Asymptotics of some integrals involving modified Bessel and hyper-Bessel functions
aa r X i v : . [ m a t h . C A ] F e b Asymptotics of some integrals involving modified Besseland hyper-Bessel functions
R. B. Paris
Division of Computing and Mathematics,Abertay University, Dundee DD1 1HG, UK
Abstract
We investigate the asymptotic expansion of integrals analogous to Ball’s integral Z ∞ (cid:18) Γ(1 + ν ) | J ν ( x ) | ( x/ ν (cid:19) n dx for large n in which the Bessel function J ν ( x ) is replaced by the modified Bessel functions I ν ( x ) and K ν ( x ) together with appropriate exponential factors e ∓ x , respectively.The above integral with J ν ( x ) replaced by a hyper-Bessel function of the type recentlydiscussed in Aktas et al. [The Ramanujan J., 2019] and taken over a finite interval deter-mined by the first positive zero of the function is also considered for n → ∞ . We give theleading asymptotic behaviour of the hyper-Bessel function for x → + ∞ in an appendix.Numerical examples are given to illustrate the accuracy of the various expansions obtained. Mathematics subject classification (2010):
Keywords:
Ball’s integral, modified Bessel functions, hyper-Bessel function, asymptoticexpansions
1. Introduction
The asymptotic expansion of Ball’s integral [2] for large positive values of n Z ∞ (cid:18) Γ(1 + ν ) | J ν ( x ) | ( x/ ν (cid:19) n x ν − dx, n ≥ , ν ≥ , (1.1)where J ν ( x ) is the Bessel function of the first kind, was investigated by Kerman et al. in [4].In a recent note [6] it was shown that the above integral could be replaced by Z j ν, (cid:18) Γ(1 + ν ) J ν ( x )( x/ ν (cid:19) n x ν − dx (1.2)1 R. B. Paris to within exponentially small terms when n is large, where j ν, is the first positive zero of J ν ( x ).The large- n expansion was found in the form2 ν − (1 + ν ) ν Γ( ν ) ∞ X k =0 ( − ) k c k n k + ν ( n → ∞ ) , where the leading coefficient c = 1; explicit values of c k for k ≤ k ≤ I ν ( x ) and K ν ( x ) and also the hyper-Bessel function J σ ,...,σ m ( x ) definedin Section 4. In Section 2, we consider the expansion of the integral I n = Z ∞ (cid:18) e − x Γ(1 + ν ) I ν ( x )( x/ ν (cid:19) n dx ( ν > − )for n → ∞ . Since I ν ( x ) ∼ e x / √ πx as x → + ∞ , it is necessary to add the factor e − x tocancel the exponential growth of I ν ( x ). The integrand is then of O( x − ν − / ) as x → ∞ so that I n converges for ν > − . In Section 3, a similar process is adopted to determine the large- n expansion of K n = Z ∞ (cid:18) e x Γ( ν ) ( x/ ν K ν ( x ) (cid:19) − n dx ( ν > ) , where n >
0. From the small and large argument behaviours K ν ( x ) ∼
12 Γ( ν )( x/ − ν ( x → , K ν ( x ) ∼ r π x e − x ( x → + ∞ ) , it is seen that the integrand has the value unity at x = 0 and is O( x ν − / ) as x → ∞ , therebynecessitating the condition ν > for convergence.In the final section, we consider an integral analogous to (1.1) in which the classical Besselfunction J ν ( x ) is replaced by a hyper-Bessel function. A significant difference, however, is thatthe interval of integration cannot be taken as [0 , ∞ ) on account of the asymptotic structure of theparticular hyper-Bessel function under consideration. It is necessary to take a finite integrationinterval analogous to that in (1.2) determined by the first positive zero of the function.
