Definite Integrals using Orthogonality and Integral Transforms
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.
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nasplib_isofts_kiev_ua-123456789-1487252025-02-09T16:03:19Z Definite Integrals using Orthogonality and Integral Transforms Cohl, H.S. Volkmer, H. 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. 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. 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. 2012 Article Definite Integrals using Orthogonality and Integral Transforms / H.S. Cohl, H. Volkmer // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 26A42; 33C05; 33C10; 33C45; 35A08 DOI: http://dx.doi.org/10.3842/SIGMA.2012.077 https://nasplib.isofts.kiev.ua/handle/123456789/148725 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
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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. |
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Definite Integrals using Orthogonality and Integral Transforms |
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Definite Integrals using Orthogonality and Integral Transforms |
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Definite Integrals using Orthogonality and Integral Transforms |
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Definite Integrals using Orthogonality and Integral Transforms |
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definite integrals using orthogonality and integral transforms |
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Definite Integrals using Orthogonality and Integral Transforms / H.S. Cohl, H. Volkmer // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 15 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT cohlhs definiteintegralsusingorthogonalityandintegraltransforms AT volkmerh definiteintegralsusingorthogonalityandintegraltransforms |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 077, 10 pages
Definite 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 Definite 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
x→r±
F (x);
see [15, p. 456]:
Theorem 1. Let F : (0,∞)→ C be such that∫ ∞
0
√
x |F (x)| dx <∞, (1)
and let ν ≥ −1
2 . Then
1
2
(F (r + 0) + F (r − 0)) =
∫ ∞
0
uJν(ur)
∫ ∞
0
xF (x)Jν(ux) dx du (2)
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
mailto:howard.cohl@nist.gov
http://hcohl.sdf.org
mailto:volkmer@uwm.edu
http://dx.doi.org/10.3842/SIGMA.2012.077
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 > −1
2 ,
then ∫ ∞
0
e−kaJν(kb)Jν(kc)dk =
1
π
√
bc
Qν−1/2
(
a2 + b2 + c2
2bc
)
, (3)
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) := Q0
ν(z). If we apply Theorem 1 to the
function F : (0,∞)→ C defined by
F (k) :=
π
√
c
k
e−kaJν(kc), (4)
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 > −1
2 , then∫ ∞
0
Jν(kb)Qν−1/2
(
a2 + b2 + c2
2bc
)√
b db =
π
√
c
k
e−kaJν(kc),
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∫ ∞
0
ke−kaJν(kb)Jν(kc)dk
=
−2a
π
√
bc (a2 + (b+ c)2)1/2 (a2 + (b− c)2)1/2
Q1
ν−1/2
(
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∫ ∞
0
Jν(kb)
(a2 + (b+ c)2)1/2 (a2 + (b− c)2)1/2
Q1
ν−1/2
(
a2 + b2 + c2
2bc
)√
b db = −π
√
c
2a
e−kaJν(kc).
Proof. By applying Theorem 1 to the function F : (0,∞)→ C defined by
F (k) := −π
√
c
2a
e−kaJν(kc),
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, λ′ ∈ (0,∞), µ, µ′ ∈ (0,∞), µ 6= µ′, k ∈ (0,∞). Then∫ ∞
0
Qm−1/2(χ)Jm(kλ)
√
λ dλ = 2π
√
λ′µµ′Jm(kλ′)Im(kµ<)Km(kµ>),
Definite Integrals using Orthogonality and Integral Transforms 3
where
χ =
4λ2µ2 + 4λ′2µ′2 + (λ2 − λ′2 + µ′2 − µ2)2
8λλ′µµ′
> 1
and µ≶ := min
max{µ, µ
′}.
Proof. We apply Theorem 1 to the function F : (0,∞)→ C defined by
F (k) := 2π
√
λ′µµ′Jm(kλ′)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,∞),∫ ∞
0
Jm(kλ)Jm(kλ′)Im(kµ<)Km(kµ>)kdk =
Qm−1/2(χ)
2π
√
λλ′µµ′
. �
2.2 Application of Fourier cosine transform
Theorem 4. Let a, b ∈ (0,∞) with b ≤ a, k ∈ (0,∞), Re ν > −1
2 . Then∫ ∞
0
Qν−1/2
(
a2 + b2 + z2
2ab
)
cos(kz) dz = π
√
ab Iν(kb)Kν(ka). (6)
Proof. According to [7, (6.672.4)] we have the integral relation∫ ∞
0
Iν(kb)Kν(ka) cos(kz)dk =
1
2
√
ab
Qν−1/2
(
a2 + b2 + z2
2ab
)
,
where a, b ∈ (0,∞) with b < a, z > 0, Re ν > −1
2 . We obtain the desired result from Theorem 1
with F : (0,∞)→ C defined such that
F (k) := π
√
ab
k
Iν(kb)Kν(ka)
and ν = −1
2 . Furthermore, if one makes the replacement z 7→ z/(
√
2a), k 7→
√
2ka in [7,
(7.162.6)], namely∫ ∞
0
Qν−1/2(1 + z2) cos(kz)dz =
π√
2
Iν
(
k√
2
)
Kν
(
k√
2
)
,
where k ∈ (0,∞) and Re ν > −1
2 , then we see that (6) holds also for any a, b ∈ (0,∞) with
a = b. �
3 Definite integrals from orthogonality relations
3.1 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)])∫ π
0
Pmn (cos θ)Pmn′(cos θ) sin θdθ =
2
2n+ 1
(n+m)!
