εm
π
∫ π
0
(cosψ +A/B) cos(mψ)dψ√(
1−A
B − cosψ
) (
1+A
B + cosψ
) .
If we substitute x = cosψ, then this integral can be converted into
G1/2
m (θ, θ′, θ2, θ
′
2) =
εm
π
∫ 1
−1
(x+A/B)Tm(x)dx√
(1− x2)
(
1−A
B − x
) (
1+A
B + x
) . (3.16)
Since (x+A/B)Tm(x) is a polynomial in x, it is sufficient to solve the integral∫ 1
−1
xpdx√
(1− x2)
(
1−A
B − x
) (
1+A
B + x
) , (3.17)
which by definition is an elliptic integral since it involves the square root of a quartic in x
multiplied by a rational function of x.
Let F : [0, π2 ] × [0, 1) → R be Legendre’s incomplete elliptic integral of the first kind which
can be defined through the definite integral [24, Section 19.2(ii)]
F (ϕ, k) :=
∫ ϕ
0
dθ√
1− k2 sin2 θ
,
K : [0, 1) → [1,∞) is Legendre’s complete elliptic integral of the first kind given by K(k) :=
F (π2 , k), E : [0, 1] → [1, π2 ] and Π : [0,∞) \ {1} × [0, 1) → R are Legendre’s complete elliptic
integrals of the second and third kind, respectively, defined as
E(k) :=
∫ π/2
0
√
1− k2 sin2 θdθ, Π(α2, k) :=
∫ π/2
0
dθ√
1− k2 sin2 θ(1− α2 sin2 θ)
.
For n ∈ N0, let (·)n : C→ C, denote the Pochhammer symbol (rising factorial), which is defined
by (a)n := (a)(a+ 1) · · · (a+ n− 1).
Theorem 3.2. Let A, B be defined by (3.13), (3.14) and α, k be defined by (3.21). The azimuthal
Fourier coefficients for a fundamental solution of Laplace’s equation on the unit hypersphere S3
expressed in standard hyperspherical coordinates (2.1) is given by
G1/2
m (θ, θ′, θ2, θ
′
2) =
2Bεm
π
√
(1−A+B)(1 +A−B)
m+1∑
p=0
p∑
j=0
ap
(B − 1 +A)j
Bp(1−A)j−p
(
p
j
)
Vj(α, k),
10 H.S. Cohl and R.M. Palmer
where
ap =
A(m)m(−m2 )m−p
2
(−m+1
2 )m−p
2
2mB(1
2)m(1−m)m−p
2
(m−p2 )!
if p = m− 2
⌊
m
2
⌋
, . . . ,m,
(m)m(−m2 )m−p+1
2
(−m+1
2 )m−p+1
2
2m(1
2)m(1−m)m−p+1
2
(m−p+1
2 )!
if p = m− 2
⌊
m
2
⌋
+ 1, . . . ,m+ 1,
(3.18)
V0(α, k) = K(k), V1(α, k) = Π(α2, k),
V2(α, k) =
1
2(α2 − 1)(k2 − α2)
[
(k2 − α2)K(k) + α2E(k)
+ (2α2k2 + 2α2 − α4 − 3k2)Π(α2, k)
]
,
and the values of Vj(α, k) for j > 3 can be computed using the recurrence relation [3, equa-
tion (336.00-03)]
Vj+3(α, k) =
1
2(j + 2)(α2 − 1)(k2 − α2)
[
(2j + 1)k2Vj(α, k)
+ 2(j + 1)(α2k2 + α2 − 3k2)Vj+1(α, k)
+ (2j + 3)(α4 − 2α2k2 − 2α2 + 3k2)Vj+1(α, k)
]
.
Proof. We can directly compute (3.16) using [3, equation (255.17)]. If we define
a :=
1−A
B
, b := 1, c := −1, y := −1, d := −1 +A
B
, (3.19)
(clearly d < y ≤ c < b < a), then we can express the Fourier coefficient (3.16) as a linear
combination of integrals∫ 1
−1
xpdx√
(a− x)(b− x)(x− c)(x− d)
=
2√
(a− c)(b− d)
∫ u1
0
(
1− α2
1sn2u
1− α2sn2u
)p
du, (3.20)
where 0 ≤ p ≤ m+1. In this expression, snu is a Jacobi elliptic function. Byrd and Friedman [3]
give a procedure for computing (3.20) for all p ∈ N0. These integrals will be given in terms
of complete elliptic integrals of the first three kind. We have the following definitions from [3,
Section 255, equations (255.17), (340.00)]:
0 < α2 =
b− c
a− c
< k2, g =
2√
(a− c)(b− d)
, ϕ = sin−1
√
(a− c)(b− y)
(b− c)(a− y)
,
α2
1 =
a(b− c)
b(a− c)
, k2 =
(b− c)(a− d)
(a− c)(b− d)
, u1 = F (ϕ, k).
For our specific choices in (3.19), these reduce to
α2 =
2B
1−A+B
, g =
2B√
(1 +A+B)(1−A+B)
, ϕ =
π
2
,
α2
1 =
2(1−A)
1−A+B
, k2 =
4B
(1 +A+B)(1−A+B)
, u1 = K(k). (3.21)
The required integration formula (cf. [3, equation (340.04)]) is given by∫ 1
−1
xpdx√
(a− x)(b− x)(x− c)(x− d)
=
g(bα2
1)p
α2p
p∑
j=0
(α2 − α2
1)j
α2j
1
(
p
j
)
Vj(α, k).
