Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

0.2 MB·English

by  Howard S. Cohl

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7(2011), 108, 14 pages
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, Gaithersburg, Maryland, USA
E-mail: [email protected]
‡ Department of Mathematics, University of Auckland, 38 Princes Str., Auckland, New Zealand
2
URL: http://hcohl.sdf.org
1
0
Received August 18, 2011,in final form November 22, 2011; Published online November 29, 2011
2
http://dx.doi.org/10.3842/SIGMA.2011.108
n
a
J Abstract. Due to the isotropy of d-dimensional hypersphericalspace, one expects there to

9 exist a spherically symmetric fundamental solution for its corresponding Laplace–Beltrami
2 operator. The R-radius hypersphere Sd with R > 0, represents a Riemannian manifold
R
with positive-constantsectional curvature. We obtain a spherically symmetric fundamental
]
h solution of Laplace’s equation on this manifold in terms of its geodesic radius. We give
p severalmatchingexpressionsforthis fundamentalsolutionincluding a definite integralover
- reciprocalpowersofthetrigonometricsine,finite summationexpressionsovertrigonometric
h
functions,Gausshypergeometricfunctions,andintermsoftheassociatedLegendrefunction
t
a
of the second kind on the cut (Ferrers function of the second kind) with degree and order
m
givenby d/2 1 and 1 d/2 respectively, with real argument between plus and minus one.
[ − −
Key words: hyperspherical geometry; fundamental solution; Laplace’s equation; separation
3
of variables; Ferrers functions
v
9
2010 Mathematics Subject Classification: 35A08; 35J05;32Q10;31C12; 33C05
7
6
3
1 Introduction
.
8
0
1 We compute closed-form expressions of a spherically symmetric Green’s function (fundamental
1 solution of Laplace’s equation) for a d-dimensional Riemannian manifold of positive-constant
:
v sectional curvature, namely the R-radius hypersphere with R > 0. This problem is intimately
i
X related to the solution of the Poisson equation on this manifold and the study of spherical har-
r monics which play an important role in exploring collective motion of many-particle systems
a
in quantum mechanics, particularly nuclei, atoms and molecules. In these systems, the hy-
perradius 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 [37], and later invoked to greater effect in nuclear and atomic physics, respectively,
by Delves [5] and Smith [31]. The relevance of this representation to two-electron excited states
of the helium atom was noted by Cooper, Fano & Prats [4]; Fock [10] had previously shown that
the hyperspherical representation was particularly efficient 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 [38] (nuclear structure), Fano [9] and Lin [24] (atomic
structure), and Pack & Parker [28] (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 mathematical aspects of this paper can be
found in [22, 33, 35]. Some historical references on this topic include [17, 23, 29, 30, 36].
2 H.S. Cohl
This paper is organized as follows. In Section 2 we describe hyperspherical geometry and its
correspondingmetric,globalgeodesicdistancefunction,Laplacianandhypersphericalcoordinate
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’ fundamental solution of the Laplace’s equation on
the R-radius hypersphere.
Throughoutthis paper we rely on the following definitions. For a ,a ,... C, if i,j Z and
1 2
∈ ∈
j j
j < i then a = 0 and a = 1. The set of natural numbers is given by N := 1,2,3,... ,
n n
{ }
n=i n=i
the set N0 :P= 0,1,2,... =QN 0 , and the set Z := 0, 1, 2,... .
{ } ∪{ } { ± ± }
2 Hyperspherical geometry
The Euclidean inner product for Rd+1 is given by (x,y) = x y +x y + +x y . The variety
0 0 1 1 d d
···
(x,x) = x2+x2+ +x2 = R2, for x Rd+1 and R > 0, defines the R-radius hypersphere Sd.
0 1 ··· d ∈ R
We denote the unit radius hypersphere by Sd := Sd. Hyperspherical space in d-dimensions, de-
1
noted by Sd, is a maximally symmetric, simply connected, d-dimensional Riemannian manifold