(n−m)!
δn,n′ , (7)
4 H.S. Cohl and H. Volkmer
where m,n, n′ ∈ N0, and m ≤ n, m ≤ n′. 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)]
|Pmn (cos θ)| ≤ (m+ n)!
n!
(8)
and if θ ∈ (0, π) then [10, p. 203]
|Pmn (cos θ)| < 2
(n+m)!
n!
(πn)−1/2 (csc θ)m+1/2. (9)
If µ ∈ C, ξ > 0 are fixed and 0 ≤ ν → +∞, we also have the following asymptotic formulas for
the associated Legendre functions
Pµν (cosh ξ) = (2π sinh ξ)−1/2
Γ(ν + µ+ 1)
Γ(ν + 3
2)
e(ν+
1
2
)ξ
(
1 +O
(
ν−1
))
, (10)
Qµν (cosh ξ) =
(
π
2 sinh ξ
)1/2 Γ(ν + µ+ 1)
Γ(ν + 3
2)
e−(ν+
1
2
)ξ+iπµ
(
1 +O
(
ν−1
))
, (11)
Pµν (i sinh ξ) = (2π cosh ξ)−1/2
Γ(ν + µ+ 1)
Γ(ν + 3
2)
e(ν+
1
2
)ξ+iπν/2
(
1 +O
(
ν−1
))
, (12)
Qµν (i sinh ξ) =
(
π
2 cosh ξ
)1/2 Γ(ν + µ+ 1)
Γ(ν + 3
2)
e−(ν+
1
2
)ξ−iπ(ν+1)/2+iπµ
(
1 +O
(
ν−1
))
. (13)
These asymptotic formulae follow from representations of Legendre functions by Gauss hyper-
geometric 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, r′ ∈ (0,∞),
r 6= r′, θ′ ∈ (0, π). Then∫ π
0
(
χ2 − 1
)(ν+1)/4
Q
−(ν+1)/2
m−1/2 (χ)Pmn (cos θ)(sin θ)(ν+2)/2dθ
=
i
√
π
2(ν+1)/2(sin θ′)ν/2
(
r2> − r2<
rr′
)(ν+2)/2
Q−(ν+2)/2
n
(
r2 + r′2
2rr′
)
Pmn (cos θ′), (14)
where
χ =
r2 + r′2 − 2rr′ cos θ cos θ′
2rr′ sin θ sin θ′
(15)
and r≶ := min
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),
(
χ2 − 1
)(ν+1)/4
(sin θ)ν/2Q
−(ν+1)/2
m−1/2 (χ) =
i
√
π
2(ν+3)/2
(sin θ′)−ν/2
(
r2> − r2<
rr′
)(ν+2)/2
×
∞∑
n=m
(2n+ 1)
(n−m)!
(n+m)!
Q−(ν+2)/2
n
(
r2 + r′2
2rr′
)
Pmn (cos θ)Pmn (cos θ′), (16)
where χ > 1 is given by (15). By (8) and (11) the infinite series is uniformly convergent for
θ ∈ [0, π]. Therefore, if we multiply both sides of (16) by sin θPmn′(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, r′ ∈ (0,∞), r 6= r′, θ′ ∈ (0, π). Then∫ π
0
Qm−1/2(χ)Pmn (cos θ)
√
sin θdθ =
2π
√
sin θ′
2n+ 1
Pmn (cos θ′)
(
r<
r>
)n+1/2
,
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,∞), θ′ ∈ (0, π). Then∫ π
0
Qm−1/2(χ)Pmn (cos θ)
√
sin θ dθ = 2π(−1)m
(n−m)!
(n+m)!