Fourier and Gegenbauer Expansions on the Hypersphere 11
With our specific values, we have∫ 1
−1
xpdx√
(1−A
B − x)(1− x)(x+ 1)(x+ 1+A
B )
=
2B√
(1 +A+B)(1−A+B)
(
1−A
B
)p p∑
j=0
(
A+B − 1
1−A
)j (p
j
)
Vj(α, k). (3.22)
Since we have the integral (3.17), we would like to express (x + A/B)Tm(x) in a sum over
powers of x. This reduces to determining the coefficients ap such that
(x+A/B)Tm(x) =
m+1∑
p=0
apx
p. (3.23)
The Chebyshev polynomial Tm(x) is expressible as a finite sum in terms of the binomial (1−x)m
(see [24, equation (15.9.5)]) as Tm(x) = 2F1(−m,m; 1
2 ; 1−x
2 ). We can further expand these powers
for all m ∈ N0 to obtain
Tm(x) =
(m)m
2m
(
1
2
)
m
bm2 c∑
k=0
(−m
2
)
k
(−m+1
2
)
k
(1−m)kk!
xm−2k. (3.24)
(Note that (m)m can only be written in terms of factorials for m ≥ 1.) Examining (3.24)
multiplied by x + A/B produces the coefficients ap in (3.23) as (3.18). Combining (3.16),
(3.22), and (3.23) completes the proof. �
Utilizing the above procedure to obtain the azimuthal Fourier coefficients of a fundamental
solution of Laplace’s equation on S3
R in terms of complete elliptic integrals, let us directly
compute the m = 0 component. In this case (3.16) reduces to
G
1/2
0 (θ, θ′, θ2, θ
′
2) =
1
π
∫ 1
−1
(x+A/B)dx√
(1− x)(1 + x)(1−A
B − x)(1+A
B + x)
.
Therefore, using the above formulas, we have
G3
R(x,x′)
∣∣
m=0
=
1
4πR
G
1/2
0 (θ, θ′, θ2, θ
′
2) =
(V0(α, k) + (B − 1 +A)V1(α, k))
2π2R
√
(1−A+B)(1 +A−B)
=
1
2π2R
{
K(k) + [cos θ cos θ′ + sin θ sin θ′ cos(θ2 − θ′2)− 1]Π
(
α2, k
)}
× [1 + cos θ cos θ′ + sin θ sin θ′ cos(θ2 + θ′2)]1/2
× [1− cos θ cos θ′ − sin θ sin θ′ cos(θ2 + θ′2)]1/2.
The Fourier expansion for a fundamental solution of Laplace’s equation in three-dimensional
Euclidean space in standard spherical coordinates x = (r sin θ cosφ, r sin θ sinφ, r cos θ) is given
by (2.7) and [7, equation (1.3)] as
n3(x,x′) =
1
‖x− x′‖
=
1
π
√
rr′ sin θ sin θ′
∞∑
m=−∞
eim(φ−φ′)Qm−1/2
(
r2 + r′2 − 2rr′ cos θ cos θ′
2rr′ sin θ sin θ′
)
,
12 H.S. Cohl and R.M. Palmer
where n3 := 4πN 3 is normalized as in (3.1) [8]. By [1, equation (8.13.3)],
N 3(x,x′)
∣∣
m=0
=
1
2π2
√
r2 + r′2 − 2rr′ cos(θ + θ′)
K
(√
4rr′ sin θ sin θ′
r2 + r′2 − 2rr′ cos(θ + θ′)
)
,
which is the flat-space limit of G3
R(x,x′)
∣∣
m=0
as θ, θ′ → 0+, R→∞.
4 Gegenbauer polynomial expansions on the d-sphere
The Gegenbauer polynomial Cµl : C → C, l ∈ N0, µ ∈ (−1
2 ,∞) \ {0}, can be defined in terms
of the Gauss hypergeometric function as
Cµl (x) :=
(2µ)l
l!
2F1
(
−l, 2µ+ l
µ+ 1
2
;
1− x
2
)
.
Note that in the following theorem, we are using the notation (1.1).
Theorem 4.1. The Gegenbauer polynomial expansion for a fundamental solution of Laplace’s
equation on the R-radius, d-dimensional hypersphere SdR for d ≥ 3 expressed in standard hyper-
spherical coordinates (2.1) is given by
GdR(x,x′) =
Γ(d/2)
(d− 2)Rd−2(sin θ sin θ′)d/2−1
×
(−1)d/2−1
2πd/2
∞∑
l=0
(−1)l(2l + d− 2)P
−(d/2−1+l)
d/2−1 (cos θ<)
× Q
d/2−1+l
d/2−1 (cos θ>)C
d/2−1
l (cos γ) if d is even,
(−1)(d−3)/2
4πd/2−1
∞∑
l=0
(−1)l(2l + d− 2)P
−(d/2−1+l)
d/2−1 (cos θ<)
× P
d/2−1+l
d/2−1 (cos θ>)C
d/2−1
l (cos γ) if d is odd.
(4.1)
Theorem 3.1 can be obtained by starting with Theorem 4.1 and taking the limit as d→ 2.
Proof. The completeness relation for hyperspherical harmonics in standard hyperspherical co-
ordinates (2.1) is given by
∞∑
l=0
∑
K
Y K
l (φ, θ2, . . . , θd−1)Y K
l (φ′, θ′2, . . . , θ
′
d−1) =
δ(φ− φ′)δ(θ2 − θ′2) · · · δ(θd−1 − θ′d−1)
sind−2 θ′d−1 · · · sin θ′2
.