2. An integral involving the modified Bessel function I ν ( x )The first integral we consider is the analogue of (1.1) where the function | J ν ( x ) | is replaced bythe modified Bessel function I ν ( x ), viz. I n = Z ∞ (cid:18) e − x Γ(1 + ν ) I ν ( x )( x/ ν (cid:19) n dx ( ν > − ) , (2.1)where n > x = 0 and is amonotonically decreasing function. This can be seen by letting y ( x ) = e − x ( x ) − ν I ν ( x ), whence y ′ ( x ) = − e − x ( x ) − ν { I ν ( x ) − I ν +1 ( x ) } < , since for x > ν > − it is known that I ν ( x ) > I ν +1 ( x ) (see [3]). symptotic expansion of an integral ψ ( x ) = x − log (cid:18) Γ(1 + ν ) I ν ( x )( x/ ν (cid:19) = x − log ∞ X k =0 ( x/ k (1 + ν ) k k ! . Since ψ (0) = 0 and ψ ( ∞ ) = ∞ , the change of variable τ = ψ ( x ) yields I n = Z ∞ e − nτ dxdτ dτ, (2.2)where τ = ψ ( x ) = x − x ν ) + x ν ) (2 + ν ) − x ν ) (2 + ν )(3 + ν ) + · · · valid in x < j ν, . Inversion of this series with the help of Mathematica then produces x = τ + τ ν ) + τ ν ) + (8 + 3 ν ) τ ν ) (2 + ν ) + (8 + ν ) τ ν ) (2 + ν ) + · · · , whence we obtain the expansion dxdτ = ∞ X k =0 A k τ k ( τ < τ ) . (2.3)The first few coefficients A k are A = 1 , A = 12(1 + ν ) , A = 38(1 + ν ) ,A = 8 + 3 ν ν ) (2 + ν ) , A = 5(8 + ν )128(1 + ν ) (2 + ν ) , A = 142 + 11 ν − ν ν ) (2 + ν )(3 + ν ) ,A = 7(272 + 8 ν − ν − ν )3072(1 + ν ) (2 + ν ) (3 + ν ) , A = 2656 − ν − ν − ν + 19 ν ν ) (2 + ν ) (3 + ν )(4 + ν ) ,A = 3(6816 − ν − ν + 1025 ν + 119 ν )32768(1 + ν ) (2 + ν ) (3 + ν )(4 + ν ) , . . . . The expansion (2.3) holds in τ < τ , where τ = | ψ ( ± ij ′ ν, ) | since x = ± ij ′ ν, is the nearestpoint in the mapping x τ where dx/dτ is singular. The quantity j ′ ν, is the second positivezero of J ′ ν ( x ).From (2.2) and (2.3), straightforward integration yields I n ∼ ∞ X k =0 A k Z ∞ e − nτ τ k dτ = ∞ X k =0 k ! A k n k +1 . Then we obtain: The circle of convergence is determined by the nearest singularity of ψ ( x ) that occurs at x = ± ij ν, , since I ν ( z ) on the imaginary z -axis behaves like J ν ( | z | ). R. B. Paris
Table 1:
Values of the absolute relative error in the computation of I n when n = 100 for different values ofthe order ν and truncation index k using the expansion in (2.4). k ν = 0 ν = 3 / ν = 10 5 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − Theorem 1 . For ν > − and n → ∞ the following expansion holds I n ∼ n (cid:26) ν ) n + 34(1 + ν ) n + 3(8 + 3 ν )8(1 + ν ) (2 + ν ) n + 15(8 + ν )16(1 + ν ) (2 + ν ) n + 15(142 + 11 ν − ν )32(1 + ν ) (2 + ν )(3 + ν ) n + 105(272 + 8 ν − ν − ν )64(1 + ν ) (2 + ν ) (3 + ν ) n + · · · (cid:27) . (2.4)In Table 1 we show values of the absolute relative error in the computation of I n againsttruncation index for different values of ν .A similar integral is given byˆ I n = Z ∞ (cid:18) Γ(1 + ν ) I ν ( x )( x/ ν (cid:19) − n dx ( ν ≥ , where n >
