Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 8 (2012), 077, 10 pages
Def inite Integrals using Orthogonality
and Integral Transforms?
Howard S. COHL
†
and Hans VOLKMER
‡
†
Applied and Computational Mathematics Division, Information Technology Laboratory,
National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA
E-mail: howard.cohl@nist.gov
URL: http://hcohl.sdf.org
‡
Department of Mathematical Sciences, University of Wisconsin-Milwaukee,
P.O. Box 413, Milwaukee, WI, 53201, USA
E-mail: volkmer@uwm.edu
Received July 31, 2012, in final form October 15, 2012; Published online October 19, 2012
http://dx.doi.org/10.3842/SIGMA.2012.077
Abstract. We obtain definite integrals for products of associated Legendre functions with
Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind
using orthogonality and integral transforms.
Key words: definite integrals; associated Legendre functions; Bessel functions; Chebyshev
polynomials of the first kind
2010 Mathematics Subject Classification: 26A42; 33C05; 33C10; 33C45; 35A08
1
Introduction
In [3] and [6] (see also [4]), we present some definite integral and infinite series addition theorems
which arise from expanding fundamental solutions of elliptic equations on Rd in axisymmetric
coordinate systems which separate Laplace’s equation. We utilize orthogonality and integral
transforms to obtain new definite integrals from some of these addition theorems.
2
Def inite integrals from integral transforms
2.1
Application of Hankel’s transform
We use the following result where for x ∈ (0, ∞) we define
F (r ± 0) := lim F (x);
x→r±
see [15, p. 456]:
Theorem 1. Let F : (0, ∞) → C be such that
Z ∞
√
x |F (x)| dx < ∞,
(1)
0
and let ν ≥ − 12 . Then
1
(F (r + 0) + F (r − 0)) =
2
Z
∞
Z
uJν (ur)
0
∞
xF (x)Jν (ux) dx du
(2)
0
provided that the positive number r lies inside an interval in which F (x) has finite variation.
?
This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”.
The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html
2
H.S. Cohl and H. Volkmer
As an illustration for the method of integral transforms, we give the following example.
According to [15, (13.22.2)] (see also [7, (6.612.3)]), we have for Re a > 0, b, c > 0, Re ν > − 12 ,
then
2
Z ∞
a + b2 + c2
1
,
(3)
e−ka Jν (kb)Jν (kc)dk = √ Qν−1/2
2bc
π bc
0
where Jν : C \ (−∞, 0] → C, for order ν ∈ C is the Bessel function of the first kind defined
in [12, (10.2.2)] and Qµν : C \ (−∞, 1] → C for ν + µ ∈
/ −N with degree ν and order µ, is the
associated Legendre function of the second kind defined in [12, (14.3.7), § 14.21]. The Legendre
function of the second kind Qν : C\(−∞, 1] → C for ν ∈
/ −N is defined in terms of the zero-order
associated Legendre function of the second kind Qν (z) := Q0ν (z). If we apply Theorem 1 to the
function F : (0, ∞) → C defined by
√
π c −ka
F (k) :=
e Jν (kc),
(4)
k
then condition (1) is satisfied. If we use (4) in (2) then we obtain the following result. If
Re a > 0, c > 0, Re ν > − 12 , then
Z
∞
Jν (kb) Qν−1/2
0
√
a2 + b2 + c2 √
π c −ka
e Jν (kc),
b db =
2bc
k
which is actually given in [13, (2.18.8.11)].
Hardy [8, (33.16)] derives an interesting extension of (3) (see also [15, p. 389] and [2, p. 17]).
We apply the Whipple formula [12, (14.9.17), § 14.21] to Hardy’s extension to obtain
Z ∞
ke−ka Jν (kb)Jν (kc)dk
0
−2a
= √
π bc (a2 + (b + c)2 )1/2 (a2 + (b −
Q1
1/2 ν−1/2
2
c) )
a2 + b2 + c2
2bc
,
(5)
for Re a > 0, b, c > 0, Re ν > −1. It is mentioned in [15, p. 389] that (5) can be derived from (3)
by differentiation with respect to a. Using this integral and Theorem 1, we prove the following
theorem.