Therefore, through (2.12), we can write
δg(x,x
′) =
δ(θ − θ′)
Rd sind−1 θ′
∞∑
l=0
∑
K
Y K
l (φ, θ2, . . . , θd−1)Y K
l (φ′, θ′2, . . . , θ
′
d−1). (4.2)
Since GdR is harmonic on its domain for fixed φ′ ∈ [−π, π), θ, θ′, θ′2, . . . , θ
′
d−1 ∈ [0, π], its restriction
is in C2(Sd−1), and therefore has a unique expansion in hyperspherical harmonics, namely
GdR(x,x′) =
∞∑
l=0
∑
K
uKl (θ, θ′, φ′, θ′2, . . . , θ
′
d−1)Y K
l (φ, θ2, . . . , θd−1), (4.3)
Fourier and Gegenbauer Expansions on the Hypersphere 13
where uKl : [0, π]2 × [−π, π) × [0, π]d−2 → C. If we substitute (4.2), (4.3) into (2.10) and
use (2.5), (2.8), we obtain
−
∞∑
l=0
∑
K
Y K
l (φ, θ2, . . . , θd−1)
[
∂2
∂θ2
+ (d− 1) cot θ
∂
∂θ
− l(l + d− 2)
sin2 θ
]
× uKl (θ, θ′, φ′, θ′2, . . . , θ
′
d−1).
=
∞∑
l=0
∑
K
Y K
l (φ, θ2, . . . , θd−1)Y K
l (φ′, θ′2, . . . , θ
′
d−1)
δ(θ − θ′)
Rd sind−1 θ′
. (4.4)
This indicates the ansatz ul : [0, π]2 → R given by
uKl (θ, θ′, φ′, θ′2, . . . , θ
′
d−1) = ul(θ, θ
′)Y K
l (φ′, θ′2, . . . , θ
′
d−1). (4.5)
What is derived subsequently is an explicit form of a fundamental solution for the associated
Legendre equation. From (4.3), (4.5), the expression for a fundamental solution of the Laplace–
Beltrami operator in standard hyperspherical coordinates on the hypersphere is given by
GdR(x,x′) =
∞∑
l=0
ul(θ, θ
′)
∑
K
Y K
l (φ, θ2, . . . , θd−1)Y K
l (φ′, θ2, . . . , θ′d−1). (4.6)
We use the addition theorem for hyperspherical harmonics [28] given by∑
K
Y K
l (x̂)Y K
l (x̂′) =
Γ(d/2)
2πd/2(d− 2)
(2l + d− 2)C
d/2−1
l (cos γ), (4.7)
where γ is the separation angle (see Section 2.1). The above equation (4.6) can now be simplified
using (4.7); therefore,
GdR(x,x′) =
Γ(d/2)
2πd/2(d− 2)
∞∑
l=0
ul(θ, θ
′)(2l + d− 2)C
d/2−1
l (cos γ). (4.8)
Now we compute the exact expression of ul(θ, θ
′). By separating out the radial part in (4.4)
and using (4.5), we obtain the differential equation
∂2ul(θ, θ
′)
∂θ2
+ (d− 1) cot θ
∂ul(θ, θ
′)
∂θ
− l(l + d− 2)ul(θ, θ
′)
sin2 θ
= − δ(θ − θ′)
Rd sind−1 θ′
. (4.9)
Away from θ = θ′, solutions to the differential equation (4.9) must be given by linearly indepen-
dent solutions to the homogeneous equation, which are given by (2.13), (2.14). These solutions
are given in terms of Ferrers functions of the first and second kind. The selection of linear
independent pairs of Ferrers functions depends on the dimension of the space. We make use of
the Wronskian relations [24, equation (14.2.6)], [20, p. 170], to obtain
W
{
P
−(d/2−1+l)
d/2−1 (cos θ′),Q
d/2−1+l
d/2−1 (cos θ′)
}
=
(−1)d/2−1+l
sin2 θ′
, (4.10)
W
{
P
−(d/2−1+l)
d/2−1 (cos θ′),P
d/2−1+l
d/2−1 (cos θ′)
}
=
2(−1)(d−3)/2+l
π sin2 θ′
, (4.11)
for d even and d odd respectively. The Ferrers functions provide a linearly independent set of
solutions for the homogeneous problem provided that the Wronskians are non-zero and well-
defined. The first Wronskian relation (4.10) is well-defined for d even, but is not defined for d
14 H.S. Cohl and R.M. Palmer
odd. The second Wronskian relation is well-defined for d odd, but is not defined for d even. So
unlike the corresponding problem in hyperbolic geometry [6, equation (70)], on the hypersphere
we must choose our linearly dependent solutions differently depending on the evenness or oddness
of the dimension. Therefore, the solution to (4.9) is given by
ul(θ, θ
′) =
C
(sin θ sin θ′)d/2−1
P
−(d/2−1+l)
d/2−1 (cos θ<)Q
d/2−1+l
d/2−1 (cos θ>) if d is even,
D
(sin θ sin θ′)d/2−1
P
−(d/2−1+l)
d/2−1 (cos θ<)P
d/2−1+l
d/2−1 (cos θ>) if d is odd,
(4.12)
such that ul(θ, θ
′) is continuous at θ = θ′ and C,D ∈ R.