R
with positive-constant sectional curvature (given by 1/R2, see for instance [22, 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 Sd can be parametrized using subgroup-type coor-
R
dinate systems, i.e., those which correspond to a maximal subgroup chain O(d) (see for
⊃ ···
instance [18, 20]). The isometry group of the space Sd is the orthogonal group O(d). Hyper-
R
spherical space Sd, can be identified with the quotient space O(d)/O(d 1). The isometry
R −
group O(d) acts transitively on Sd. There exist separable coordinate systems on the hyper-
R
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 Section IX.5 in [35].) Polyspherical
coordinates on Sd all sharetheproperty that they aredescribed by (d+1)-variables: R [0, )
R ∈ ∞
plus d-angles each being given by the values [0,2π), [0,π], [ π/2,π/2] or [0,π/2] (see [18, 19]).
−
In our context, a useful subset of polyspherical coordinate are geodesic polar coordinates
(θ,x) (see for instance [27]). These coordinates, which parametrize points on Sd, have origin
R
at O = (R,0,...,0) Rd+1 and are given by a ‘radial’ parameter θ [0,π] which parametrizes
∈ ∈
poibnts along a geodesic curve emanating from O in a direction x Sd−1. Geodesic polar
∈
coordinate systems partition Sd into a family of (d 1)-dimensional hyperspheres, each with
R −
a ‘radius’ θ := θd (0,π), on which all possible hyperspherical coordbinate systems for Sd−1 may
∈
be used (see for instance [35]). One then must also consider the limiting case for θ = 0,π to
fill out all of Sd. Standard hyperspherical coordinates (see [21, 25]) are an example of geodesic
R
polar coordinates, and are given by
x = Rcosθ,
0
x = Rsinθcosθ ,
1 d−1
x = Rsinθsinθ cosθ ,
2 d−1 d−2
(1)
···························
x = Rsinθsinθ cosθ ,
d−2 d−1 2
···
x = Rsinθsinθ sinθ cosφ,
d−1 d−1 2
···
x = Rsinθsinθ sinθ sinφ,
d d−1 2
···
θ [0,π] for i 2,...,d , θ := θ , and φ [0,2π).
i d
∈ ∈{ } ∈
Fundamental Solution of Laplace’s Equation in Hyperspherical Geometry 3
In order to study a 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 Sd are
R
simplygiven byarclengths, angles betweentwo arbitraryvectors, fromtheorigin intheambient
Euclidean space (see for instance [22, p. 82]). Any parametrization of the hypersphereSd, must
R
have (x,x) = x2 + +x2 = R2, with R > 0. The distance between two points x,x′ Sd on
0 ··· d ∈ R
the hypersphere is given by
(x,x′) 1
d(x,x′) = Rγ = Rcos−1 = Rcos−1 (x,x′) . (2)
(x,x)(x′,x′) R2
(cid:18) (cid:19) (cid:18) (cid:19)
This is evident from the fact that the geodesics on Sd are great circles, i.e., intersections of Sd
R R
withplanesthroughtheoriginoftheambientEuclideanspace,withconstantspeedparametriza-
tions.
In any geodesic polar coordinate system, the geodesic distance between two points on the
submanifold is given by
1
d(x,x′) = Rcos−1 (x,x′) = Rcos−1 cosθcosθ′+sinθsinθ′cosγ , (3)
R2
(cid:18) (cid:19)
(cid:0) (cid:1)
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
d−2 d−2 i−1
cosγ = cos(φ φ′) sinθ sinθ ′+ cosθ cosθ ′ sinθ sinθ ′. (4)
i i i i j j
−
i=1 i=1 j=1
Y X Y
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 dvol (see for instance Section 3.4
g
in[16]),invariantundertheisometrygroupSO(d),oftheRiemannianmanifoldSd. Forinstance,
R
in standard hyperspherical coordinates (1) on Sd the volume measure is given by
R
dvol = Rdsind−1θdθdω := Rdsind−1θdθ sind−2θ sinθ dθ dθ . (5)
g d−1 2 1 d−1
··· ···
Thedistance r [0, ) along a geodesic, measured from theorigin, 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 [21]), we examine the limit of zero curvature. In order to do this, we take the
limit θ 0+ and R of the volume measure (5) which produces
→ → ∞
r
dvol Rd−1sind−1 drdω rd−1drdω,
g
∼ R ∼
(cid:16) (cid:17)
which is the Euclidean measure in Rd, expressed in standard Euclidean hyperspherical coordi-
nates. This measure is invariant under the Euclidean motion group E(d).
Itwill beusefulbelow to express theDirac delta function on Sd. TheDirac delta function on
R