Theorem 2. Let Re a > 0, c > 0, Re ν > −1. Then
2
√
Z ∞
Jν (kb)
a + b2 + c2 √
π c −ka
1
Qν−1/2
b db = −
e Jν (kc).
2bc
2a
0
(a2 + (b + c)2 )1/2 (a2 + (b − c)2 )1/2
Proof . By applying Theorem 1 to the function F : (0, ∞) → C defined by
√
π c −ka
e Jν (kc),
F (k) := −
2a
using (5), we obtain the desired result.
Now, we give another example of how an integral expansion for a fundamental solution of
Laplace’s equation on R3 in parabolic coordinates can be used to prove a new definite integral.
Theorem 3. Let m ∈ N0 , λ0 ∈ (0, ∞), µ, µ0 ∈ (0, ∞), µ 6= µ0 , k ∈ (0, ∞). Then
Z ∞
p
√
Qm−1/2 (χ)Jm (kλ) λ dλ = 2π λ0 µµ0 Jm (kλ0 )Im (kµ< )Km (kµ> ),
0
Definite Integrals using Orthogonality and Integral Transforms
3
where
χ=
and µ≶ :=
4λ2 µ2 + 4λ0 2 µ0 2 + (λ2 − λ0 2 + µ0 2 − µ2 )2
>1
8λλ0 µµ0
min
0
max {µ, µ }.
Proof . We apply Theorem 1 to the function F : (0, ∞) → C defined by
p
F (k) := 2π λ0 µµ0 Jm (kλ0 )Im (kµ< )Km (kµ> ),
where Iν : C \ (−∞, 0] → C, for ν ∈ C, is the modified Bessel function of the first kind defined
in [12, (10.25.2)] and Kν : C \ (−∞, 0] → C, for ν ∈ C, is the modified Bessel function of the
second kind defined in [12, (10.27.4)]. Again, we see that condition (1) is satisfied. We obtain
the desired result from [6, (41)], namely, for λ ∈ (0, ∞),
Z ∞
Qm−1/2 (χ)
Jm (kλ)Jm (kλ0 )Im (kµ< )Km (kµ> )kdk = √ 0 0 .
2π λλ µµ
0
2.2
Application of Fourier cosine transform
Theorem 4. Let a, b ∈ (0, ∞) with b ≤ a, k ∈ (0, ∞), Re ν > − 12 . Then
2
Z ∞
√
a + b2 + z 2
Qν−1/2
cos(kz) dz = π ab Iν (kb)Kν (ka).
2ab
0
(6)
Proof . According to [7, (6.672.4)] we have the integral relation
2
Z ∞
1
a + b2 + z 2
√
Qν−1/2
Iν (kb)Kν (ka) cos(kz)dk =
,
2ab
2 ab
0
where a, b ∈ (0, ∞) with b < a, z > 0, Re ν > − 21 . We obtain the desired result from Theorem 1
with F : (0, ∞) → C defined such that
r
ab
F (k) := π
Iν (kb)Kν (ka)
k
√
√
and ν = − 12 . Furthermore, if one makes the replacement z 7→ z/( 2a), k 7→ 2ka in [7,
(7.162.6)], namely
Z ∞
k
k
π
2
Kν √
,
Qν−1/2 (1 + z ) cos(kz)dz = √ Iν √
2
2
2
0
where k ∈ (0, ∞) and Re ν > − 21 , then we see that (6) holds also for any a, b ∈ (0, ∞) with
a = b.
3
3.1
Def inite integrals from orthogonality relations
Degree orthogonality for associated Legendre functions
with integer degree and order
We take advantage of the degree orthogonality relation for the Ferrers function of the first kind
with integer degree and order, namely (cf. [7, (7.112.1)])
Z π
2 (n + m)!
m
Pm
δn,n0 ,
(7)
n (cos θ)Pn0 (cos θ) sin θdθ =
2n + 1 (n − m)!
