A Generalisation For The Infinite Integral Over Three Spherical Bessel Functions
aa r X i v : . [ m a t h - ph ] A ug A GENERALISATION FOR THE INFINITE INTEGRALOVER THREE SPHERICAL BESSEL FUNCTIONS
R. Mehrem a Associate LecturerThe Open University in the North West351 Altrincham RoadSharston, Manchester M22 4UNUnited Kingdom
A. HOHENEGGER b Max-Planck-Institut f¨ur KernphysikSaupfercheckweg 169117 HeidelbergGermany a Email: [email protected] Email: [email protected]. BSTRACT
A new formula is derived that generalises an earlier result for the infinite integralover three spherical Bessel functions. The analytical result involves a finite sum overassociated Legendre functions, P ml ( x ) of degree l and order m . The sum allows forvalues of | m | that are greater than l . A generalisation for the associated Legendrefunctions to allow for any rational m for a specific l is also shown. . Introduction The calculations of infinite integrals that involve a product of spherical Besselfunctions have been the focus of many papers, amongst which are references [1-22]. Inparticular, reference [12] showed a derivation for an analytical evaluation of an infiniteintegral over three spherical Bessel functions, given by the form I ( λ , λ , λ ; k k k ) ≡ ∞ Z r j λ ( k r ) j λ ( k r ) j λ ( k r ) dr. (1 . I ( λ ; λ , λ , λ ; k k k ) ≡ ∞ Z r − λ j λ ( k r ) j λ ( k r ) j λ + λ ( k r ) dr. (1 . λ = 0, only works when | λ − λ | ≤ λ ≤ λ + λ , i.e. when the integer indices λ , λ and λ satisfy the triangular condition, and λ + λ + λ is even. However, unlike theearlier result where the triangular condition forces k , k and k to form the sides of atriangle, this generalised integral can have non-zero values when the k ’s do not form thesides of a triangle. Our new result is only derived for the case the k ’s do form the sidesof a triangle. We also show the general result, i.e. for any values of k , k and k , when λ = 1. This new result involves finite sums over the associated Legendre function, P ml ( x ), of degree l and order m , whereas the earlier result involved sums over theLegendre polynomial, P l ( x ). The sums can involve values for | m | that are larger than . An extension for the associated Legendre functions which produces a formula thatallows for such values is shown in appendix A. An application is shown to evaluate anintegral involving four spherical Bessel functions and is used in the angular integrationof the homogeneous and velocity isotropic Boltzmann equation. This integral, which ishighly oscillatory and needs special treatment when evaluated numerically, is reduced toan integral over an associated Legendre function and a Legendre polynomial combinedwith algebraic factors which can easily be evaluated numerically. A comparison is shownin appendix B between our analytical result and the analytical evaluation carried outin reference [8].
2. Generalising the Integral Over Three Spherical Bessel Functions
An earlier result [12] showed that an infinite integral over three spherical Besselfunctions can be written as λ λ λ ! I ( λ , λ , λ ; k k k ) = πβ (∆)4 k k k i λ + λ − λ × (2 λ + 1) / (cid:18) k k (cid:19) λ λ X L =0 (cid:18) λ L (cid:19) / (cid:18) k k (cid:19) L X l (2 l + 1) λ λ − L l ! × λ L l ! ( λ λ λ L λ − L l ) P l (∆) , (2 . k + k − k ) / k k and β (∆) = θ (1 − ∆) θ (1 + ∆) with θ the Heavisidefunction in half-maximum convention. P l ( x ) is a Legendre polynomial of order l , λ λ λ ! is a 3j symbol and ( λ λ λ L λ − L l ) is a 6j symbol which can befound in any standard angular momentum text [23, 24]. Note that the summand in l vanishes unless | λ − ( λ − L ) | ≤ l ≤ λ + λ − L . Now, multiply both sides by k λ +23 and integrate over k from 0 to K , where K can have any positive value. The eft hand side involves the integral (see [25], eq. 5.52-1, page 661) K Z k λ +23 j λ ( k r ) dk = 1 r K λ +2 j λ +1 ( Kr ) . (2 . J ≡ K Z β (∆) k P l (∆) dk . (2 . k dk = − k k d ∆. So, when k = 0 then k = k and ∆ = 1. Also, when k = K , ∆ = ∆ ′ , where ∆ ′ = ( k + k − K ) / k k . Hence J = k k Z ∆ ′ β (∆) P l (∆) d ∆ . (2 . ′ >
1, i.e.
