Noname manuscript No. (will be inserted by the editor)
A Remarkable Identity Involving Bessel Functions Diego E. Dominici · Peter M.W. Gill · Taweetham Limpanuparb
arXiv:1103.0058v1 [math.CA] 1 Mar 2011
the date of receipt and acceptance should be inserted later
Abstract We consider a new identity involving integrals and sums of Bessel functions. The identity provides new ways to evaluate integrals of products of two Bessel functions. The identity is remarkably simple and powerful since the summand and integrand are of exactly the same form and the sum converges to the integral relatively fast for most cases. A proof and numerical examples of the identity are discussed.
1 Introduction The Newtonian kernel K(r, r 0 ) =
1 , |r − r 0 |
r, r 0 ∈ R3 ,
(1)
is ubiquitous in mathematical physics and is essential to an understanding of both gravitation and electrostatics [9]. It is central in classical mechanics [6] but plays an equally important role in quantum mechanics [11] where it mediates the dominant two-particle interaction in electronic Schr¨odinger equations of atoms and molecules. Although the Newtonian kernel has many beautiful mathematical properties, the fact that it is both singular and long-ranged is awkward and expensive from a computational point of view [8] and this has led to a great deal of research into effective methods for its treatment. Of the many schemes that D. E. Dominici Technische Universit¨ at Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany Permanent address: Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr., New Paltz, NY 12561-2443, USA E-mail:
[email protected] P. M. W. Gill, T. Limpanuparb Research School of Chemistry, Australian National University, ACT 0200, Australia E-mail:
[email protected]
2
have been developed, Ewald partitioning [3], multipole methods [7] and Fourier transform techniques [15] are particularly popular and have enabled the simulation of large-scale particulate and continuous systems, even on relatively inexpensive computers. A recent alternative [4, 5, 13, 19] to these conventional techniques is to resolve (1), a non-separable kernel, into a sum of products of one-body functions K(r, r 0 ) =
∞ X l X
Ylm (r)Ylm (r 0 )Kl (r, r0 ) =
∞ l X X
φnlm (r)φnlm (r 0 )
n,l=0 m=−l
l=0 m=−l
(2) where Ylm (r) is a real spherical harmonic [14, 14.30.2] of the angular part of three dimensional vector r, Z∞ Jl+1/2 (kr)Jl+1/2 (kr0 ) √ dk, Kl (r, r ) = 4π k rr0 0
0
Jl (z) is a Bessel function of the first kind [14, 10.2.2], and r = |r|. The resolution (2) is computationally useful because it decouples the coordinates r and r 0 and allows the two-body interaction integral ZZ E[ρa , ρb ] = ρa (r)K(r, r 0 )ρb (r 0 )drdr 0 , (3) between densities ρa (r) and ρb (r) to be recast as E[ρa , ρb ] =
∞ X ∞ X l X
Anlm Bnlm ,
(4)
n=0 l=0 m=−l
where Anlm is a one-body integral of the product of ρa (r) and φnlm (r). If the one-body integrals can be evaluated efficiently and the sum converges rapidly, (4) may offer a more efficient route to E[ρa , ρb ] than (3). The key question is how best to obtain the Kl resolution Kl (r, r0 ) =
∞ X
Knl (r)Knl (r0 ).
n=0
Previous attempts [4, 5, 19] yielded complicated Knl whose practical utility is questionable but, recently, we have discovered the remarkable identity Z 0
∞
∞ X Jν (at)Jν (bt) Jν (an)Jν (bn) dt = εn t n n=0
where a, b ∈ [0, π], ν = 12 , 32 , 52 , . . . and εn is defined by 1 , n=0 εn = 2 . 1, n ≥ 1
(5)
(6)
3
This yields [12] the functions r φnlm (r) =
4πεn Jl+1/2 (rn) Ylm (r), rn
and these provide a resolution which is valid provided r < π. The aim of this Communication is to prove an extended version of the identity (5) and demonstrate its viability in approximating the integral of Bessel functions. 2 Preliminaries The Bessel function of the first kind Jν (z) is defined by [20, 3.1 (8)] Jν (z) =
∞ X
n
z ν+2n (−1) . Γ (ν + n + 1) n! 2 n=0
(7)
It follows from (7) that Jν (z)
z −ν
2 is an entire function of z and we have z −ν 1 lim Jν (z) = . z→0 2 Γ (ν + 1) Gauss’ hypergeometric function is defined by [1, 2.1.2] X ∞ (a)k (b)k z k a, b F , ; z = 2 1 c (c)k k!