×
√
sinhσ sinhσ′ sin θ′ Pmn (cos θ′)Pmn (coshσ<)Qmn (coshσ>), (17)
where
χ =
cosh2 σ + cosh2 σ′ − sin2 θ − sin2 θ′ − 2 coshσ coshσ′ cos θ cos θ′
2 sinhσ sinhσ′ sin θ sin θ′
(18)
and σ≶ := min
max{σ, σ
′}.
Proof. We start with the following addition theorem for the associated Legendre function of
the second kind (see [6, (37)]), namely
Qm−1/2(χ) = π(−1)m
√
sinhσ sinhσ′ sin θ sin θ′
∞∑
n=m
(2n+ 1)
[
(n−m)!
(n+m)!
]2
× Pmn (cos θ)Pmn (cos θ′)Pmn (coshσ<)Qmn (coshσ>), (19)
where χ ≥ 1 is defined by (18). Note that χ = 1 only if σ = σ′ and θ = θ′, and in that case
Qm−1/2(χ) has a logarithmic singularity. If σ 6= σ′ then (8), (10), (11) show that the series
in (19) is uniformly convergent for θ ∈ [0, π]. Therefore, if we multiply both sides of (19) by√
sin θPmn′(cos θ) and integrate over θ ∈ [0, π], then by (7) we have obtained (17). If σ = σ′ 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
∞∑
n=m
{
(n−m)!
(n+m)!
φn(θ′)Pmn (coshσ<)Qmn (coshσ>)
}2
<∞, (20)
where φn ∈ L2(0, π) forms an orthonormal basis and is defined as
φn(θ) :=
√
sin θ
√
2n+ 1
2
(n−m)!
(n+m)!
Pmn (cos θ),
for n = m,m+ 1, . . . . Then by the asymptotics of Pmn and Qmn (cf. (10), (11))
Pmn (coshσ<)Qmn (coshσ>) = O
(
n2m−1
)
as n→∞.
Also from the estimate of Pmn (9), φn(θ′) = O (1) . Therefore,
(n−m)!
(n+m)!
φn(θ′)Pmn (coshσ<)Qmn (coshσ>) = O
(
n−1
)
,
and this implies (20) because
∞∑
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,∞), θ′ ∈ [0, π]. Then∫ π
0
Qm−1/2(χ)Pmn (cos θ)
√
sin θdθ = 2πi(−1)m
(n−m)!
(n+m)!
×
√
coshσ coshσ′ sin θ′Pmn (cos θ′)Pmn (i sinhσ<)Qmn (i sinhσ>), (21)
where
χ =
sinh2 σ + sinh2 σ′ + sin2 θ + sin2 θ′ − 2 sinhσ sinhσ′ cos θ cos θ′
2 coshσ coshσ′ sin θ sin θ′
. (22)
Proof. We start with oblate spheroidal coordinates on R3, namely
x = a coshσ sin θ cosφ, y = a coshσ sin θ sinφ, z = a sinhσ cos θ,
where a > 0, σ ∈ [0,∞), θ ∈ [0, π], φ ∈ [0, 2π). The reciprocal distance between two points
x,x′ ∈ R3 expanded in terms of the separable harmonics in this coordinate system is given in
[9, (41), p. 218], namely
1
‖x− x′‖
=
i
a
∞∑
n=0
(2n+ 1)
n∑
m=0
(−1)mεm
[
(n−m)!
(n+m)!
]2
cos(m(φ− φ′))
× Pmn (cos θ)Pmn (cos θ′)Pmn (i sinhσ<)Qmn (i sinhσ>),
where εm := 2 − δm,0 is the Neumann factor [11, p. 744] commonly occurring in Fourier cosine
series, with σ′ ∈ [0,∞), θ′ ∈ [0, π], φ′ ∈ [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
1
‖x− x′‖
=
1
πa
√
coshσ coshσ′ sin θ sin θ′
∞∑
m=0
εm cos(m(φ− φ′))Qm−1/2(χ),
where χ > 1 is given by (22), we obtain the following addition theorem for the associated
Legendre function of the second kind
Qm−1/2(χ) = iπ(−1)m
√
coshσ coshσ′ sin θ sin θ′
∞∑
n=m
(2n+ 1)
[
(n−m)!
(n+m)!