0
4
H.S. Cohl and H. Volkmer
where m, n, n0 ∈ N0 , and m ≤ n, m ≤ n0 . We are using the associated Legendre function of the
first kind (on-the-cut), Pµν : (−1, 1) → C, for ν, µ ∈ C, the Ferrers function of the first kind,
which is defined in [12, (14.3.1)].
The following estimates for the Ferrers function of the first kind with integer degree and
order will be useful. If θ ∈ [0, π] and m, n ∈ N0 then [14, § 5.3, (19)]
|Pm
n (cos θ)| ≤
(m + n)!
n!
(8)
and if θ ∈ (0, π) then [10, p. 203]
(n + m)!
(πn)−1/2 (csc θ)m+1/2 .
(9)
n!
If µ ∈ C, ξ > 0 are fixed and 0 ≤ ν → +∞, we also have the following asymptotic formulas for
the associated Legendre functions
|Pm
n (cos θ)| < 2
Pνµ (cosh ξ) = (2π sinh ξ)−1/2
Γ(ν + µ + 1) (ν+ 1 )ξ
−1
2
1
+
O
ν
,
e
Γ(ν + 32 )
1/2
Γ(ν + µ + 1) −(ν+ 1 )ξ+iπµ
2
e
1 + O ν −1 ,
3
Γ(ν + 2 )
Γ(ν + µ + 1) (ν+ 1 )ξ+iπν/2
e 2
1 + O ν −1 ,
Pνµ (i sinh ξ) = (2π cosh ξ)−1/2
3
Γ(ν + 2 )
1/2
π
Γ(ν + µ + 1) −(ν+ 1 )ξ−iπ(ν+1)/2+iπµ
2
1 + O ν −1 .
Qµν (i sinh ξ) =
e
3
2 cosh ξ
Γ(ν + 2 )
Qµν (cosh ξ) =
π
2 sinh ξ
(10)
(11)
(12)
(13)
These asymptotic formulae follow from representations of Legendre functions by Gauss hypergeometric functions; see [1, (8.1.1), (8.10.4–5)].
Theorem 5. Let n, m ∈ N0 , with n ≥ m, ν ∈ C \ {2m, 2m + 2, 2m + 4, . . .}, r, r0 ∈ (0, ∞),
r 6= r0 , θ0 ∈ (0, π). Then
Z π
(ν+1)/4 −(ν+1)/2
(ν+2)/2
χ2 − 1
Qm−1/2 (χ)Pm
dθ
n (cos θ)(sin θ)
0
!
2
√
2 (ν+2)/2
r> − r<
r2 + r0 2
i π
−(ν+2)/2
0
(14)
= (ν+1)/2
Qn
Pm
n (cos θ ),
0
0
ν/2
0
rr
2rr
2
(sin θ )
where
χ=
and r≶ :=
r2 + r0 2 − 2rr0 cos θ cos θ0
2rr0 sin θ sin θ0
(15)
min
0
max {r, r }.
Proof . We start with the following addition theorem for the associated Legendre function of
the second kind (see [3]), namely for θ ∈ (0, π),
2
√
2 (ν+2)/2
(ν+1)/4
i π
2
ν/2 −(ν+1)/2
0 −ν/2 r> − r<
χ −1
(sin θ) Qm−1/2 (χ) = (ν+3)/2 (sin θ )
rr0
2
!
∞
X
(n − m)! −(ν+2)/2 r2 + r0 2
m
0
×
(2n + 1)
Qn
Pm
(16)
n (cos θ)Pn (cos θ ),
0
(n
+
m)!
2rr
n=m
where χ > 1 is given by (15). By (8) and (11) the infinite series is uniformly convergent for
0
θ ∈ [0, π]. Therefore, if we multiply both sides of (16) by sin θ Pm
n0 (cos θ), where n ∈ N0 and
integrate over θ ∈ (0, π) we obtain (14).