In order to determine the constants C, D, we first define
vl(θ, θ
′) := (sin θ sin θ′)(d−1)/2ul(θ, θ
′). (4.13)
Thus (4.9) can be given by
∂2vl(θ, θ
′)
∂θ2
− vl(θ, θ
′)
4
[
−5(d− 1)2 +
5d2 − 12d+ 7 + 4l2 + 4ld− 8l
sin2 θ
]
= −δ(θ − θ
′)
Rd−2
,
which we then integrate over θ from θ′ − ε to θ′ + ε and take the limit as ε→ 0+. This provides
a discontinuity condition for the derivative of vl(θ, θ
′) with respect to θ evaluated at θ = θ′,
namely
lim
ε→0+
∂vl(θ, θ
′)
∂θ
∣∣∣∣θ′+ε
θ′−ε
=
−1
Rd−2
. (4.14)
After inserting (4.12) with (4.13) into (4.14), substituting z = cos θ′, evaluating at θ = θ′, and
using (4.10), (4.11), we obtain C = (−1)d/2−1+lR2−d, D = 1
2π(−1)(d−3)/2+lR2−d. Finally
ul(θ, θ
′) =
(−1)d/2−1+l
Rd−2(sin θ sin θ′)d/2−1
P
−(d/2−1+l)
d/2−1 (cos θ<)Q
d/2−1+l
d/2−1 (cos θ>) if d is even,
π(−1)(d−3)/2+l
2Rd−2(sin θ sin θ′)d/2−1
P
−(d/2−1+l)
d/2−1 (cos θ<)P
d/2−1+l
d/2−1 (cos θ>) if d is odd,
and therefore through (4.8), this completes the proof. �
4.1 Addition theorem for the azimuthal Fourier coefficient on S3
R
As in [6, Section 5.1], one may compute multi-summation addition theorems for the azimuthal
Fourier coefficients on SdR by relating the azimuthal Fourier coefficient of GdR(x,x′) defined
by (3.2) to its Gegenbauer polynomial expansion (4.1). This is accomplished by using the ad-
dition theorem for hyperspherical harmonics (4.7) which expresses the Gegenbauer polynomial
with degree l ∈ N0 in (4.1) as a sum over the space of degenerate quantum numbers, of a mul-
tiplicative product of separated harmonics as a function of geodesic polar coordinates on the
hypersphere, such as (2.1). For d = 3 one obtains a single-summation addition theorem using
the notation (1.1).
Corollary 4.1. Let m ∈ Z, θ, θ′, θ2, θ
′
2 ∈ [0, π]. Then the azimuthal Fourier coefficient of
a fundamental solution of Laplace’s equation on S3
R is given by
G1/2
m (θ, θ′, θ2, θ
′
2) =
πεm
2
√
sin θ sin θ′
(4.15)
×
∞∑
l=|m|
(−1)l(2l + l)
(l −m)!
(l +m)!
Pml (cos θ2)Pml (cos θ′2)P
−(1/2+l)
1/2 (cos θ<)P
1/2+l
1/2 (cos θ>).
Fourier and Gegenbauer Expansions on the Hypersphere 15
Proof. By expressing (4.1) for d = 3, we obtain
G3
R(x,x′) =
1
8R
√
sin θ sin θ′
∞∑
l=0
(−1)l(2l+1)P
−(1/2+l)
1/2 (cos θ<)P
1/2+l
1/2 (cos θ>)Pl(cos γ), (4.16)
since the Gegenbauer polynomial C
1/2
l is the Legendre polynomial Pl. By using the addition
theorem for hyperspherical harmonics (4.7) with d = 3 on has
Pl(cos γ) =
l∑
m=−l
(l −m)!
(l +m)!
Pml (cos θ2)Pml (cos θ′2)eim(φ−φ′). (4.17)
By inserting (4.17) in (4.16) and reversing the order of the two summation symbols, we can
compare the resulting expression with (3.12) obtaining an expression for G
1/2
m : [0, π]4 → R, in
standard hyperspherical coordinates (2.1). This completes the proof of the corollary. �
The above addition theorem expresses the azimuthal Fourier coefficients G
1/2
m given by Theo-
rem 3.2 as the infinite sum contribution from all meridional modes for a given m value. Al-
ternatively, Theorem 3.2 gives the azimuthal Fourier coefficients as a finite sum of complete
elliptic integrals of the first, second and third kind which encapsulate the contribution from
the infinite meridional sum given by Corollary 4.1. By truncating the infinite sum over the
meridional modes, one obtains what is referred to in Euclidean space, as a multipole method for
the potential problem. The multipole method is well-known to be slowly convergent, even for
moderate departure from spherical symmetry. On the other hand, even for extremely deformed
axisymmetric density distributions, Theorem 3.2 gives the exact solution to the potential prob-
lem in a single m term! Note that the addition theorem (4.15) reduces to the corresponding
result [7, equation (2.4)] in the flat-space limit.
4.2 Spherically symmetric contribution to a fundamental solution
In the next section, we will take advantage of the following result to evaluate Newtonian poten-
tials for spherically symmetric density distributions.
Corollary 4.2. Let R > 0, d = 2, 3, . . ., HdR : [0, π] → R be defined such that HdR(Θ(x̂, x̂′)) :=
GdR(x,x′) with geodesic distances measured from the origin. Then
GdR(x,x′)
∣∣
l=0
= HdR(θ>).
Note that we are using the notation (1.1).
Proof. In dimension d = 2 the result immediately follows from Theorem 3.1. For d ≥ 3 odd,
using Theorem 4.1, one has
GdR(x,x′)
∣∣
l=0
=
(−1)(d−3)/2Γ
(
d
2
)
4πd/2−1Rd−2(sin θ sin θ′)d/2−1
P
−(d/2−1)
d/2−1 (cos θ<)P
d/2−1
d/2−1(cos θ>),
and when combined with [24, equations (14.5.18), (14.9.2)], the result follows. For d ≥ 4 even,
using Theorem 4.1, one has
GdR(x,x′)
∣∣
l=0
=
(−1)d/2−1Γ
(
d
2
)
2πd/2Rd−2(sin θ sin θ′)d/2−1
P
−(d/2−1)
d/2−1 (cos θ<)Q
d/2−1
d/2−1(cos θ>),
which when combined with [24, equations (14.5.18), (14.9.1)] completes the proof. �
16 H.S. Cohl and R.M. Palmer
5 Applications
The expansion formulas that we derive in the previous sections directly provide a mechanism
to obtain rapidly convergent solutions to Newtonian potential problems in d-dimensional hy-
perspherical geometry. One may use these expansion formulae to obtain exact solutions either
through numerical computation, or through symbolic derivation, taking advantage of symmetries
in density distributions.