K < | k − k | , then J = 0. If − ≤ ∆ ′ ≤
1, then J = k k β (∆ ′ ) (1 − ∆ ′ ) / P − l (∆ ′ ) , (2 . P − ml ( x ) = (1 − x ) − m/ Z x ... Z x P l ( x ) ( dx ) m , (2 . P ml ( x ) is the associated Legendre function of the first kind of degree l and order m . If ∆ ′ < −
1, then J = k k θ [ K − ( k + k )] Z − P l (∆) d ∆ = 2 k k θ [ K − ( k + k )] δ l, . (2 . ence, the result is λ λ λ ! I (1 ; λ , λ , λ ; k k K ) = π β (∆ ′ )4 K i λ + λ − λ × (2 λ + 1) / (cid:18) k K (cid:19) λ (1 − ∆ ′ ) / λ X L =0 (cid:18) λ L (cid:19) / (cid:18) k k (cid:19) L × X l (2 l + 1) λ λ − L l ! λ L l ! ( λ λ λ L λ − L l ) P − l (∆ ′ )+ ( − λ π k λ k λ K λ +2 √ λ + 1(2 λ + 1) (2 λ + 1) (cid:18) λ λ (cid:19) / θ [ K − ( k + k )] δ λ , λ + λ . (2 . .
8) applies for any values of k , k and K . The second term on theright hand side of (2 .
8) is also consistent with equation 6.578-4 of reference [25] when λ = λ + λ . This equation applies for K > k + k and can be written in the form ∞ Z r λ − λ − λ +1 j λ ( k r ) j λ ( k r ) j λ +1 ( Kr ) dr = 2 λ − λ − λ − π / × k λ k λ K λ +2 Γ( λ + 3 / λ + 3 /
2) Γ( λ + 3 / . (2 . K satisfies the triangular condition, then the second term in (2 . K back to k ) I ( λ ; λ , λ , λ ; k k k ) = πβ (∆)4 k k k i λ + λ − λ (2 λ + 1) / (cid:18) k k (cid:19) λ × (cid:18) k k k (cid:19) λ (1 − ∆ ) λ/ λ λ λ ! − λ X L =0 (cid:18) λ L (cid:19) / (cid:18) k k (cid:19) L × X l (2 l + 1) λ λ − L l ! λ L l ! ( λ λ λ L λ − L l ) P − λl (∆) , (2 . λ , λ , λ and λ are all greater than or equal to 0 to ensure convergence of he integral. Equation (2 .
10) is a new result that generalizes the infinite integral overthree spherical Bessel functions. In general λ can exceed the value for l . In this casethe definition of P ml ( x ) needs to be extended for | m | > l in a way compatible witheq. (2 . .
10) are presented in appendix B. For somesmall values of the indices these have been compared with a different existing result forthe integral as will also be discussed there.
3. Special Cases and Identities
Equation (2 .
10) can be reduced to a known integral [3] for the case λ = 0, i.e. λ = λ ≡ λ ′ with the result I ( λ ; λ ′ , λ ′ , k k k ) = π β (∆)4 k k k (cid:18) k k k (cid:19) λ (1 − ∆ ) λ/ P − λλ ′ (∆) , (3 . k , k and k satisfy the triangular condition | k − k | ≤ k ≤ k + k . (3 . λ = λ = 0 and λ = λ = λ ′ in eq. (2 .
10) and equate it to eq. (3 .
1) aftersetting λ = 0 and interchanging k and k , the following sum rule for the Legendrepolynomial is obtained β ( η ) λ X L =0 (cid:18) λ L (cid:19) (cid:18) − k k (cid:19) L P L ( η ) = β ( η ′ ) (cid:18) k k (cid:19) λ P λ ( η ′ ) , (3 . η = ( k + k − k ) / k k and η ′ = ( k + k − k ) / k k . Equation (3 .