Definite Integrals using Orthogonality and Integral Transforms
5
Corollary 1. Let n, m ∈ N0 with n ≥ m, r, r0 ∈ (0, ∞), r 6= r0 , θ0 ∈ (0, π). Then
√
n+1/2
Z π
√
2π sin θ0 m
r<
m
0
Qm−1/2 (χ)Pn (cos θ) sin θdθ =
,
Pn (cos θ )
2n + 1
r>
0
where χ > 1 is given by (15).
Proof . Substitute ν = −1 in (14) and use [12, (14.5.17)].
Theorem 6. Let m, n ∈ N0 with n ≥ m, σ, σ 0 ∈ (0, ∞), θ0 ∈ (0, π). Then
Z π
√
m (n − m)!
Qm−1/2 (χ)Pm
n (cos θ) sin θ dθ = 2π(−1)
(n + m)!
0
√
m
0
m
× sinh σ sinh σ 0 sin θ0 Pn (cos θ )Pn (cosh σ< )Qm
n (cosh σ> ),
(17)
where
χ=
and σ≶ :=
cosh2 σ + cosh2 σ 0 − sin2 θ − sin2 θ0 − 2 cosh σ cosh σ 0 cos θ cos θ0
2 sinh σ sinh σ 0 sin θ sin θ0
(18)
min
0
max {σ, σ }.
Proof . We start with the following addition theorem for the associated Legendre function of
the second kind (see [6, (37)]), namely
∞
X
√
(n − m)! 2
m
0
0
Qm−1/2 (χ) = π(−1)
sinh σ sinh σ sin θ sin θ
(2n + 1)
(n + m)!
n=m
m
0
m
m
× Pm
n (cos θ)Pn (cos θ )Pn (cosh σ< )Qn (cosh σ> ),
(19)
where χ ≥ 1 is defined by (18). Note that χ = 1 only if σ = σ 0 and θ = θ0 , and in that case
Qm−1/2 (χ) has a logarithmic singularity. If σ 6= σ 0 then (8), (10), (11) show that the series
in
uniformly convergent for θ ∈ [0, π]. Therefore, if we multiply both sides of (19) by
√ (19) is
m
sin θ Pn0 (cos θ) and integrate over θ ∈ [0, π], then by (7) we have obtained (17). If σ = σ 0 then
one may use (9), (10), (11) and the orthogonality relation (7) to show that the series in (19) (as
a series of functions in the variable θ ∈ (0, π)) converges in L2 (0, π). That is
2
∞ X
(n − m)!
0
m
m
φn (θ )Pn (cosh σ< )Qn (cosh σ> ) < ∞,
(20)
(n + m)!
n=m
where φn ∈ L2 (0, π) forms an orthonormal basis and is defined as
s
√
2n + 1 (n − m)! m
P (cos θ),
φn (θ) := sin θ
2 (n + m)! n
for n = m, m + 1, . . . . Then by the asymptotics of Pnm and Qm
n (cf. (10), (11))
2m−1
Pnm (cosh σ< )Qm
as n → ∞.
n (cosh σ> ) = O n
0
Also from the estimate of Pm
n (9), φn (θ ) = O (1) . Therefore,
(n − m)!
−1
φn (θ0 )Pnm (cosh σ< )Qm
,
n (cosh σ> ) = O n
(n + m)!
and this implies (20) because
∞
P
n=1
1
n2
< ∞. Therefore, we again obtain (17).
6
H.S. Cohl and H. Volkmer
Theorem 7. Let m, n ∈ N0 with 0 ≤ m ≤ n, σ, σ 0 ∈ (0, ∞), θ0 ∈ [0, π]. Then
Z π
√
m (n − m)!
Qm−1/2 (χ)Pm
n (cos θ) sin θdθ = 2πi(−1)
(n + m)!