An anonymous referee has suggested the following in regard to applications. One potential
source for applications of explicit fundamental solutions on hyperspheres is the mathemati-
cal geosciences (see for instance [11]). Furthermore, fundamental solutions of elliptic partial
differential equations on hyperspheres are often used as tools in numerical analysis under the
heading of spherical basis functions (SBFs) and that explicit formulas and series representations
are essential for practical implementations. The use of SBFs in data-fitting and numerical solu-
tion of partial differential equations (via collocation and Galerkin methods) is plentiful (see for
instance [21, 23]).
5.1 Applications to Newtonian potential theory
In this section we derive hyperspherical counterparts to some of the most elementary problems
in classical Euclidean Newtonian potential theory.
Consider Poisson’s equation −∆Φ = ρ in Euclidean space Rd with Cartesian coordinates
(x1, . . . , xd) and ∆ :=
d∑
n=1
∂2
∂x2n
. One then has for ρ : Rd → R integrable (or even more generally
ρ ∈ (D(Rd))′) and Φ ∈ C2(Rd), an integral solution to Poisson’s equation
Φ(x) =
∫
Rd
N d(x,x′)ρ(x)dx′, (5.1)
where N d is given by (2.7). Furthermore, the total Newtonian binding energy E ∈ R, for
a density distribution ρ is given by (cf. [4, equation (16), p. 64])
E =
1
2
∫
Rd
ρ(x)Φ(x)dx. (5.2)
Now consider Poisson’s equation on SdR which is given by
−∆Φ(x) = ρ(x), (5.3)
where ρ : SdR → R integrable (or even more generally ρ ∈ (D(SdR))′), Φ ∈ C2(SdR), and ∆ is
given as in (2.5). An integral solution to (5.3) is given through (2.4), (2.9) as
Φ(x) =
∫
Sd
R
GdR(x,x′)ρ(x′)d vol′g, (5.4)
and the total Newtonian binding energy (5.2) for ρ is given analogously as
E =
1
2
∫
Sd
R
ρ(x)Φ(x)d volg .
Using our above derived azimuthal Fourier and Gegenbauer expansions for GdR, we can exploit
symmetries in ρ to derive closed-form expressions for Φ and E which reduce to their counterparts
in Euclidean space.
The following three examples show direct applications for the expansions of a fundamental
solution of Laplace’s equation on hyperspheres given in this paper. The first example shows an
Fourier and Gegenbauer Expansions on the Hypersphere 17
application of the azimuthal Fourier expansion for a fundamental solution of Laplace’s equation
on S2
R, namely Theorem 3.1. The second example shows an application for the azimuthal
Fourier expansion for a fundamental solution of Laplace’s equation on S3
R, which is derived
from Theorem 3.2. The third example shows an application for the Gegenbauer expansion for
a fundamental solution of Laplace’s equation on S3
R which is derived from Theorem 4.1.
Example 5.1. Newtonian potential of a uniform density 2-ball with geodesic radius Rθ0 in S2
R
(hereafter, uniform density 2-disc). Consider a uniform density 2-disc in S2
R defined using (2.1)
for all φ ∈ [−π, π) such that
ρ(x) :=
{
ρ0 if θ ∈ [0, θ0],
0 if θ ∈ (θ0, π],
(5.5)
where |ρ0| > 0. Due to the axial symmetry in ρ (invariance under rotations centered about
the origin of S2
R at θ = 0), the only non-zero contribution to Φ is from the n = 0 term in
Theorem 3.1. This term is given by Corollary 4.2 and (3.5) as
G2
R(x,x′)
∣∣
n=0
=
1
2π
log cot
θ>
2
. (5.6)
Applying (5.6) in (5.4) with (5.5), using elementary trigonometric integration, one obtains
Φ(x) :=
ρ0R
2
(
log cot
θ
2
− cos θ0 log cot
θ0
2
+ log
sin θ
sin θ0
)
if θ ∈ [0, θ0],
ρ0R
2(1− cos θ0) log cot
θ
2
if θ ∈ (θ0, π],
(5.7)
for all φ ∈ [−π, π).
In comparison, consider the problem to obtain the Newtonian potential for an uniform density
2-disc with radius r0 in R2, with points parametrized using polar coordinates (r cosφ, r sinφ).
Using (3.11) we see that N 2(x,x′)
∣∣
n=0
= − 1
2π log r>, which yields through (5.1)
Φ(x) :=
−
ρ0
4
(
r2 − r2
0 + 2r2
0 log r0
)
if r ∈ [0, r0],
−1
2
ρ0r
2
0 log r if r ∈ (r0,∞),
(5.8)
for all φ ∈ [−π, π). The potential on the uniform density 2-disc (5.7) reduces to (5.8) in the
Euclidean space. This is accomplished by performing an asymptotic expansion as θ, θ0 → 0+
in the flat-space limit R → ∞. Note however that in this two-dimensional example, such as
in other two-dimensional problems (see for example [9, p. 151]), we must subtract a constant
(cR := 1
2ρ0r
2
0 log(2R)) which tends to infinity. This is required in the flat-space limit R → ∞
for Φ in (5.7) to obtain the finite limit (5.8). The potential Φ satisfies (5.3), and therefore, so
does any solution Φ + cR, with cR constant. Using the geodesic distances r ∼ Rθ, r0 ∼ Rθ0,
the result follows using Maclaurin expansions for the logarithmic and trigonometric functions
in (5.7).
For the uniform density 2-disc, the total Newtonian binding energy is
E2-disc =
π
4
ρ2
0R
4
(
(1− 4 cos θ0 + cos(2θ0)) log cot
θ0
2
− 4 log cos
θ0
2
− 2 log sin θ0 + 2 cos(θ0) + 2(log 2− 1)
)
. (5.9)
For the uniform density 2-disc embedded in R2, the total Newtonian binding energy is
ER2-2-disc =
π
16
ρ2
0r
4
0
[
−4 log r0 + 1 +
r2
0
6R2
(
2 log
r0
2
− 1
)]
, (5.10)
18 H.S. Cohl and R.M. Palmer
where we have included the lowest-order correction term due to curvature originating from (5.9).
