MMIT-CTP/4872
Spherical Bessel Functions
Teboho A. Moloi ∗ Department of Physics, Nelson Mandela University, Port Elizabeth, 6031, South Africa (Dated: February 5, 2021)We examine indefinite integral involving of arbitrary power x , multiplied by three spherical Besselfunctions of the first kind j h , j k , and j l with integer order h, k, l ≥ I. INTRODUCTION
In the past, numerous studies have included integrating the zero to four spherical function of Bessel asform of applications from different fields such as astrophysics, particle physics, nuclear physics etc. [4, 14],had a way of measuring these integrals using a triangle in an effort to achieve precise numerical integrationsof these combinations. And a very interesting method was also developed involving an exponential, twoBessel and a polynomial equation introduced by [13]. All of these methods seem restrictive, and [16], thenpresented an alternative solution with no restrictions. Where the relation between spherical function andfunction of Bessel were taken up and the trigonometric identities was applied to the integral. They focusedon the estimation of the infinite integrals that involve polynomial multiplied by three spherical functions ofthe Bessel and an exponent. In this text, we have followed this approach and ways of applying such integralconditions to promote numerical analysis.Specifically, we focus on indefinite integral of the form I nmhkl ( x ; α, β, µ ) = (cid:90) dxx n e − mx j h ( αx ) j k ( βx ) j l ( µx ) . (1)Here j h , j k , and j l are spherical Bessel functions of the first kind, this integral converge when h, k, l ≥
0, and α, β, µ are real numbers.This paper has the following structure; we provide the general literature on Bessel function in II. Insection. III we introduce some of the important equations required in our results. We provide some results insection. IV. We finally, conclude in section.V.
II. SPHERICAL BESSEL FUNCTIONS LITERATURE
Spherical Bessel functions are well-known to account for the problems with circular symmetry. In sphericalcoordinate, if one solve Helmholtz’s and Laplacian’s equation the solution yield the following differentialequation d ydx + 2 x dydx + (cid:20) l ( l + 1) x (cid:21) y = 0 . (2)Spherical Bessel functions with indices that are not integers are usually less important to implement, here wepresume that index l to be an integral. The solution to expression above 2 results in the spherical Bessel- ∗ [email protected] a r X i v : . [ m a t h . G M ] F e b nd Neumann-function, j l ( kr ) and n l ( kr ) they are both respectively defined as follows; j l ( x ) = (cid:114) π x J l +1 / ( x ) , n l ( x ) = (cid:114) π x N l +1 / ( x ) = ( − l +1 (cid:114) π x J − l − / ( x ) . (3)Here we make note that j l ( x ) are Bessel functions of the first kind and n l ( x ) are the second kind, we thencan relate the first and second kind by the following expression h (1) l ( x ) = (cid:114) π x H (1) l +1 / ( x ) = j l ( x ) + in l ( x ) , h (2) l ( x ) = (cid:114) π x H (2) l +1 / ( x ) = j l ( x ) − in l ( x ) , (4)with h (1) l ( x ) and h (2) l ( x ) are Spherical Hankel functions and their counter-parts are Hankel functions H (1) l and H (2) l , now as we can see from the information provided above j l and n l are spherical Bessel functions andtheir counter-parts are Bessel functions. The Bessel functions and spherical Bessel functions are related—thiscan be shown by the function √ xj l ( x ) and √ xn l ( x ) both satisfy the Bessel functions. From the series solution,with the conventional normalization (for more details see [1]) one can show that y l − + y l +1 = 2 l + 1 x , ly l − − ( l + 1) y l +1 = (2 l + 1) dy dx , (5) ddx (cid:20) x l +1 y l ( x ) (cid:21) = x l +1 y l − ( x ) , ddx (cid:20) x − y l ( x ) (cid:21) = x − y l +1 ( x ) , (6)where y l can be any of the following functions j , n l , h (2) l , and h (2) l . This recurrence relations in turn leadsback to the differential equations. The spherical Bessel function can be computed by indiction on l whichleads to Rayleigh’s formulas; j l ( x ) = ( − − x ) l (cid:18) x ddx (cid:19) sin( x ) x , n l ( x ) = ( − − x ) l (cid:18) x ddx (cid:19) cos( x ) x , (7) h (1) l ( x ) = ( − − x ) l (cid:18) x ddx (cid:19) ix e ix , h (2) l ( x ) = ( − − x ) l (cid:18) x ddx (cid:19) ix e − ix . (8)In this paper, we focus on spherical Bessel functions of the first, second and third kind for integer l ≥ l = 0 , , j = sin( x ) x , j = sin( x ) x − cos( x ) x , j = (cid:18) x − x (cid:19) sin( x ) − x cos( x ) , (9) n = cos( x ) x , j = − cos( x ) x − sin( x ) x , j = (cid:18) x − x (cid:19) cos( x ) − x sin( x ) , (10) h (1)0 = − ix e ix , h (1)1 = e ix (cid:18) x − ix (cid:19) , h (1)2 = e ix (cid:18) ix − x − ix (cid:19) . (11)Furthermore, from the Rayleigh expressions provided in (7) we can easily extract limiting behaviors; forexamples for x (cid:28) l , the solution have the following behavior j l ≈ l l (2 l + 1) x l = x l (2 l + 1) , n l ∼ l lx l +1 = (2 l − x l +1 , (12)and for x (cid:29) l we then have j l ≈ x sin (cid:18) x − lπ/ (cid:19) , n l ≈ − x cos (cid:18) x − lπ/ (cid:19) , h (1) l ∼ ( − l e ix x . (13)Now lets introduce the closure relation of the spherical Bessel functions, starting with the non-trivial formulawe get e ikz = ∞ (cid:88) l =0 (2 l + 1) i l j l ( kr ) P l (cos θ ) . (14)2here P l is the Legendre polynomial of order l , if the wave vector is pointing at the direction than the positive z-axis, then the above expression (14) can be generalized; we make a note that Y l ( θ, φ ) = (cid:112) (2 l + 1) / πP l (cos θ ),we find e i(cid:126)k · (cid:126)x = 4 π ∞ (cid:88) l =0 i l j ( kr ) l (cid:88) m = − l Y m ∗ ( θ (cid:126)k φ (cid:126)k ) Y m ( θ (cid:126)k φ (cid:126)k ) . (15)Lets now normalize the delta function, the usefulness of this will be seen later as consequence of the identityof (15) is the inner-product of the two spherical Bessel functions. Solving this we begin with the following (cid:90) d(cid:126)x e i(cid:126)k · (cid:126)x e − i(cid:126)k (cid:48) · (cid:126)x = (2 π ) δ ( (cid:126)k − (cid:126)k (cid:48) ) , (16)following from (15) the right hand side yields (cid:90) d(cid:126)x e i(cid:126)k · (cid:126)x e − i(cid:126)k (cid:48) · (cid:126)x = (cid:88) l,m (4 π ) (cid:90) dr r j l ( kr ) j ( k (cid:48) r ) Y m ∗ ( θ (cid:126)k ) Y m ( θ (cid:126)k ) . (17)While on the right side, we get the following(2 π ) δ ( (cid:126)k − (cid:126)k (cid:48) ) = (2 π ) k sin θ δ ( (cid:126)k − (cid:126)k (cid:48) ) δ ( (cid:126)θ − (cid:126)θ (cid:48) ) δ ( (cid:126)φ − (cid:126)φ (cid:48) ) , (18)combining the above information and making note of the fact that (cid:80) l,m Y m ∗ l (Ω (cid:126)k ) Y ml ( (cid:126)k ) = δ (Ω (cid:126)k − Ω (cid:48) (cid:126)k ), wearrive at (cid:90) ∞ dr r j l ( kr ) j ( k (cid:48) r ) = π k δ ( (cid:126)k − (cid:126)k (cid:48) ) , (19)with some mathematics we also consider orthogonality relation which reads (cid:90) ∞−∞ j k ( x ) j l ( x ) dx = π l + 1 δ kl , for k, l ∈ N , (20)here δ kl is Kronecker delta. It is useful to mention that infinite integrals over one, two and three Besselfunctions over the years have gained interest and they are well-known [2–11]. The integrals are generallyexpressed in accordance with the prescribed functions of the Bessel functions and some coefficients of thesefunctions are established from a finite series, the terms of which are obtained from recurrence relation whichinvolve polynomials.Below we introduce the two most important recursion relation which are always fulfilled by spherical Besselfunctions— which normally act as a connector between contiguous l , as well as the derivatives ∂ x y l andvarious y l : they require a bit of trail and error, which are given by; y l = ± ∂ x y l − ( x ) + l ± x y l ± ( x ) (21) y l = ± y l ± ( x ) + 2 l ± x y l ± x. (22) III. PRELIMINARIES