]2
× Pmn (cos θ)Pmn (cos θ′)Pmn (i sinhσ<)Qmn (i sinhσ>). (23)
If we multiply both sides of (23) by
√
sin θPmn′(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,∞), θ′ ∈ (0, π). Then∫ π
0
Qm−1/2(χ)Pmn (cos θ)
√
sin θdθ =
2π
√
sin θ′
2n+ 1
Pmn (cos θ′)e−(n+1/2)(σ>−σ<), (24)
where if we define s = coshσ, s′ = coshσ′, τ = cos θ, τ ′ = cos θ′, then
χ =
sin2 θ(s′ − τ ′)2 + sin2 θ′(s− τ)2 +
[
(s′ − τ ′) sinhσ − (s− τ) sinhσ′
]2
2 sin θ sin θ′(s− τ)(s′ − τ ′)
. (25)
Definite Integrals using Orthogonality and Integral Transforms 7
Proof. We start with bispherical coordinates on R3, namely
x =
a sin θ cosφ
coshσ − cos θ
, y =
a sin θ sinφ
coshσ − cos θ
, z =
a sinhσ
coshσ − cos θ
,
where a > 0, σ ∈ [0,∞), θ ∈ [0, π], φ ∈ [0, 2π). The reciprocal distance between two points
x,x′ ∈ R3 expanded in terms of the separable harmonics in this coordinate system is given in
[9, (9), p. 222], namely
1
‖x− x′‖
=
1
a
√
(coshσ − cos θ)(coshσ′ − cos θ′)
∞∑
n=0
e−(n+1/2)(σ>−σ<)
×
n∑
m=0
εm
(n−m)!
(n+m)!
Pmn (cos θ)Pmn (cos θ′) cos(m(φ− φ′)),
where σ′ ∈ [0,∞), θ′ ∈ [0, π], φ′ ∈ [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
1
‖x− x′‖
=
√
(coshσ − cos θ)(coshσ′ − cos θ′)
πa
√
sin θ sin θ′
∞∑
m=0
εm cos(m(φ− φ′))Qm−1/2(χ),
where χ ≥ 1 is given by (25), we obtain the following addition theorem for the associated
Legendre function of the second kind
Qm−1/2(χ) = π
√
sin θ sin θ′
∞∑
n=m
(n−m)!
(n+m)!
e−(n+1/2)(σ>−σ<)Pmn (cos θ)Pmn (cos θ′). (26)
If σ = σ′ and θ = θ′, 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
sides of (26) by
√
sin θPmn′(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)])∫ π
0
Pmn (cos θ)Pm
′
n (cos θ)
1
sin θ
dθ =
1
m
(n+m)!
(n−m)!
δm,m′ , (27)
with m ≥ 1.
Theorem 9. Let m ∈ N, n ∈ N0 with 1 ≤ m ≤ n, θ′ ∈ [0, π], φ, φ′ ∈ [0, 2π). Then∫ π
0
Pn(cos γ)Pmn (cos θ)
1
sin θ
dθ =
2
m
Pmn (cos θ′) cos(m(φ− φ′)),
where
cos γ = cos θ cos θ′ + sin θ sin θ′ cos(φ− φ′).
8 H.S. Cohl and H. Volkmer
Proof. We start with the addition theorem for spherical harmonics (cf. [12, (14.18.1)]), namely
Pn(cos γ) =
n∑
m=−n
(n−m)!
(n+m)!
Pmn (cos θ)Pmn (cos θ′)eim(φ−φ′), (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
Pn(z) := 2F1
(
−n, n+ 1
1
;
1− z
2
)
.
We then take advantage of the order orthogonality relation for the Ferrers functions of the first
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 first kind
Here we take advantage of orthogonality from Chebyshev polynomials of the first kind (cf. [12,
§ 18.3])∫ π
0
Tm(cos θ)Tn(cos θ)dθ =
π
εn
δm,n, (29)
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])
Tn(z) = 2F1
(
−n, n
1
2
;
1− z
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,∞). Then∫ π
0
Qm−1/2(χ) cos(nψ)dψ = π(−1)m
√
sinhσ sinhσ′
×
Γ
(
n−m+ 1
2
)
Γ
(
n+m+ 1
2
)Pmn−1/2(coshσ<)Qmn−1/2(coshσ>), (30)
where
χ = cothσ cothσ′ − cschσ cschσ′ cosψ. (31)
Proof. We start with toroidal coordinates on R3, namely
x =
a sinhσ cosφ
coshσ − cosψ
, y =
a sinhσ sinφ
coshσ − cosψ
, z =
a sinψ
coshσ − cosψ
,
Definite Integrals using Orthogonality and Integral Transforms 9
where a > 0, σ ∈ (0,∞), ψ, φ ∈ [0, 2π). The reciprocal distance between two points x,x′ ∈ R3
is given algebraically by
1
‖x− x′‖
=
1
a
√
(coshσ − cosψ)(coshσ′ − cosψ′)
2 sinhσ sinhσ′
×
[
coshσ coshσ′ − cos(ψ − ψ′)
sinhσ sinhσ′
− cos(φ− φ′)
]−1/2
,
where (σ′, ψ′, φ′) are the toroidal coordinates corresponding to the point x′. Using Heine’s
reciprocal square root identity (see for instance [5, (3.11)])
1√
z − x
=
√
2
π
∞∑
m=0
εmQm−1/2(z)Tm(x),
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
‖x− x′‖
=
1
πa
√
(coshσ − cosψ)(coshσ′ − cosψ′)
sinhσ sinhσ′
∞∑
m=0
εm cos(m(φ− φ′))Qm−1/2(χ),