(8)
k=0
where (u)k is the Pochhammer symbol (or rising factorial), given by (u)k = u (u + 1) · · · (u + k − 1) . The series (8) converges absolutely for |z| < 1 [1, 2.1.1]. If Re (c − a − b) > 0, we have [1, 2.2.2] Γ (c) Γ (c − a − b) a, b F ; 1 = . (9) 2 1 c Γ (c − a) Γ (c − b) Many special functions can be defined in terms of the hypergeometric func(λ) tion. In particular, the Gegenbauer (or ultraspherical) polynomials Cn (t) are defined by [10, 9.8.19] (2λ)n −n, n + 2λ 1 − x Cn(λ) (x) = ; , (10) 2 F1 λ + 12 n! 2 with n ∈ N0 and N0 = {0, 1, . . .} . In the sequel, we will use the following Lemmas.
4
Lemma 1 For k ∈ N0 , we have 1 −µ+2k (k + 1 − µ)k (x) = 1 − x2 2 k! 1 1 + k, − µ+k 2 2 2 × 2 F1 ; x , |x| < 1. 1 (µ−2k)
C2k
(11)
2
Proof Using the formula [1, 3.1.12] Γ a + b + 21 Γ 12 x+1 2a, 2b a, b 2 F F ; = ; x 2 1 2 1 1 a + b + 21 2 Γ a + 12 Γ b + 12 2 1 1 Γ a + b + 12 Γ − 12 a + 2, b + 2 2 ;x −x 2 F1 3 Γ (a) Γ (b) 2 in (10), we obtain 22µ−2k−1 Γ µ − 2k + 12 Γ (µ − k) −k, µ − k 2 F ; x , 2 1 1 Γ 12 − k Γ (2µ − 4k) (2k)! 2 (12) 1 = 0 for k = 0, 1, . . . . since Γ (−k) Applying Euler’s transformation [1, 2.2.7] a, b c − a, c − b c−a−b ; x = (1 − x) ;x 2 F1 2 F1 c c (µ−2k)
C2k
(x) =
to (12), we get 22µ−2k−1 Γ µ − 2k + 21 Γ (µ − k) Γ 12 − k Γ (2µ − 4k) (2k)! 1 1 + k, 12 − µ + k 2 2 2 −µ+2k 2 ;x , × 1−x 2 F1 1 (µ−2k)
C2k
(x) =
2
and (11) follows since 22µ−2k−1 Γ µ − 2k + 21 Γ (µ − k) (k + 1 − µ)k = . 1 k! Γ 2 − k Γ (2µ − 4k) (2k)! Lemma 2 Let the function h(x; a) be defined by µ−2k− 12 µ (µ−2k) x x2 A (a) 1 − C2k ( a ), 2 k a h(x; a) = 0, a ≤ x ≤ π
0≤x
2k −
1 2
(−1) (2k)!Γ (µ − 2k) 22µ−2k−1 , a2k+1 Γ (2µ − 2k)
and k ∈ N0 .