the Riemannian manifold Sd with metric g is defined for an open set U Sd with x,x′ Sd
R ⊂ R ∈ R
such that
1 if x′ U,
δ (x,x′)dvol = ∈ (6)
ZU g g (0 if x′ ∈/ U.
For instance, using (5) and (6), in standard hyperspherical coordinates on Sd (1), we see that
R
the Dirac delta function is given by
δ (x,x′) = δ(θ−θ′) δ(θ1 −θ1′)···δ(θd−1 −θd′−1).
g Rdsind−1θ′ sinθ′ sind−2θ′
2··· d−1
4 H.S. Cohl
2.1 Laplace’s equation 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 thatconverts thestandardCartesian coordinates intheambient spacetoacoordinate
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
1 ∂ ∂
∆ = g gij , (7)
g ∂ξi | | ∂ξj
iX,j=1 | | (cid:18)p (cid:19)
p
where g = det(g ), the metric is given by
ij
| | | |
d
ds2 = g dξidξj, (8)
ij
i,j=1
X
and
d
g gij = δj,
ki k
i=1
X
j
where δ 0,1 is the Kronecker delta
i ∈{ }
1 if i = j,
j
δ := (9)
i
(0 if i = j,
6
for i,j Z. The relationship between the metric tensor G = diag(1,...,1) in the ambient
ij
∈
space and g of (7) and (8) is given by
ij
d ∂xk ∂xl
g (ξ) = G .
ij kl ∂ξi ∂ξj
k,l=0
X
The Riemannian metric in a geodesic polar coordinate system on the submanifold Sd is
R
given by
ds2 = R2 dθ2+sin2θ dγ2 , (10)
(cid:0) (cid:1)
where an appropriate expression for γ in a curvilinear coordinate system is given. If one com-
bines(1), (4),(7)and(10), theninageodesicpolarcoordinatesystem, Laplace’s equationonSd
R
is given by
1 ∂2f ∂f 1
∆f = +(d 1)cotθ + ∆Sd−1f = 0, (11)
R2 ∂θ2 − ∂θ sin2θ
(cid:20) (cid:21)
where ∆Sd−1 is the corresponding Laplace–Beltrami operator on Sd−1.
Fundamental Solution of Laplace’s Equation in Hyperspherical Geometry 5
3 A Green’s function on the hypersphere
3.1 Harmonics in geodesic polar coordinates
The harmonics 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 [18, 19]. 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–
BeltramioperatoronSd−1 whichsatisfythefollowingeigenvalue problem(seeforinstance(12.4)
and Corollary 2 to Theorem 10.5 in [32])
∆Sd−1YlK(x) =−l(l+d−2)YlK(x), (12)
where x ∈ Sd−1,bYlK(x) are normalizedbangular hyperspherical harmonics, l ∈ N0 is the angular
momentum quantum number, and K stands for the set of (d 2)-quantum numbers identifying
−
degenebrate harmonicsbfor each l and d. The degeneracy
(d 3+l)!
(2l+d 2) −
− l!(d 2)!
−
(see (9.2.11) in [35]), tells you how many linearly independent solutions exist for a particular l
value and dimension d. The angular hyperspherical harmonics are normalized such that
YK(x)YK′(x)dω = δl′δK′,
l l′ l K
Sd−1
Z
where dω is thebRiemanbnian (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 Chapter IX in [35] and
Chapter 11 [8]). The generalized Kronecker delta symbol δK′ (cf. (9)) is defined such that it
K
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 ‘radial’ solutions of Laplace’s equation on Sd, which satisfy the following
R
ordinary differential equation (cf. (11) and (12))
d2u du l(l+d 2)
+(d 1)cotθ − u = 0. (13)
dθ2 − dθ − sin2θ
Four solutions of this ordinary differential equation ud,l,ud,l : ( 1,1) C are given by
1± 2± − →
1
d,l ±(d/2−1+l)
u (cosθ):= P (cosθ),
1± (sinθ)d/2−1 d/2−1
and
1
d,l ±(d/2−1+l)
u (cosθ):= Q (cosθ), (14)
2± (sinθ)d/2−1 d/2−1
where Pµ,Qµ : ( 1,1) C are Ferrers functions of the first and second kind (associated
ν ν
− →
Legendre functions of the first and second kind on the cut). The Ferrers functions of the first
6 H.S. Cohl
and second kind (see Chapter 14 in [26]) can be defined respectively in terms of a sum over two
Gauss hypergeometric functions, for all ν,µ C such that ν +µ N,
∈ 6∈ −
Pµ(x) := 2µ+1 sin π(ν +µ) Γ ν+µ2+2 x(1 x2)−µ/2 F 1−ν −µ, ν −µ+2;3;x2
ν √π 2 Γ(cid:16)ν−µ+1(cid:17) − 2 1(cid:18) 2 2 2 (cid:19)
h i 2
(cid:16) (cid:17)
+ 2µ cos π(ν +µ) Γ ν+µ2+1 (1 x2)−µ/2 F −ν −µ, ν −µ+1;1;x2
√π 2 Γ(cid:16)ν−µ+2(cid:17) − 2 1(cid:18) 2 2 2 (cid:19)
h i 2
(cid:16) (cid:17)
(cf. (14.3.11) in [26]), and
Γ ν+µ+2
Qµ(x) := √π2µcos π(ν +µ) 2 x(1 x2)−µ/2 F 1−ν −µ,ν −µ+2; 3;x2
ν 2 Γ(cid:16)ν−µ+1(cid:17) − 2 1(cid:18) 2 2 2 (cid:19)
h i 2
(cid:16) (cid:17)
Γ ν+µ+1