3) isconsistent with the result of reference [26]. In a triangle of sides k , k and k , η is thecosine of the angle facing side k and η ′ is the cosine of the angle facing side k . . Applications One application is an integral over four spherical Bessel functions which arises inthe angular integration of the homogeneous and velocity isotropic Boltzmann equation[19,20] I ( L ; N + M − L, , N, M ; k k k k ) ≡ ∞ Z r − L j N + M − L ( k r ) j ( k r ) j N ( k r ) j M ( k r ) dr, (4 . L , N and M are non-negative integers with N + M ≥ L and k , k , k and k form the sides of a quadrilateral. Using the closure relation for the Bessel functions[2,19], this integral can be written as2 π ∞ Z K dK I ( L ; N + M − L, , N + M − L ; k k K ) × I (0 ; N, M, N + M ; k k K ) . (4 . . I ( L ; N + M − L, , N + M − L ; k k K ) = π β ( ξ )4 k k K (cid:18) k K (cid:19) N + M − L × (cid:18) k k K (cid:19) L (1 − ξ ) L/ N + M − L X L =0 ( − k /k ) L (cid:18) N + M − L L (cid:19) P − L L ( ξ ) , (4 . ξ = ( k + k − K ) / k k and using (cid:18) l l ′ (cid:19) / l l − l ′ l ′ ! = ( − l √ l + 1 (cid:18) ll ′ (cid:19) . (4 . he resulting integral is I ( L ; N + M − L, , N, M ; k k k k ) = π k k k k × ( k k ) N + M k L p N + M ) + 1 N M N + M ! − × N + M − L X L =0 N + M X L ′ =0 ( − k /k ) L ( k /k ) L ′ (cid:18) N + 2 M L ′ (cid:19) / (cid:18) N + M − L L (cid:19) X l (2 l + 1) × N N + M − L ′ l ! M L ′ l ! ( N M N + M L ′ N + M − L ′ l ) × S ( k k k k ; LM N L ′ l ) , (4 . ξ ′ = ( k + k − K ) / k k , S ( k k k k ; LM N L ′ l ) = ∞ Z β ( ξ ) β ( ξ ′ ) K N + M ) (1 − ξ ) L/ P − L L ( ξ ) P l ( ξ ′ ) dK, (4 .
5. Conclusions
A new generalised analytical formula for the infinite integral over three sphericalBessel functions was shown. It involves finite sums over the associated Legendrefunction combined with angular momentum coupling coefficients 3j and 6j symbols.The sums involved values of the order which exceeded the degree, which is in conflictwith the usual definition of the associated Legendre functions. An extension for theassociated Legendre functions is shown in appendix A. PPENDIX A: A Generalised Solutionto the Associated Legendre Equation
The associated Legendre function of the first kind, P ml ( x ) is a solution of thedifferential equation [25](1 − x ) d P ml ( x ) dx − x dP ml ( x ) dx + [ l ( l + 1) − m − x ] P ml ( x ) = 0 , ( A. P ml ( x ) = ( − m (1 − x ) m/ l l ! d l + m dx l + m ( x − l , ( A. l ≥ m (with | m | ≤ l ). In this appendix, itwill be shown that another solution exists for this differential equation that extends theassociated Legendre function to any rational | m | ( | m | < ∞ ) for integer l . This solutionis irregular at x = 1 when m > l and irregular at x = − m < l . The formula canbe used to derive closed form expressions for P ml for a specific l and any rational m .If we set l = 0 in eq. ( A. P m ( x ),is a solution: P m ( x ) = a m (cid:16) − x x (cid:17) − m/ , ( A. a m is a coefficient that depends on m . Now, if we assume that the dependenceon l can be separated, then in general P ml can be written as P ml ( x ) = A l,m ( x ) P m ( x ) , ( A. A l,m ( x ) is a coefficient that depend on l , m and x . To determine the coefficient(up to a constant), substitute eq. ( A.