0
√
m
0
m
× cosh σ cosh σ 0 sin θ0 Pn (cos θ )Pn (i sinh σ< )Qm
n (i sinh σ> ),
(21)
where
χ=
sinh2 σ + sinh2 σ 0 + sin2 θ + sin2 θ0 − 2 sinh σ sinh σ 0 cos θ cos θ0
.
2 cosh σ cosh σ 0 sin θ sin θ0
(22)
Proof . We start with oblate spheroidal coordinates on R3 , namely
x = a cosh σ sin θ cos φ,
y = a cosh σ sin θ sin φ,
z = a sinh σ cos θ,
where a > 0, σ ∈ [0, ∞), θ ∈ [0, π], φ ∈ [0, 2π). The reciprocal distance between two points
x, x0 ∈ R3 expanded in terms of the separable harmonics in this coordinate system is given in
[9, (41), p. 218], namely
∞
n
X
iX
(n − m)! 2
1
m
=
(2n + 1)
(−1) m
cos(m(φ − φ0 ))
kx − x0 k
a
(n + m)!
n=0
m=0
m
m
× Pn (cos θ)Pn (cos θ0 )Pnm (i sinh σ< )Qm
n (i sinh σ> ),
where m := 2 − δm,0 is the Neumann factor [11, p. 744] commonly occurring in Fourier cosine
series, with σ 0 ∈ [0, ∞), θ0 ∈ [0, π], φ0 ∈ [0, 2π). Note that the corresponding expression given
in [6, § 5.2] is given incorrectly (see [4]). By reversing the order of summations in the above
expression and comparing with the Fourier cosine expansion for the reciprocal distance between
two points, namely
∞
X
1
1
√
=
m cos(m(φ − φ0 ))Qm−1/2 (χ),
kx − x0 k
πa cosh σ cosh σ 0 sin θ sin θ0 m=0
where χ > 1 is given by (22), we obtain the following addition theorem for the associated
Legendre function of the second kind
m
Qm−1/2 (χ) = iπ(−1)
√
cosh σ cosh σ 0 sin θ sin θ0
∞
X
(n − m)!
(2n + 1)
(n + m)!
n=m
2
m
0
m
m
× Pm
n (cos θ)Pn (cos θ )Pn (i sinh σ< )Qn (i sinh σ> ).
(23)
√
If we multiply both sides of (23) by sin θ Pm
n0 (cos θ) and integrate over θ ∈ [0, π], then by (7) we
have obtained (21). We justify the interchange of integral and infinite sum as before by using
the asymptotic formulas (12), (13).
Theorem 8. Let m, n ∈ N0 with 0 ≤ m ≤ n, σ, σ 0 ∈ (0, ∞), θ0 ∈ (0, π). Then
√
Z π
√
2π sin θ0 m
m
Qm−1/2 (χ)Pn (cos θ) sin θdθ =
P (cos θ0 )e−(n+1/2)(σ> −σ< ) ,
2n + 1 n
0
where if we define s = cosh σ, s0 = cosh σ 0 , τ = cos θ, τ 0 = cos θ0 , then
2
sin2 θ(s0 − τ 0 )2 + sin2 θ0 (s − τ )2 + (s0 − τ 0 ) sinh σ − (s − τ ) sinh σ 0
χ=
.