√π2µ−1sin π(ν +µ) 2 (1 x2)−µ/2 F −ν −µ,ν −µ+1;1;x2 (15)
− 2 Γ(cid:16)ν−µ+2(cid:17) − 2 1(cid:18) 2 2 2 (cid:19)
h i 2
(cid:16) (cid:17)
(cf. (14.3.12) in [26]). The Gauss hypergeometric function F :C C (C N ) z C:
2 1 0
× × \− ×{ ∈
z < 1 C, can be defined in terms of the infinite series
| | } →
∞
(a) (b)
F (a,b;c;z) := n nzn
2 1
(c) n!
n
n=0
X
(see (15.2.1) in [26]), 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 conditio-
− − ∈ ∞
nally if z = 1 and Re(c a b) ( 1,0], and diverges if Re(c a b) ( , 1]. For z C
6 − − ∈ − − − ∈ −∞ − ∈
and n N , the Pochhammer symbol (z) (also referred to as the rising factorial) is defined as
0 n
∈
(cf. (5.2.4) in [26])
n
(z) := (z+i 1).
n
−
i=1
Y
The Pochhammer symbol (rising factorial) is expressible in terms of gamma functions as (5.2.5)
in [26]
Γ(z+n)
(z) = ,
n
Γ(z)
for all z C N . The gamma function Γ : C N C (see Chapter 5 in [26]) is an
0 0
∈ \ − \ − →
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 (5.2.1) in [26])
∞
Γ(z) := tz−1e−tdt,
Z0
Rez > 0. The Euler reflection formula allows one to obtain values of the gamma function in
the left-half complex plane (cf. (5.5.3) in [26]), namely
π
Γ(z)Γ(1 z) = ,
− sinπz
Fundamental Solution of Laplace’s Equation in Hyperspherical Geometry 7
0 < Rez < 1, for Rez = 0, z = 0, and then for z shifted by integers using the following
6
recurrence relation (see (5.5.1) in [26])
Γ(z+1) = zΓ(z).
Animportantformulawhichthegammafunctionsatisfiesistheduplicationformula(i.e., (5.5.5)
in [26])
22z−1 1
Γ(2z) = Γ(z)Γ z+ , (16)
√π 2
(cid:18) (cid:19)
provided 2z N .
0
6∈ −
Due to the fact that the space Sd is homogeneous with respect to its isometry group, the
R
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 associated Legendre functions of the second kind on the
cut with argument given by cosθ. This associated Legendre function naturally fits our require-
ments because it is singular at θ = 0, whereas the associated Legendre functions of the first
kind, with the same argument, is regular at θ = 0. We require there to exist a singularity at
the origin of a fundamental solution of Laplace’s equation on Sd, since it is a manifold and
R
must behave locally like a Euclidean fundamental solution of Laplace’s equation which also has
a singularity at the origin.
3.2 Fundamental solution of the Laplace’s equation on the hypersphere
In computing a fundamental solution of the Laplacian on Sd, we know that
R
∆ d(x,x′) = δ (x,x′), (17)
− SR g
where g is the Riemannian metric on Sd (e.g., (10)) and δ is the Dirac delta function on the
R g
manifold Sd (e.g., (6)). In general since we can add any harmonic function to a fundamental
R
solution of Laplace’s equation and still have a 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 d on the hyperspherewith pure
SR
‘radial’, θ := d(x,x′) (e.g., (3)), and constant angular dependence due to the influence of the
point-like nature of the Dirac delta function in (17). For a spherically symmetric solution to
the Laplace equation, the corresponding ∆Sd−1 term in (11) vanishes since only the l = 0 term
survives in (13). In other words we expect there to exist a fundamental solution of Laplace’s
equation on Sd such that d(x,x′) = f(θ) (cf. (3)), where R is a parameter of this fundamental
R SR
solution.
We have proven that on the R-radius hypersphere Sd, a Green’s function for the Laplace
R
operator (fundamental solution of Laplace’s equation) can be given as follows.
Theorem 1. Let d 2,3,... . Define :(0,π) R as
d
∈ { } I →
π/2 dx
(θ) := ,
Id sind−1x
Zθ
x,x′ Sd, and d : (Sd Sd) (x,x) :x Sd R defined such that
∈ R SR R× R \{ ∈ R}→
Γ(d/2)
d(x,x′):= (θ),
SR 2πd/2Rd−2Id
8 H.S. Cohl
where θ := cos−1([x,x′]) is the geodesic distance between x and x′ on the unit radius hyper-
sphere Sd, with x = x/R, x′ = x′/R, then d is a fundamental solution for ∆ where ∆ is the
SR −
Laplace–Beltrami opberbator on Sd. Moreover, b b
R
b b
d/2−1
(d 3)!! θ (2k 2)!! 1
− logcot +cosθ − if d even,
(d 2)!! 2 (2k 1)!!sin2kθ
 − (cid:20) Xk=1 − (cid:21)