0. With the standard substitution ψ ( x ) = log (Γ(1 + ν ) I ν ( x ) / ( x/ ν ) and change ofvariable τ = ψ ( x ), we find τ = x ν ) − x ν ) (2 + ν ) + x ν ) (2 + ν )(3 + ν ) + · · · , which upon inversion yields the expansion12 √ ν dxdτ = ∞ X k =0 ˆ A k τ k ( τ < τ ) . The first few coefficients ˆ A k areˆ A = 1 , ˆ A = 34(2 + ν ) , ˆ A = − ν )96(2 + ν ) (3 + ν ) , ˆ A = − ν − ν )128(2 + ν ) (3 + ν )(4 + ν ) , ˆ A = 75404 + 262439 ν + 182205 ν − ν − ν ν ) (3 + ν ) (4 + ν )(5 + ν ) , ˆ A = 11(127864 − ν − ν − ν + 8605 ν + 48361 ν ν ) (3 + ν ) (4 + ν )(5 + ν )(6 + ν ) . symptotic expansion of an integral x = 0. Routine evaluation thenproduces the asymptotic expansionˆ I n ∼ (1 + ν ) / ∞ X k =0 ˆ A k Γ( k + ) n k +1 / ( n → ∞ ) . (2.5)
3. An integral involving the modified Bessel function K ν ( x )In this section we consider an analogous integral to (2.1) with the modified Bessel function ofthe second kind, also known as the Macdonald function K ν ( x ), namely K n = Z ∞ (cid:18) e x Γ( ν ) ( x/ ν K ν ( x ) (cid:19) − n dx ( ν > ) , (3.1)where again n > ν is not an integer. The quantity in brackets in (3.1) is monotonically increasing for ν > , sincewith y ( x ) = x ν e x K ν ( x ) we have y ′ ( x ) = x ν e x { K ν ( x ) − K ν − ( x ) } > . The fact that the quantity in braces is positive follows from the result [5, (10.32.9)] K ν ( x ) − K ν − ( x ) = Z ∞ e − x cosh t { cosh νt − cosh( ν − t } dt > , ν > . With the changes of variable ψ ( x ) = x + log { x/ ν K ν ( x ) / Γ( ν ) } and τ = ψ ( x ), so that x ∈ [0 , ∞ ) maps to τ ∈ [0 , ∞ ) when ν > , we have K n = Z ∞ e − nψ ( x ) dt = Z ∞ e − nτ dxdτ dτ. To proceed further we require the inversion of the mapping τ x . To do this in generalterms is complicated so we prefer to carry out this procedure for specific values of ν . From thedefinition K ν ( x ) = π πν { I − ν ( x ) − I ν ( x ) } , it is seen that, provided ν = 1 , , . . . ,2Γ( ν ) ( x/ ν K ν ( x ) = 1 + x − ν ) + x − ν )(2 − ν ) + · · · + (cid:18) x (cid:19) ν Γ( − ν )Γ( ν ) (cid:18) x ν ) + x ν )(2 + ν ) + · · · (cid:19) . Thus when ν = 2 /
3, for example, we find τ = x + g (cid:18) x (cid:19) / + 34 x − g (cid:18) x (cid:19) / − g (cid:18) x (cid:19) / + (cid:18) − g (cid:19) x + · · · , R. B. Paris where g := Γ( − / / Γ(2 / x = τ − gτ / / + g τ / · / + ( g −
34 ) τ + 5 g · / (13 g + 162) τ / + · · · valid in a neighbourhood of τ = 0.However, it is found much easier to deal with this inversion process with the coefficientsexpressed in numerical form rather than in algebraic form. In this manner we obtain afterdifferentiation with respect to τ dxdτ = ∞ X k =0 B k ( ) τ k/ , where the coefficients B k ( ) are listed in Table 2 for k ≤
6. Then K n ∼ ∞ X k =0 B k ( ) n k/ Z ∞ e − w w k/ dw = ∞ X k =0 B k ( ) n k/ Γ( k/ ν = ) (3.2)as n → ∞ .We present the series expansion for dx/dτ for the two cases ν = 6 / ν = 4 /