2 sin θ sin θ0 (s − τ )(s0 − τ 0 )
(24)
(25)
Definite Integrals using Orthogonality and Integral Transforms
7
Proof . We start with bispherical coordinates on R3 , namely
x=
a sin θ cos φ
,
cosh σ − cos θ
y=
a sin θ sin φ
,
cosh σ − cos θ
z=
a sinh σ
,
cosh σ − cos θ
where a > 0, σ ∈ [0, ∞), θ ∈ [0, π], φ ∈ [0, 2π). The reciprocal distance between two points
x, x0 ∈ R3 expanded in terms of the separable harmonics in this coordinate system is given in
[9, (9), p. 222], namely
∞
X
1
1p
=
(cosh σ − cos θ)(cosh σ 0 − cos θ0 )
e−(n+1/2)(σ> −σ< )
0
kx − x k
a
n=0
×
n
X
m=0
m
(n − m)! m
0
0
P (cos θ)Pm
n (cos θ ) cos(m(φ − φ )),
(n + m)! n
where σ 0 ∈ [0, ∞), θ0 ∈ [0, π], φ0 ∈ [0, 2π). By reversing the order of summations in the above
expression and comparing with the Fourier cosine expansion for the reciprocal distance between
two points, namely
p
∞
(cosh σ − cos θ)(cosh σ 0 − cos θ0 ) X
1
√
=
m cos(m(φ − φ0 ))Qm−1/2 (χ),
kx − x0 k
πa sin θ sin θ0
m=0
where χ ≥ 1 is given by (25), we obtain the following addition theorem for the associated
Legendre function of the second kind
∞
X
√
(n − m)! −(n+1/2)(σ> −σ< ) m
0
Qm−1/2 (χ) = π sin θ sin θ0
e
Pn (cos θ)Pm
n (cos θ ).
(n
+
m)!
n=m
(26)
If σ = σ 0 and θ = θ0 , then χ = 1, and Qm−1/2 (χ) has a logarithmic singularity. Note that the
corresponding expression
given in [6, § 6.1, (45)] is given incorrectly (see [4]). If we multiply both
√
m
sides of (26) by sin θ Pn0 (cos θ) and integrate over θ ∈ [0, π], then by (7) we have obtained (24).
We justify the interchange of integral and infinite sum in the same way as in the proof of
Theorem 6.
3.2
Order orthogonality for associated Legendre functions
with integer degree and order
In this subsection we take advantage of the order orthogonality relation for the Ferrers function
of the first kind with integer degree and order (cf. [12, (14.17.8)])
Z π
1
1 (n + m)!
m0
Pm
dθ =
δm,m0 ,
(27)
n (cos θ)Pn (cos θ)
sin θ
m (n − m)!
0
with m ≥ 1.
Theorem 9. Let m ∈ N, n ∈ N0 with 1 ≤ m ≤ n, θ0 ∈ [0, π], φ, φ0 ∈ [0, 2π). Then
Z π
1
2
Pn (cos γ)Pm
dθ = Pm
(cos θ0 ) cos(m(φ − φ0 )),
n (cos θ)
sin θ
m n
0
where
cos γ = cos θ cos θ0 + sin θ sin θ0 cos(φ − φ0 ).
8
H.S. Cohl and H. Volkmer
Proof . We start with the addition theorem for spherical harmonics (cf. [12, (14.18.1)]), namely
n
X
(n − m)! m
0 im(φ−φ0 )
Pn (cos γ) =
Pn (cos θ)Pm
,
n (cos θ )e
(n + m)!
m=−n
(28)
where Pn : C → C, for n ∈ N0 , is the Legendre polynomial which can be defined in terms of the
terminating Gauss hypergeometric series (see for instance [12, Chapters 15, 18]) as follows
−n, n + 1 1 − z
;
.
Pn (z) := 2 F1
1
2
We then take advantage of the order orthogonality relation for the Ferrers functions of the first
0
kind with integer degree and order. If we multiply both sides of (28) by (sin θ)−1 Pm
n (cos θ) and
integrate over θ ∈ (0, π), by using (27) we obtain the desired result.
Theorem 9, originating from (28), is the only example of a definite integral that we could find
using the order orthogonality relation for the Ferrers functions of the first kind (27). Therefore
we highly suspect that this result is previously known, and include it mainly for completeness
sake. It would however be very interesting to find another example using this orthogonality
relation.
3.3
Orthogonality for Chebyshev polynomials of the f irst kind
Here we take advantage of orthogonality from Chebyshev polynomials of the first kind (cf. [12,
§ 18.3])
Z π
π
Tm (cos θ)Tn (cos θ)dθ = δm,n ,
(29)
n
0
where Tn : C → C, for n ∈ N0 , is the Chebyshev polynomial of the first kind which can be
defined in terms of the terminating Gauss hypergeometric series (see [12, Chapter 18])
!