4) back into eq. ( A. quation for A l,m is(1 − x ) d A l,m ( x ) dx + (2 m − x ) dA l,m ( x ) dx + l ( l + 1) A l,m ( x ) = 0 . ( A. A l,m as the Jacobi polynomial, [25] P − m, ml ( x ), defined by P a, bl ( x ) = ( − l l l ! (1 − x ) − a (1 + x ) − b d l dx l [(1 − x ) l + a (1 + x ) l + b ] . ( A. P ml ( x ) = b l,m (cid:16) − x x (cid:17) − m/ P − m, ml ( x ) , ( A. b l,m is a constant that depends on l and m . Using the normalisation for theassociated Legendre functions Z − [ P ml ( x )] dx = 22 l + 1 ( l + m )!( l − m )! , ( A. Z − (1 − x ) a (1 + x ) b [ P a, bl ( x )] dx = 2 a + b +1 ( l + a )!( l + b )! l !( a + b + 2 l + 1)( l + a + b )! , ( A. A.
7) and integrate it over x from − A.
8) to obtain an expression for b l,m : b l,m = l ! | l − m | ! , ( A. | l − m | ! is introduced to allow m to go from −∞ to ∞ , andthe expression reduces to ( l − m )! for − l ≤ m ≤ l . Equation ( A.
7) then becomes P ml ( x ) = ( − l l | l − m | ! (cid:16) − x x (cid:17) m/ d l dx l [(1 − x ) l (cid:16) x − x (cid:17) m ] , ( A. l ≥ −∞ < m < ∞ . hen − l ≤ m ≤ l , equations ( A.
11) and ( A.
2) are equal, leading to the identity d l + m dx l + m (1 − x ) l = ( − m l !( l − m )! (1 + x ) − m d l dx l [(1 − x ) l (cid:16) x − x (cid:17) m ] , ( A. − l ≤ m ≤ l .Another property that can easily be found is P ml ( − x ) = ( − l | l + m | ! | l − m | ! P − ml ( x ) , ( A. l and any rational m .Using equations ( A.
7) and ( A. P ml ( x ) = l ! | l − m | ! (cid:16) − x x (cid:17) − m/ P − m, ml ( x ) . ( A. n + 1) ( n + α + β + 1)(2 n + α + β ) P ( α, β ) n +1 ( x ) = (2 n + α + β + 1) × [(2 n + α + β )(2 n + α + β + 2) x + α − β ] P ( α, β ) n ( x ) − n + α )( n + β ) × (2 n + α + β + 2) P ( α, β ) n − ( x ) , ( A. | l − m + 1 | P ml +1 ( x ) = (2 l + 1) x P ml ( x ) − ( l − m ) | l − m | P ml − ( x ) . ( A. A.
11) one finds the following closed-form xpressions for the associated Legendre functions at specific l ’s and any rational m P m ( x ) = 1 | m | ! (cid:16) x − x (cid:17) m/ , ( A. P m ( x ) = 1 | − m | ! ( x − m ) (cid:16) x − x (cid:17) m/ , ( A. P m ( x ) = 1 | − m | ! (3 x − xm − m ) (cid:16) x − x (cid:17) m/ , ( A. P m ( x ) = 1 | − m | ! (15 x − x m − x + 6 xm + 4 m − m ) (cid:16) x − x (cid:17) m/ , ( A. P m ( x ) = 1 | − m | ! (105 x − x m − x + 45 x m + 55 xm − xm + 9 − m + m ) (cid:16) x − x (cid:17) m/ . ( A. APPENDIX B: Explicit Expressions
Here we provide some explicit expressions for the integral (1 .
2) and compare tothose obtained from a result given in [8], which reads in our notation I ( λ ; λ , λ , λ ; k k k ) = − π i λ + λ + λ × (cid:16) sgn ( k + k + k ) F m m m λ λ λ λ ( k , k , k )+ ( − λ sgn ( − k + k + k ) F m m m λ λ λ λ ( − k , k , k )+ ( − λ sgn ( k − k + k ) F m m m λ λ λ λ ( k , − k , k )+ ( − λ + λ sgn ( − k − k + k ) F m m m λ λ λ λ ( − k , − k , k ) (cid:17) . ( B. ith F m m m λ λ λ λ ( k , k , k ) = λ X m =0 ( λ + m )! ( − m ( λ − m )! m ! (2 k ) m +1 λ X m =0 ( λ + m )! ( − m ( λ − m )! m ! (2 k ) m +1 × λ + λ X m =0 ( λ + λ + m )! ( − m ( λ + λ − m )! m ! (2 k ) m +1 ( k + k + k ) m + m + m + λ ( m + m + m + λ )! . ( B. k , k and k which satisfy the triangular conditionand using symmetries of F m m m λ λ λ λ , ( B.