In the flat-space limit R→∞, (5.9) reduces to (5.10), with the term corresponding to cR being
subtracted off.
Example 5.2. Newtonian potential of a uniform density circular curve segment with geodesic
length 2Rϕ with ϕ ∈ (0, π] in S3
R. Consider a uniform density circular curve segment using Hopf
coordinates (2.15) on S3
R. Recall ϑ ∈ [0, π2 ], φ1, φ2 ∈ [−π, π). The density distribution ρ is given
by the uniform density circular curve segment
ρ(x) :=
ρ0δ(ϑ)
R2 sinϑ cosϑ
if φ1 ∈ [−ϕ,ϕ),
0 otherwise.
(5.11)
This uniform density distribution is rotationally-invariant about the φ2 angle. In order to make
a connection with the corresponding Euclidean space problem, consider the flat-space limit
R → ∞ of the density distribution. Recall from Section 2.4 that ϑ ∼ r
R , φ1 ∼ z
R , with ϕ ∼ L
R ,
L > 0. One has since δ(ax) = δ(x)
|a| , that
ρ(x) =
ρ0δ(ϑ)
R2 sinϑ cosϑ
∼
ρ0δ(
r
R)
R2 sin
(
r
R
)
cos
(
r
R
) ∼ ρ0δ(
r
R)
Rr
=
ρ0δ(r)
r
,
for all z ∈ [−L,L]. This is the axisymmetric density distribution of a uniform density line
segment centered on the origin, along the z-axis in R3.
In order to solve for the Newtonian potential of this density distribution (5.11) on S3
R using
Hopf coordinates, one must obtain the Riemannian structure, namely
ds2 = R2
(
dϑ2 + cos2 ϑdφ2
1 + sin2 ϑdφ2
2
)
.
This yields the metric coefficients and therefore in Hopf coordinates on S3
R one has the Rieman-
nian volume measure [19, p. 29]
d volg = R3 cosϑ sinϑdϑdφ1dφ2.
and the geodesic distance through (2.2)
d(x,x′) = R cos−1
(
cosϑ cosϑ′ cos(φ1 − φ′1) + sinϑ sinϑ′ cos(φ2 − φ′2)
)
.
A fundamental solution of Laplace’s equation is given through (3.12) as
G3
R(x,x′) =
A/B + cosψ
4πR
√(
1−A
B − cosψ
) (
1+A
B + cosψ
) ,
where A : [0, π2 ]2 × [−π, π)2 → [0, 1], B : [0, π2 ]2 → [0, 1], are defined as
A(ϑ, ϑ′, φ1, φ
′
1) := cosϑ cosϑ′ cos(φ1 − φ′1), B(ϑ, ϑ′) := sinϑ sinϑ′.
The azimuthal Fourier coefficients of g3 = 4πRG3
R are therefore given as an integral of the same
form as that given in standard hyperspherical coordinates, cf. (3.16),
G1/2
m (ϑ, ϑ′, φ1, φ
′
1) =
εm
π
∫ 1
−1
(x+A/B)Tm(x)dx√
(1− x2)
(
1−A
B − x
) (
1+A
B + x
) ,
Fourier and Gegenbauer Expansions on the Hypersphere 19
where x = cos(φ2 − φ′2). Since our density distribution is axisymmetric about the φ2 angle, the
only contribution to the Newtonian potential originates from the m = 0 term. We can evaluate
this term using the methods of Section 3.2 which results in
G
1/2
0 (ϑ, ϑ′, φ1, φ
′
1) =
2
π
[
K(k)√
(1−A+B)(1 +A+B)
+
(A+B − 1)Π(α2, k)√
(A+B + 1)(1−A+B)
]
.
The Newtonian potential for the density distribution (5.11) is given through (5.4). The ϑ′
integration selects ϑ′ = 0 in the integrand which implies that A = cosϑ cos(φ1 − φ′1), B = 0,
k = α2 = 0, and since K(0) = Π(0, 0) = π
2 ,
G
1/2
0 (ϑ, 0, φ1, φ
′
1) =
A√
1−A2
=
cotϑ cos(φ1 − φ′1)√
1 + cot2 ϑ sin2(φ1 − φ′1)
.
The Newtonian potential of the uniform density circular curve segment is given by
Φ(x) =
ρ0 cotϑ
2
∫ ϕ
−ϕ
cos(φ1 − φ′1)dφ′1√
1 + cot2 ϑ sin2(φ1 − φ′1)
.
With the substitution u = cotϑ sin(φ1 − φ′1), one finally has
Φ(x) =
ρ0
2
∫ cotϑ sin(φ1+ϕ)
cotϑ sin(φ1−ϕ)
du√
1 + u2
=
ρ0
2
log
∣∣∣∣∣sin(φ1 + ϕ) +
√
tan2 ϑ+ sin2(φ1 + ϕ)
sin(φ1 − ϕ) +
√
tan2 ϑ− sin2(φ1 + ϕ)
∣∣∣∣∣ ,
since the antiderivative of (1 + z2)−1/2 is sinh−1 z = log |z +
√
z2 + 1|. In the flat-space limit
R→∞ we have
Φ(x) =
ρ0
2
log
∣∣∣∣∣z + L+
√
r2 + (z + L)2
z − L+
√
r2 + (z − L)2
∣∣∣∣∣ ,
which is the potential in cylindrical coordinates of a uniform density line segment on the z′-axis
with z′ ∈ [−L,L].