In addition to work presented in ref. [9] we provide the exponential-integral which take a form Z n ( x ) = (cid:90) dxx n e x , (23)3here n denotes an integer. Evaluating this yields a recursion relation, which can be obtained by performingintegration by parts in Eq. (23), which will output the following expression Z n ( x ) = x n e x − nZ n − ( x ) . (24)It follows that the recursion will allow us to step down by n for integer values above zero and the is somekind of cut–off at integers strictly equal to zero, and there will be a step up of n for integer values belowzero and the is some kind of cut–off at integers strictly equal to negative one [9], leading to definite integralswhich where introduced by Sch¨ o milch and Arndt called the sine-integral, cosine-integral shown in ref. [9] andthe exponential-integral [12] with the following form Ei ( x ) = (cid:90) − x ∞ e − u u du. (25)Where Ei ( x ) is a special function, this exponential-integral was introduced evaluated for all real values bySch¨ o milch which is related to Logarithm-integral in this mannerLi( x ) = (cid:90) du log u , relation , Li e x = Ei ( x ) , (26)We provide a schematic for of the special function Ei ( x ), We see from Fig. 1 that the non-conical function − − − − x − − − E i ( x ) Figure 1. Exponential-integral evaluated at specific range. dubbed the exponential function, in scale x < x >
0, and it has aunique zero. We also notice that it is concave on scales from negative infinity to zero and also on scales [0 , IV. INTEGRALS OF THREE SPHERICAL BESSELS OF DIFFERENT ORDER, EXPONENT,AND POLYNOMIAL
We focus on the integral of the form I nmhkl ( x ; α, β, µ ) = (cid:90) dxx n e − mx j h ( αx ) j k ( βx ) j l ( µx ) , (27)4ith n, m ∈ Z , h, k, l ∈ N , and α, β, µ ∈ R . These form of Bessel functions are previously, robustly investigatedin [12–15]. It is easy to see that if one consider the scenario where α = 0, or β = 0, or µ = 0 the integralEq. (27) become finite, so we look at scenarios where α (cid:54) = 0, β (cid:54) = 0, and µ (cid:54) = 0. The general solution of theintegral become [16] I nmhkl ( x ; α, β, µ ) = παβµ Γ(3 − n ) sin( πn ) (cid:26) − [ m + ( α + β + µ ) ] (2 − n )2 sin (cid:20) ( n −
2) arctan (cid:18) ( α + β + µ ) m (cid:19)(cid:21) + [ m + ( µ + β − α ) ] (2 − n )2 sin (cid:20) ( n −
2) arctan (cid:18) ( µ + β + α ) m (cid:19)(cid:21) + [ m + ( µ + α − β ) ] (2 − n )2 sin (cid:20) ( n −
2) arctan (cid:18) ( µ + α − β ) m (cid:19)(cid:21) + [ m + ( α + β − µ ) ] (2 − n )2 sin (cid:20) ( n −
2) arctan (cid:18) ( α + β − µ ) m (cid:19)(cid:21)(cid:27) + c (28)Here Γ is a Gamma function. We then extend work by [16] by looking at different simple cases, such as when h, k, l are all set to zero, or if m = i or if m = 0 and so forth. Now if h = k = l = 0 we get the following I nm ( x ; α, β, µ ) = (cid:90) dxx n e − mx j ( αx ) j ( βx ) j ( µx ) . (29)Taking into account the trigonometric identity belowsin( αx ) sin( βx ) sin( µx ) = − sin[ x ( α + β + µ )] + sin[ x ( α − β + µ )] + sin[ x ( α + β − µ )] . (30)Putting all information together we arrive at I nm ( x ; α, β, µ ) = − i αβµ [ A ( α, β, µ ) + B ( α, β, µ ) + C ( α, β, µ ) + c, (31)where A , B , and C are defined as follows A = x n +1 [ ix ( α − β + µ − im )] − n − Γ[ n + 1 , i ( α − β + µ − im ) x ] − x n +1 [ x ( m − i ( α − β + µ ))] − n − Γ[ n + 1 , ( m − i ( α − β + µ )) x ] , (32) B = x n +1 [ ix ( α + β − µ − im )] − n − Γ[ n + 1 , i ( α + β − µ − im ) x ] − x n +1 [ x ( m − i ( α + β − µ ))] − n − Γ[ n + 1 , ( m − i ( α + β − µ )) x ] , (33) C = − (cid:26) x n +1 [ ix ( α + β + µ − im )] − n − Γ[ n + 1 , i ( α + β + µ − im ) x ] − x n +1 [ x ( m − i ( α + β + µ ))] − n − Γ[ n + 1 , ( m − i ( α + β + µ )) x ] (cid:27) . (34)We can express the above functions in the different form, such that A = x n +1 { [ E n i ( α − β + µ ) x ] − E n [( m − i ( α − β + µ )) x ] (35)Now provided Eqs. (32)—(34) we notice that terms with im if we consider a case we m = i reduce to one,while for cases where m = 0 all the terms with m vanishes. V. CONCLUSIONS
In this paper, we revisited the indefinite integral of the power of x which we multiplied with three sphericalBessel function and an exponential. In this study, we adopted the general and elementary method whichintroduced by [16], we extended this work by providing some limitation on spherical Bessel functions of thefirst kind with different order. This expressions allow for accurate computations, where the general andelementary method might not be computed smoothly.5 CKNOWLEDGMENTS
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