1) can be simplified to I ( λ ; λ , λ , λ ; k , k , k ) = − π β (∆) i λ + λ + λ × λ X m =0 ( − m ( λ + m )!( λ − m )! m ! (2 k ) m +1 λ X m =0 ( − m ( λ + m )!( λ − m )! m ! (2 k ) m +1 λ + λ X m =0 ( − m ( λ + λ + m )!( λ + λ − m )! m ! (2 k ) m +1 m + m + m + λ )! × (cid:20) ( k + k + k ) m + m + m + λ − ( − λ + m c m + m + m + λ − ( − λ + m c m + m + m + λ − ( − λ + λ + m c m + m + m + λ (cid:21) , ( B. c = ( − k + k + k ), c = ( k − k + k ) and c = ( k + k − k ). For numericalcomputations this expression is more suitable than ( B.
1) in which almost cancellationsbetween the different terms including sgn functions can lead to large relative errors.For some small values of λ , λ , λ and λ this yields the following explicit results: I (0; 0 , , k k k ) = 14 πβ (∆) k k k ,I (0; 1 , , k k k ) = − πβ (∆) k k k (cid:16) − k k + k + 2 k c − k c − k c + k − k c − k c + 2 k c − k + c + c + c (cid:17) , (0; 1 , , k k k ) = − πβ (∆) k k k (cid:16) − k k + k + 2 k c − k c − k c + k − c k + 2 k c − k c − k + c + c + c (cid:17) ,I (1; 0 , , k k k ) = − πβ (∆) k k k (cid:16) k − k c − k c + 2 k c − k − k k − k + c + c + c (cid:17) ,I (1; 1 , , k k k ) = 164 πβ (∆) k k k (cid:16) k k + 4 k c − k c k + c + 8 k k c − k k c − k k c + 12 k k c − k k c + 6 k k − k c − k c + 4 k c + 4 k c + 4 k c + 4 k c − k c − k c − k k + 8 k k + c + c + 3 k − k − k (cid:17) ,I (1; 1 , , k k k ) = 1192 πβ (∆) k k k (cid:16) − k c − k k + 24 k k k c − k c k + c + 12 k k c − k k c − k c k − k k c + 12 k k c − k k c + 12 k k c − k k c + 24 k k k c + 24 k k k c − k c − k k − k c − k c + 4 k c + 4 k c − k c − k c − k k + 4 k c + c + c + 3 k + 3 k + 3 k (cid:17) . ( B. . I ( λ ; λ , λ , λ ; k k k ) = πβ (∆)2 λ +2 ( λ + λ + λ + λ )! × k − ( λ +1) k − ( λ +1) k − ( λ + λ +1) A ( λ ; λ , λ , λ ) , ( B. ith A (0; 0 , ,
0) = 1 A (0; 1 , ,
1) = − k ( − k + k ∆) ,A (0; 1 , ,
0) = 2 k k ∆ ,A (1; 0 , ,
0) = 2 k k (1 − ∆) ,A (1; 1 , ,
1) = − k k ( − k + k ∆ + k ) (1 − ∆) ,A (1; 1 , ,
0) = 6 k k (∆ + 1) (1 − ∆) ,A (2; 0 , ,
0) = 4 (∆ − k k ,A (2; 1 , ,
1) = −
16 (∆ − ( − k + 2 k + k ∆) k k ,A (2; 1 , ,
0) = 16 (∆ − (2 + ∆) k k . To test our new result we have compared several explicit analytic expressions obtainedfrom (2 .
10) and ( B.
3) respectively (including the ones given above), and found fullagreement. The advantage of equations ( B.
5) over ( B.
4) is their compact form interms of generalized Legendre functions. This property will be exploited in [28] wherewe study the integral ∞ Z e − r/u r n j λ ( k r ) j λ ( k r ) dr. ( B. eferences
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