3. Thecoefficients B k ( ν ) are computed using the two lines of Mathematica commands below and aregiven in Table 2. f := ( x / ) ν BesselK [ ν, x ] / Gamma ( ν ); S = N [ Series [ x + Log [ f ] , { x , , m } ] , ] D [ InverseSeries [ S , τ ] , τ ]where m is an integer that determines how far we carry out the expansion process. Theasymptotic expansion of K n is then computed as above. When ν = 6 /
5, we haveTable 2:
The coefficients B k ( ν ) for 1 ≤ k ≤ B ( ν ) = 1) for different values of the order ν . k ν = 2 / ν = 6 / ν = 4 /
31 +1 . . . . − . − . . . . . − . − . . . . . . . dxdτ = 1 + B ( ) τ + B ( ) τ / + B ( ) τ + B ( ) τ / + B ( ) τ / + B ( ) τ + · · · and the asymptotic expansion K n ∼ n (cid:26) B ( ) n + B ( ) n / Γ( )+ 2 B ( ) n + B ( ) n / Γ( )+ B ( ) n / Γ( )+ 6 B ( ) n + · · · (cid:27) ; (3.3) symptotic expansion of an integral Values of the absolute relative error in the computation of K n when n = 100 for different values ofthe order ν and truncation index k using the expansions in (3.2)–(3.4). k ν = 2 / ν = 6 / ν = 4 /
30 3 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − when ν = 4 /
3, we have dxdτ = 1 + B ( ) τ + B ( ) τ / + B ( ) τ + B ( ) τ / + B ( ) τ + B ( ) τ / + · · · and the asymptotic expansion K n ∼ n (cid:26) B ( ) n + B ( ) n / Γ( )+ 2 B ( ) n + B ( ) n / Γ( )+ 6 B ( ) n + B ( ) n / Γ( )+ · · · (cid:27) (3.4)as n → ∞ .In Table 3 we present the absolute relative error in the computation of the integral K n for the three values of the order ν using different truncations of the expansions in (3.2)–(3.4).Because these expansions involve inverse fractional powers of n , it is seen that the rate of decayof the relative error with increasing truncation index is rather slow.
4. An integral involving the hyper-Bessel function
The particular hyper-Bessel function we shall use to replace the Bessel function J ν ( x ) in theintegral (1.1) is defined by [1] J σ ,...,σ m ( x ) = ( x/ ( m + 1)) σ + ··· + σ m Q mj =1 Γ( σ j + 1) F m (cid:18) −− σ +1 , . . . , σ m +1 ; − (cid:18) xm + 1 (cid:19) m +1 (cid:19) . (4.1)Here F m denotes the generalised hypergeometric function with m denominator parameters F m ( z ) = ∞ X k =0 σ +1) k . . . ( σ m +1) k z k k ! ( σ j > − , ≤ j ≤ m )and ( a ) k = Γ( a + k ) / Γ( a ) = a ( a + 1) . . . ( a + k −
1) is Pochhammer’s symbol for the risingfactorial. When m = 1, σ = ν , the definition (4.1) reduces to the classical Bessel function J ν ( x ), viz. J ν ( x ) = ( x ) ν Γ(1 + ν ) F (cid:18) −− ν ; − (cid:18) x (cid:19) (cid:19) = ( x ) ν Γ(1 + ν ) ∞ X k =0 ( − ) k ( x ) k (1 + ν ) k k ! . R. B. Paris
Before we can formulate an integral analogous to that in (1.1), it is necessary to considerthe basic properties and asymptotic behaviour of J σ ,...,σ m ( x ). In what follows we write ~σ = { σ , σ , . . . , σ m } and define the quantity µ k := m Y j =1 ( σ j + k ) − . (4.2)Both J σ ,...,σ m ( x ) (for m ≥