(13)
5
Then, h(x; a) can be represented by the Fourier cosine series h(x; a) =
∞ X
εn
n=0
Jµ (na) 2k µ n cos (nx) , 1 2 an
(14)
where εn was defined in 6. Proof For Re(σ) > − 21 , α > 0 and k ∈ N0 , we have [20, 3.32] Z1
1 − t2
σ− 21
k
(σ)
C2k (t) cos (αt) dt =
π (−1) Γ (2k + 2σ) σ Jσ+2k (α) . (2k)!Γ (σ) (2α)
(15)
0
Replacing σ = µ − 2k and α = na in (15), we obtain Z1 µ−2k− 21 (µ−2k) Jµ (na) n2k µ = 2a Aµk (a) 1 − t2 C2k (t) cos (nat) dt 1 2 an 0
or Jµ (na) 2k 2 µ n = 1 π an 2
Za h(x; a) cos (nx) dx,
(16)
0
and the result follows. 3 Main result The discontinuous integral Z∞ Jµ (at) Jν (bt) I (a, b) = dt, tλ 0
was investigated by Weber [21], Sonine [18] and Schafheitlin [17]. They proved that [20, 13.4 (2)] 2 ! aλ−ν−1 bν Γ ν+µ−λ+1 ν+µ−λ+1 ν−µ−λ+1 2 b , 2 2 2 F1 ; , I (a, b) = λ+µ−ν+1 ν + 1 a λ 2 Γ (ν + 1) Γ 2
(17) for Re(µ + ν + 1) > Re (λ) > −1
(18)
and 0 < b < a. The corresponding expression for the case when 0 < a < b, is obtained from (17) by interchanging a, b and also µ, ν. When a = b, we have [20, 13.41 (2)] ν+µ−λ+1 λ−1 a Γ Γ (λ) 2 , I (a, a) = λ+µ−ν+1 λ+ν−µ+1 ν+µ+λ+1 2λ Γ Γ Γ 2 2 2
6
provided that Re(µ + ν + 1) > Re (λ) > 0. This result also follows from Gauss’ summation formula (9) and (17). Theorem 1 If 0 < b < a < π, Re (µ) > 2k − 12 , Re (ν) > − 21 , k ∈ N0 , and ∞ X
Sk (a, b) =
εn
n=0
Jµ (an) Jν (bn) an 2k µ 1 ν , 1 2 2 an 2 bn
then, Γ k + 12 2 F1 Sk (a, b) = aΓ (ν + 1) Γ µ − k + 12 Proof Multiplying (14) by Aν0 (b) π2 1 − we get 2 Sk (a, b) = Aν0 (b) Aµk (a) π
Zb
1 2
x2 b2
ν− 21
ν− 21
x2 1− 2 b
+ k, 12 − µ + k ; ν+1
2 ! b . (19) a
and integrating from 0 to b,
x2 1− 2 a
µ−2k− 21
(µ−2k)
C2k
x ( )dx, a
0
(20) where we have used the integral representation (16). Setting x = bt and b = ωa in (20), we obtain 2b Sk (a, b) = Aν0 (b) Aµk (a) π
Z1
1 − t2
ν− 12
1 − ω 2 t2
µ−2k− 12
(µ−2k)
C2k
(ωt)dt.
0
(21) Thus, we can re-write (21) as Sk (a, b) =
22µ−2k−1 Γ µ − 2k + 12 Γ (µ − k) 2b ν A0 (b) Aµk (a) π Γ 12 − k Γ (2µ − 4k) (2k)! Z1
×
1 − t2
ν− 21
1 2 F1
0
2
+ k, 12 − µ + k 1 2
; ω 2 t2 dt,
or, using (13) and changing variables in the integral, √ k (−1) 22k π Sk (a, b) = 2k+1 a Γ µ − k + 12 Γ ν + 21 Γ 12 − k 1 Z1 + k, 21 − µ + k 2 ν− 12 − 12 2 × s (1 − s) ; ω s ds. 2 F1 1 2
0
Recalling the formula [1, Th. 2.2.4] 2 F1
Z1 Γ (c) a, b a, b c−d−1 d−1 ;x = t (1 − t) ; xt dt, 2 F1 c d Γ (d) Γ (c − d) 0
7
valid for Re (c) > Re(d) > 0, x ∈ C \ [1, ∞) , we conclude that √ k (−1) 22k π a2k+1 Γ µ − k + 12 Γ ν + 12 Γ 12 − k √ 1 1 πΓ ν + 21 2 + k, 2 − µ + k ; ω 2 . × 2 F1 ν+1 Γ (ν + 1)
Sk (a, b) =
But since [1, 1.2.1] Γ
k+
1 2
Γ
1 −k 2
k
= (−1) π,
k = 0, 1, . . . ,
the result follows. The special case of Theorem 1 in which k = 0, was derived by Cooke in [2], as part of his work on Schl¨ omilch series. Corollary 1 If 0 < b < a < π, Re (µ) > 2k − 12 , Re (ν) > − 12 , k ∈ N0 , then Z∞ ∞ X Jµ (at) Jν (bt) Jµ (an) Jν (bn) εn dt = . µ+ν−2k t nµ+ν−2k n=0 0
Proof The result follows immediately from (17) and (19), after taking λ = µ + ν − 2k. Note that since for all k ∈ N0 Re(µ + ν + 1) = Re (2k + 1 + λ) > Re (1 + λ) > Re (λ) and Re (λ) = Re (µ + ν − 2k) > −1, the conditions (18) are satisfied. Corollary 2 If 0 < a, b < π, and Re(µ) > 0, then ∞ X
Jµ (an) Jµ (bn) εn = n n=0
( Z∞ 1 Jµ (at) Jµ (bt) dt = 2µ 1 t 2µ 0
a µ b b µ a
, ,
a≤b . a≥b
Proof The result is a consequence of Corollary 1 and a special case of the integral (17) (see [20, 13.42 (1)]).