 d 3 (d−1)/2 cot2k−1θ
 − ! ,
Id(θ) = (cid:18)or 2 (cid:19) Xk=1 (2k−1)(k−1)!((d−2k−1)/2)!  if d odd,
 
 
 (d−1)/2 
 (d 3)!! (2k 3)!! 1
 − cosθ − ,
cos(θd−F2)!!1,d; 3;Xkc=o1s2θ(2k, −2)!!sin2k−1θ 
2 1
2 2 2
 (cid:18) (cid:19)
=sincods−θ2θ 2F1(cid:18)1, 3−2 d; 23;cos2θ(cid:19),

(d 2)! 1
1−d/2
− Q (cosθ).
Γ(d/2)2d/2−1 (sinθ)d/2−1 d/2−1





In the restof this section, we develop the material in order to prove this theorem.
Since a spherically symmetric choice for a fundamental solution satisfies Laplace’s equation
everywhereexceptattheorigin, wemay firstsetg = f′ in(11)andsolvethefirst-orderequation
g′+(d 1)cosθ g = 0,
−
which is integrable and clearly has the general solution
df
g(θ) = = c (sinθ)1−d, (18)
0
dθ
where c R is a constant. Now we integrate (18) to obtain a fundamental solution for the
0
∈
Laplacian on Sd
R
d(x,x′)= c (θ)+c , (19)
SR 0Id 1
where : (0,π) R is defined as
d
I →
π/2 dx
(θ) := , (20)
Id sind−1x
Zθ
and c ,c R are constants which depend on d and R. Notice that we can add any harmonic
0 1
∈
function to (19) and still have a fundamental solution of the Laplacian since a fundamental
solution of the Laplacian must satisfy
( ∆ϕ)(x′) d(x,x′)dvol′ = ϕ(x),
Sd − SR g
Z R
for all ϕ (Sd), where is the space of test functions, and dvol′ is the Riemannian (volume)
∈ S R S g
measure on Sd in the primed coordinates. Notice that our fundamental solution of Laplace’s
R
equation on the hypersphere ((19), (20)) has the property that it tends towards + as θ 0+
∞ →
Fundamental Solution of Laplace’s Equation in Hyperspherical Geometry 9
and tends towards as θ π−. Therefore our fundamental solution attains all real values.
−∞ →
As an aside, by the definition therein (see [14, 15]), Sd is a parabolic manifold. Since the
R
hypersphere Sd is bi-hemispheric, we expect that a fundamental solution of Laplace’s equation
R
on the hypersphere should vanish at θ = π/2. It is therefore convenient to set c =0 leaving us
1
with
d(x,x′)= c (θ). (21)
SR 0Id
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 [11, p. 94], [12, p. 17],
[3, p. 211], [6, p. 6]). Let d N. Define
∈
Γ(d/2)
x x′ 2−d if d= 1 or d 3,
d(x,x′)= 2πd/2(d−2)k − k ≥ (22)
G  1 log x x′ −1 if d= 2,
2π k − k