Example 5.3. Newtonian potential of a uniform density 3-ball with geodesic radius Rθ0 in S3
R
(hereafter, uniform density 3-ball). Consider a uniform density 3-ball defined using (2.1) for all
θ2 ∈ [0, π], φ ∈ [−π, π) such that (5.5) is true with |ρ0| > 0. Due to the spherical symmetry
in ρ (invariance under rotations centered about the origin of S3
R at θ = 0), the only non-zero
contribution to Φ is from the l = 0 term in Theorem 4.1. This term is given through Corollary 4.2
and (3.12) as
G3
R(x,x′)
∣∣
l=0
=
1
4πR
cot θ>. (5.12)
Applying (5.12) in (5.4) with (5.5), using elementary trigonometric integration, one obtains
Φ(x) :=
ρ0R
2
2
[θ cot θ − cos2 θ0] if θ ∈ [0, θ0],
ρ0R
2
2
[θ0 − sin θ0 cos θ0] cot θ if θ ∈ (θ0, π],
(5.13)
for all θ2 ∈ [0, π], φ ∈ [−π, π).
20 H.S. Cohl and R.M. Palmer
In comparison, consider the well-known freshman problem to obtain the Newtonian potential
for a uniform density 3-ball with radius r0 in R3. Using the generating function for Gegenbauer
polynomials [24, equation (18.12.4)] for d = 3 one obtains N 3(x,x′)
∣∣
l=0
= 1
4πr>
, which yields
Φ(x) :=
ρ0
6
(
3r2
0 − r2
)
if r ∈ [0, r0],
ρ0r
3
0
3r
if r ∈ (r0,∞).
(5.14)
The Newtonian potential of the 3-ball (5.13) reduces to (5.14) in the flat-space limit, namely
θ, θ0 → 0+ in the flat-space limit R → ∞. Using the geodesic distances r ∼ Rθ, r0 ∼ Rθ0, the
result follows using the first two terms of Laurent series for cot θ.
For the uniform density 3-ball, the total Newtonian binding energy is
E3-ball =
π
16
ρ2
0R
5 (−4θ0 + 4 sin(2θ0) + sin(4θ0)− 8θ0 cos(2θ0)) . (5.15)
Furthermore, the classic result for the 3-ball embedded in R3 is given through (5.14), namely
ER3-3-ball =
4π
15
ρ2
0r
5
0
(
1− 13
21
r2
0
R2
)
=
3M2
20πr0
(
1− 13
21
r2
0
R2
)
, (5.16)
where we have included the lowest-order correction term due to curvature. Since the total mass
of the 3-ball is given by M = 4
3πρ0r
3
0, the standard result (−3GM2/(5r0)) from Newtonian
gravity is obtained by mapping M 7→ −4πGM for the M which originated from the Newtonian
potential (by solving the Poisson equation ∆Φ = 4πGρ). If one performs a Maclaurin expansion
about θ0 in (5.15), then we see that
−4θ0 + 4 sin(2θ0) + sin(4θ0)− 8θ0 cos(2θ0) =
64
15
θ5
0 −
832
315
θ7
0 +O
(
θ9
0
)
,
and since θ0 ∼ r0/R, then (5.16) follows in the flat-space limit R→∞.
5.2 Applications to superintegrable potentials
Superintegrable potentials, and in fact any potentials such that, Φ ∈ C2(SdR) will satisfy Poisson’s
equation −∆Φ(x) = ρ(x), for a particular paired density distribution ρ : SdR → R. Therefore,
given a superintegrable Φ(x) on SdR, one may obtain a density distribution corresponding to
the superintegrable potential by evaluating the negative Laplace–Beltrami operator on the d-
dimensional hypersphere acting on the superintegrable potential. Furthermore, one may then
re-obtain the superintegrable potential from ρ(x) using (5.4). Fourier and Gegenbauer series
expansions of fundamental solutions, give you the ability to demonstrate this connection when
a superintegrable potential and the corresponding density distribution satisfy a degree of sym-
metry, such as for rotationally-invariant or spherically-symmetric density distributions.
Below we present two isotropic examples of this correspondence on SdR. However, such paired
density distributions are available for all superintegrable potentials, no matter how complicated,
by simply applying the negative Laplace–Beltrami operator to those potentials. Therefore a su-
perintegrable potential can be re-obtained by convolution with the paired density distribution
and a fundamental solution for Laplace’s equation on the d-dimensional R-radius hypersphere.
Example 5.4. Take for instance the superintegrable isotropic oscillator potential on SdR in
standard hyperspherical coordinates (2.1), given by [16, equation (1.4)]
Φ(x) = α tan2 θ, (5.17)
Fourier and Gegenbauer Expansions on the Hypersphere 21
for |α| > 0. In this case, one obtains through (2.5), the paired density distribution
ρ(x) =
−2α
R2
(
1 + tan2 θ
)(
3 tan2 θ + d
)
.
Note that this density distribution has an infinite total mass given by
M =
∫
Sd
R
ρ(x)d volg,
since through (2.9) one may obtain∫
Sd−1
R
dω =
2πd/2
Γ
(
d
2
) , (5.18)
and also using that
∫ π
0 (1 + tan2 θ)(3 tan2 θ + d) sind−1 θdθ diverges.
On the other hand, one may use (5.4) make a concrete correspondence between the superin-
tegrable potential (5.17) and the convolution of a fundamental solution of Laplace’s equation in
hyperspherical geometry. Using (5.4), (5.18), and Corollary 4.2, one has
tan2 θ =
−4πd/2Rd−2
Γ
(
d
2
) ∫
Sd
R
HdR(θ>)
(
1 + tan2 θ′
)(
3 tan2 θ′ + d
)
sind−1 θ′dθ′,
which is equivalent to
tan2 θ =
−(d− 2)!