8
Fig. 1
1 5
h
i min(a,b) 5/2 max(a,b)
(solid) and T10 (a, b) (dashed), for a =
π 2π 3π , , 4 4 4
and b ∈ [0, 2π]
0.20
0.15 a=
3Π 4
0.10 a= 0.05
a=
2Π 4
Π 4
Π
Π
3Π
4
2
4
Π
5Π
3Π
7Π
4
2
4
b 2Π
Fig. 2 Truncation error R20 (a, b) for a, b ∈ [−2π, 2π] 2Π
Π
b
0
-Π
-2 Π
0.1
0.0 -0.1 -2 Π
0
-Π
Π
2Π
a
Table 1 RN (a, b) for a = π/2 and various b and N b N 1 10 100 1000 10000
π/4 3.15E-02 -3.10E-03 -1.79E-06 1.81E-09 1.81E-12
2π/4 1.81E-01 2.02E-02 2.03E-03 2.03E-04 2.03E-05
3π/4 3.15E-02 -2.40E-03 -4.81E-07 4.86E-10 4.86E-13
4π/4 -2.37E-02 1.45E-04 -1.39E-07 -1.37E-10 -1.37E-13
5π/4 -4.12E-02 1.74E-03 7.46E-07 -7.46E-10 -7.46E-13
6π/4 -3.09E-2 -1.16E-2 -1.17E-3 -1.17E-4 -1.17E-5
9
4 Numerical results The efficient and accurate computation of the 2 F1 function in (17) can be numerically challenging [16], and we note that Corollary 1 provides a new route to its evaluation. Because the numerator is bounded |Jµ (at) Jµ (bt) | ≤ 1 [20, 13.42 (10)], the Weierstrass M-test shows that the series converges uniformly as long as µ + ν − 2k > 1 and µ, ν > 0. To explore the rate of convergence of the sum in Corollary 2, we choose ν = 5/2. The exact value of the integral is Z∞
J5/2 (at)J5/2 (bt) 1 dt = t 5
r
|a| a
r
5/2 |b| min(|a|, |b|) b max(|a|, |b|)
(22)
0
and truncation of the infinite series yields the finite sum
TN (a, b) =
N X J5/2 (an) J5/2 (bn) n n=1
(23)
and a truncation error 5/2 r r |a| |b| 1 min(|a|, |b|) TN (a, b). − RN (a, b) = 5 max(|a|, |b|) a b
(24)
The integral in (22) and sum in (23) are illustrated in Figure 1 and their difference (24) is shown in Figure 2 and Table 1. The truncation errors in Table 1 reveal that the rate of convergence of the Bessel sum is strongly dependent on the value of a and b. For b = 2π/4 and b = 6π/4, the truncation error appears to decay as 1/N but, for the other values of b, it appears to decay as 1/N 3 . Although these empirical results are interesting, we have not been able to determine analytically the decay behavior as a function of a and b. The excellent agreement in Figure 1 and apparently flat plateau in Figure 2 strongly suggest that Corollary 2 is true over the larger domain |a|+|b| < 2π. However, we have not yet managed to find a proof for this.
5 Concluding remark We provide a rigorous proof that the identity (5) is valid on the square domain a, b ∈ (0, π). A numerical study indicates that the rate of convergence of the sum in the identity is sensitive to the values of a and b and further work to quantify this would be helpful. Generalization of the identity is possible and should be explored in the future work.
10
Acknowledgement The work of D. Dominici was supported by a Humboldt Research Fellowship for Experienced Researchers from the Alexander von Humboldt Foundation. P.M.W. Gill thanks the Australian Research Council (Grants DP0984806 and DP1094170) for funding and the NCI National Facility for a generous grant of supercomputer time. T. Limpanuparb thanks the Development and Promotion of Science and Technology Talents Project for a Royal Thai Government PhD scholarship.
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