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)])
Qm−1/2(χ) = (−1)m
√
sinhσ sinhσ′
∞∑
n=0
εn cos(n(ψ − ψ′))
×
Γ
(
n−m+ 1
2
)
Γ
(
n+m+ 1
2
)Pmn−1/2(coshσ<)Qmn−1/2(coshσ>). (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
1
‖x− x′‖
=
1
πa
√
(coshσ − cosψ)(coshσ′ − cosψ′)
∞∑
m=0
(−1)mεm cos(m(φ− φ′))
×
∞∑
n=0
εn cos(n(ψ − ψ′))
Γ
(
n−m+ 1
2
)
Γ
(
n+m+ 1
2
)Pmn−1/2(coshσ<)Qmn−1/2(coshσ>)
(see also [6, § 6.2] and [4]). If we relabel ψ−ψ′ 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 Postdoc-
toral 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
References
[1] Abramowitz M., Stegun I.A., Handbook of mathematical functions with formulas, graphs, and mathematical
tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, U.S. Government Printing Office,
Washington, D.C., 1964.
[2] Askey R., Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics,
Philadelphia, Pa., 1975.
[3] Cohl H.S., Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the poly-
harmonic equation and polyspherical addition theorems, arXiv:1209.6047.
[4] Cohl H.S., Erratum: Developments in determining the gravitational potential using toroidal functions,
Astronom. Nachr. 333 (2012), 784–785.
[5] Cohl H.S., Dominici D.E., Generalized Heine’s identity for complex Fourier series of binomials, Proc. R.
Soc. Lond. Ser. A 467 (2011), 333–345, arXiv:0912.0126.
[6] Cohl H.S., Tohline J.E., Rau A.R.P., Srivastava H.M., Developments in determining the gravitational po-
tential using toroidal functions, Astronom. Nachr. 321 (2000), 363–372.
[7] Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, seventh ed., Elsevier/Academic Press,
Amsterdam, 2007.
[8] Hardy G.H., Further researches in the theory of divergent series and integrals, Trans. Cambridge Philos.
Soc. 21 (1908), 1–48.
[9] MacRobert T.M., Spherical harmonics. An elementary treatise on harmonic functions with applications, 2nd
ed., Methuen & Co. Ltd., London, 1947.
[10] Magnus W., Oberhettinger F., Soni R.P., Formulas and theorems for the special functions of mathematical
physics, 3rd ed., Die Grundlehren der mathematischen Wissenschaften, Bd. 52, Springer-Verlag, New York,
1966.
[11] Morse P.M., Feshbach H., Methods of theoretical physics, Vols. 1, 2, McGraw-Hill Book Co. Inc., New York,
1953.
[12] Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions,
Cambridge University Press, Cambridge, 2010.
[13] Prudnikov A.P., Brychkov Y.A., Marichev O.I., Integrals and series. Vol. 3. More special functions, Gordon
and Breach Science Publishers, New York, 1990.
[14] Schäfke F.W., Einführung in die Theorie der speziellen Funktionen der mathematischen Physik, Die
Grundlehren der mathematischen Wissenschaften, Bd. 118, Springer-Verlag, Berlin, 1963.
[15] Watson G.N., A treatise on the theory of Bessel functions, 2nd ed., Cambridge Mathematical Library,
Cambridge University Press, Cambridge, 1944.
http://arxiv.org/abs/1209.6047
http://dx.doi.org/10.1002/asna.201211723
http://dx.doi.org/10.1098/rspa.2010.0222
http://dx.doi.org/10.1098/rspa.2010.0222
http://arxiv.org/abs/0912.0126
http://dx.doi.org/10.1002/1521-3994(200012)321:5/6<363::AID-ASNA363>3.0.CO;2-X
1 Introduction
2 Definite integrals from integral transforms
2.1 Application of Hankel's transform
2.2 Application of Fourier cosine transform
3 Definite integrals from orthogonality relations
3.1 Degree orthogonality for associated Legendre functions with integer degree and order
3.2 Order orthogonality for associated Legendre functions with integer degree and order
3.3 Orthogonality for Chebyshev polynomials of the first kind
References
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