2d/2−2Γ
(
d
2
) ∫ π
0
sind−1 θ′
sind/2−1 θ>
Q
1−d/2
d/2−1(cos θ>)
(
1 + tan2 θ′
)(
3 tan2 θ′ + d
)
dθ′.
By evaluating the above integral, one arrives at a new definite integral valid for d = 2, 3, . . .,
namely∫ π
θ
Q
1−d/2
d/2−1(cos θ′)
(
1 + tan2 θ′
)(
3 tan2 θ′ + d
)
sind/2 θ′dθ′
= − tan θ
(
1 + tan2 θ
)
sind/2 θQ
1−d/2
d/2−1(cos θ)−
2d/2−2Γ
(
d
2
)
tan2 θ
(d− 2)!
.
Example 5.5. Take for instance the superintegrable Kepler–Coulomb potential on SdR in stan-
dard hyperspherical coordinates (2.1), also given by [16, equation (1.4)]
Φ(x) = −α cot θ, (5.19)
for |α| > 0. In this case, one obtains through (2.5), the paired density distribution
ρ(x) =
(3− d)α
R2
cot θ
(
1 + cot2 θ
)
.
Note that this density distribution has an zero total mass for all dimensions greater than two,
except for d = 3, which gives a finite total mass. To prove that the total mass vanishes for
d = 2, one may approach the endpoints of integration in a limiting fashion. For d = 3, this
superintegrable potential is a fundamental solution for Laplace’s equation on the 3-sphere (3.12)
(see for instance [27]), and therefore the paired density distribution is proportional to the Dirac
delta distribution which integrates to unity.
22 H.S. Cohl and R.M. Palmer
Use (5.4) to make a concrete correspondence between the superintegrable potential (5.19)
and the convolution of a fundamental solution of Laplace’s equation in hyperspherical geometry.
Using (5.4), (5.18), and Corollary 4.2, one has
− cot θ =
(3− d)(d− 2)!
2d/2−1Γ
(
d
2
) ∫ π
0
sind−1 θ′
sind/2−1 θ>
Q
1−d/2
d/2−1(cos θ>)
(
1 + cot2 θ′
)
cot θ′dθ′.
By evaluating the above integral, one arrives at a new definite integral valid for d = 4, 5, . . .,
namely∫ π
θ
Q
1−d/2
d/2−1(cos θ′)
(
1 + cot2 θ′
)
cot θ′ sind/2 θ′dθ′
= −sind/2−2 θ
(d− 3)
Q
1−d/2
d/2−1(cos θ) +
2d/2−1Γ
(
d
2
)
cot θ
(d− 3)(d− 2)!
.
For d = 2, in order to obtain the paired density distribution for (5.19), one must approach the
endpoints of integration in a limiting fashion. By taking the integration over θ′ ∈ [ε, π − ε], and
then taking the limit ε → 0+, the singular terms which arise cancel. Furthermore, one must
add a constant (whose gradient vanishes for all values of ε) which tends to infinity as ε → 0+
to (5.19). This is often the case in two-dimensions, see for instance Example 5.1. In this case,
it is not difficult to see that one must also add a Dirac delta distribution source term at the
origin to the density distribution. The superintegrable density-potential pair for (5.19) on S2
R is
therefore
Φ(x) = α cot θ + α cot ε+
α
sin ε
log tan
ε
2
,
and
ρ(x) = − α
R2
cot θ
(
1 + cot2 θ
)
+
2αδ(θ)
sin ε sin θ
,
with (5.4) being satisfied as ε → 0+. For d = 3, no new information is gained since in this
case the superintegrable potential (5.19) is simply a fundamental solution of Laplace’s equation
on S3
R (3.12).
Acknowledgements
Much thanks to Willard Miller and George Pogosyan for valuable discussions. We would also
like to express our gratitude to the anonymous referees and the editors for this special issue in
honour of Luc Vinet, for their significant contributions.
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1 Introduction
2 Preliminaries
2.1 Coordinates
2.2 Fundamental solution of Laplace's equation on SRd
2.3 Harmonics on the d-dimensional hypersphere
2.4 The flat-space limit from hyperspherical to Euclidean geometry
3 Fourier expansions for a fundamental solution of Laplace's equation
3.1 Fourier expansion for a fundamental solution of Laplace's equation on the 2-sphere
3.2 Fourier expansion for a fundamental solution of Laplace's equation on the 3-sphere
4 Gegenbauer polynomial expansions on the d-sphere
4.1 Addition theorem for the azimuthal Fourier coefficient on SR3
4.2 Spherically symmetric contribution to a fundamental solution
5 Applications
5.1 Applications to Newtonian potential theory
5.2 Applications to superintegrable potentials
References
|
| id | nasplib_isofts_kiev_ua-123456789-147003 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:46:36Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cohl, H.S. Palmer, R.M. 2019-02-12T18:16:31Z 2019-02-12T18:16:31Z 2015 Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry / H.S. Cohl, R.M. Palmer // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 31C12; 32Q10; 33C05; 33C45; 33C55; 35J05; 35A08; 42A16 DOI:10.3842/SIGMA.2015.015 https://nasplib.isofts.kiev.ua/handle/123456789/147003 For a fundamental solution of Laplace's equation on the R-radius d-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials. This paper is a contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet. The full collection is available at http://www.emis.de/journals/SIGMA/ESSA2014.html. Much thanks to Willard Miller and George Pogosyan for valuable discussions. We would also like to express our gratitude to the anonymous referees and the editors for this special issue in honour of Luc Vinet, for their significant contributions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry Article published earlier |
| spellingShingle | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry Cohl, H.S. Palmer, R.M. |
| title | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_full | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_fullStr | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_full_unstemmed | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_short | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_sort | fourier and gegenbauer expansions for a fundamental solution of laplace's equation in hyperspherical geometry |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147003 |
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