−n, n 1 − z
Tn (z) = 2 F1
;
.
1
2
2
The Chebyshev polynomials of the first kind satisfy the identity [12, (18.5.1)]
Tn (cos θ) = cos(nθ).
Theorem 10. Let m, n ∈ Z, σ, σ 0 ∈ (0, ∞). Then
Z π
√
Qm−1/2 (χ) cos(nψ)dψ = π(−1)m sinh σ sinh σ 0
0
Γ n − m + 21 m
P
×
(cosh σ< )Qm
n−1/2 (cosh σ> ),
Γ n + m + 12 n−1/2
(30)
where
χ = coth σ coth σ 0 − csch σ csch σ 0 cos ψ.
(31)
Proof . We start with toroidal coordinates on R3 , namely
x=
a sinh σ cos φ
,
cosh σ − cos ψ
y=
a sinh σ sin φ
,
cosh σ − cos ψ
z=
a sin ψ
,
cosh σ − cos ψ
Definite Integrals using Orthogonality and Integral Transforms
9
where a > 0, σ ∈ (0, ∞), ψ, φ ∈ [0, 2π). The reciprocal distance between two points x, x0 ∈ R3
is given algebraically by
1
1
=
0
kx − x k
a
r
×
(cosh σ − cos ψ)(cosh σ 0 − cos ψ 0 )
2 sinh σ sinh σ 0
cosh σ cosh σ 0 − cos(ψ − ψ 0 )
− cos(φ − φ0 )
sinh σ sinh σ 0
−1/2
,
where (σ 0 , ψ 0 , φ0 ) are the toroidal coordinates corresponding to the point x0 . Using Heine’s
reciprocal square root identity (see for instance [5, (3.11)])
√ ∞
1
2X
√
=
m Qm−1/2 (z)Tm (x),
π
z−x
m=0
where z > 1 and x ∈ [−1, 1], we can obtain a Fourier cosine series representation for the
reciprocal distance between two points in toroidal coordinates on R3 , namely
1
1
=
0
kx − x k
πa
r
∞
(cosh σ − cos ψ)(cosh σ 0 − cos ψ 0 ) X
m cos(m(φ − φ0 ))Qm−1/2 (χ),
sinh σ sinh σ 0
m=0
where χ > 1 is given by (31). We can further expand the associated Legendre function of the
second kind using the following addition theorem (cf. [7, (8.795.2)])
∞
X
√
n cos(n(ψ − ψ 0 ))
Qm−1/2 (χ) = (−1)m sinh σ sinh σ 0
×
Γ n−m+
Γ n+m+
n=0
1
m
2 m
1 Pn−1/2 (cosh σ< )Qn−1/2 (cosh σ> ).
2
(32)
Note that with the above addition theorem, we have the expansion of the reciprocal distance
between two points in terms of the separable harmonics in toroidal coordinates
∞
X
1
1 p
=
(cosh σ − cos ψ)(cosh σ 0 − cos ψ 0 )
(−1)m m cos(m(φ − φ0 ))
0
kx − x k
πa
m=0
∞
1
X
Γ
n
−
m
+
m
2 m
×
n cos(n(ψ − ψ 0 ))
1 Pn−1/2 (cosh σ< )Qn−1/2 (cosh σ> )
Γ
n
+
m
+
2
n=0
(see also [6, § 6.2] and [4]). If we relabel ψ − ψ 0 7→ ψ and multiply both sides of (32) by cos(nψ)
and integrate over ψ ∈ [0, π], then by (29) we have obtained (30). The interchange of infinite
sum and integral is justified by (10), (11).
Acknowledgements
This work was 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. The authors would also like to acknowledge two
anonymous referees whose comments helped improve this paper.
10
H.S. Cohl and H. Volkmer
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