then d is a fundamental solution for ∆ in Euclidean space Rd, where ∆ is the Laplace
G −
operator in 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 Sd, being a manifold, must behave locally like Euclidean space Rd. There-
R
fore for small θ we have eθ 1+θ and e−θ 1 θ and in that limiting regime
≃ ≃ −
1 dx logθ if d = 2,
−
Id(θ) ≈Zθ xd−1 ≃ θd1−2 if d ≥ 3,
which has exactly the same singularity as a Euclidean fundamental solution. Therefore the
proportionality constant c is obtained by matching locally to a Euclidean fundamental solution
0
d = c d, (23)
SR 0Id ≃ G
in a small neighborhood of the singularity at x = x′, as the curvature vanishes, i.e., R .
→ ∞
We have shown how to compute a fundamental solution of the Laplace–Beltrami operator
on the hypersphere in terms of an improper integral (20). We would now like to express this
integral in terms of well-known special functions. A fundamental solution can be computed
d
I
using elementary methods through its definition (20). In d= 2 we have
π/2 dx 1 cosθ+1 θ
(θ)= = log = logcot ,
2
I sinx 2 cosθ 1 2
Zθ −
and in d =3 we have
π/2 dx
(θ)= = cotθ.
I3 sin2x
Zθ
In d 4,5,6,7 we have
∈ { }
1 θ cosθ
(θ)= logcot + ,
I4 2 2 2sin2θ
1
(θ)= cotθ+ cot3θ,
5
I 3
3 θ 3cosθ cosθ
(θ)= logcot + + , and
I6 8 2 8sin2θ 4sin2θ
2 1
(θ)= cotθ+ cot3θ+ cot5θ.
7
I 3 5
10 H.S. Cohl
Now we prove several equivalent finite summation expressions for (θ). We wish to compute
d
I
the antiderivative I :(0,π) R, which is defined as
m
→
dx
I (x) := ,
m sinmx
Z
where m N. This antiderivative satisfies the following recurrence relation
∈
cosx (m 2)
I (x) = + − I (x), (24)
m −(m 1)sinm−1x (m 1) m−2
− −
which follows from the identity
1 1 cosx
= + cosx,
sinmx sinm−2x sinmx
and integration by parts. The antiderivative I (x) naturally breaks into two separate classes,
m
namely
n
dx (2n 1)!! x (2k 2)!! 1
= − logcot +cosx − +C, (25)
sin2n+1x − (2n)!! " 2 (2k 1)!!sin2kx#
Z k=1 −
X
and
n
(2n 2)!! (2k 3)!! 1
− cosx − +C, or
dx −(2n−1)!! k=1 (2k−2)!!sin2k−1x
= X (26)
sin2nx  n cot2k−1x
Z  (n 1)! +C,
− − (2k 1)(k 1)!(n k)!
k=1 − − −
 X


where C is a constant. The double factorial ()!! : 1,0,1,... N is defined by
· {− } →
n (n 2) 2 if n even 2,
· − ··· ≥
n!! := n (n 2) 1 if n odd 1,

· − ··· ≥
1 if n 1,0 .
∈ {− }
Note that (2n)!! = 2nn! for n N0. Thefinite summation formulae for Im(x) all follow trivially
∈
by induction using (24) and the binomial expansion (cf. (1.2.2) in [26])
n cot2kx
(1+cos2x)n = n! .
k!(n k)!
k=0 −
X
The formulae (25) and (26) are essentially equivalent to (2.515.1–2) in [13], except (2.515.2) is
in error with the factor 28k being replaced with 2k. This is also verified in the original citing
reference [34]. By applying the limits of integration from the definition of (θ) in (20) to (25)
d
I
and (26) we obtain the following finite summation expression
d/2−1
(d 3)!! θ (2k 2)!! 1
− logcot +cosθ − if d even,
(d 2)!!  2 (2k 1)!!sin2kθ
 − Xk=1 −

 d 3 (d−1)/2 cot2k−1θ 
 − ! ,
d(θ) = 2 (2k 1)(k 1)!((d 2k 1)/2)!  (27)
I  o(cid:18)r (cid:19) Xk=1 − − − −  if d odd.
 
 
 (d−1)/2 
 (d 3)!! (2k 3)!! 1
 − cosθ − ,
 (d−2)!! Xk=1 (2k−2)!!sin2k−1θ 