2) and J ν ( x ) have an infinite number of zeros on [0 , ∞ ). If the firstsuch zero of J σ ,...,σ m ( x ) is denoted by j ~σ, , it was established in [1, Theorem 4] that( m + 1) µ − / ( m +1)1 < j ~σ, < ( m + 1)( µ − µ ) − / ( m +1) . However, although these two functions possess similar zero properties, their asymptotic struc-ture is quite different. From (A.2) in the appendix, the leading asymptotic behaviour of J σ ,...,σ m ( x ) is J σ ,...,σ m ( x ) ∼ π ) − m/ ( m + 1) / (cid:18) xm + 1 (cid:19) − m/ e x cos π/ ( m +1) cos (cid:26) x sin πm + 1 + πϑm + 1 (cid:27) as x → + ∞ , where ϑ is defined in (A.1).Thus, when m ≥ x → + ∞ and conse-quently the integral (1.1) modified to incorporate the hyper-Bessel function cannot be takenover an infinite range. Accordingly, we consider the asymptotic expansion of the integral overthe finite interval [0 , j ~σ, ] viz. J n = Z j ~σ, (cid:18) J σ ,...,σ m ( x ) (cid:19) n dx, (4.3)where the normalised hyper-Bessel function J σ ,...,σ m ( x ) is defined by J σ ,...,σ m ( x ) = Q mj =1 Γ( σ j + 1)( x/ ( m + 1)) σ + ··· + σ m J σ ,...,σ m ( x )= F m (cid:18) −− σ +1 , . . . , σ m +1 ; − (cid:18) xm + 1 (cid:19) m +1 (cid:19) . (4.4)In the case m = 1, σ = ν , the integral (4.3) reduces to that in (1.2) when the factor x ν − isomitted. An equivalent factor could be added to (4.3), but we choose not to do this in order toavoid the appearance of additional parameters. Let p = m + 1 and set ψ ( x ) = − log F m ( − ( x/p ) p ), so that the integral (4.3) becomes J n = Z j ~σ, e − nψ ( x ) dx. Since ψ (0) = 0 and ψ ( j ~σ, ) = ∞ , then with the change of variable τ p = ψ ( x ) we have J n = Z ∞ e − nτ p dxdτ dτ, (4.5) symptotic expansion of an integral τ p = ψ ( x ) = − log (cid:18) − ( x/p ) p µ
1! + ( x/p ) p µ µ − ( x/p ) p µ µ µ
3! + · · · (cid:19) = µ ( x/p ) p + 12 µ ( µ − µ )( x/p ) p + 16 (2 µ − µ µ + µ µ µ )( x/p ) p + · · · valid in x < p ( j ~σ, ) /p . Inversion of this last expression with the aid of Mathematica yields( x/p ) p = τ p µ + ( µ − µ )2 µ τ p + ( µ − µ µ + 3 µ − µ µ )6 µ τ p + · · · whence x = pτµ /p (cid:26) µ − µ )2 µ τ p + ( µ − µ µ + 3 µ − µ µ )6 µ τ p + · · · (cid:27) /p . This then leads to an expansion for dx/dτ given by dxdτ = pµ /p ∞ X k =0 ( − ) k A k p k ( kp + 1) τ kp ( τ < τ ) , (4.6)where A = 1 , A = 12 (1 − γ ) ,A = 124 (cid:26) p + 3 − p + 1) γ + 3(3 p + 1) γ − pγ γ (cid:27) ,A = 148 (cid:26) p + 1 − ( p + 1)(4 p + 3) γ + 3(2 p + 1)(3 p + 1) γ − p (2 p + 1) γ γ (cid:27) ,A = 15760 (cid:26) p +5 p − p − p +1) (2 p +1) γ + 10(3+13 p +14 p ) γ (3(3 p +1) γ − pγ ) − p +1) γ ((1+9 p +20 p ) γ − p (4 p +1) γ γ + 2 p γ γ ) + γ { p +107 p +210 p ) γ − p (1+11 p +30 p ) γ γ + 40 p (5 p +1) γ γ (2 γ +3 γ ) − p γ γ γ } (cid:27) (4.7)with γ k := µ k µ = m Y j =1 σ j + 1 σ j + k ( k ≥ . The expansion (4.6) holds in τ < τ , where τ p = | ψ ( j ′ ~σ, ) | since x = j ′ ~σ, is the nearest pointin the mapping x τ where dx/dτ is singular. The quantity j ′ ~σ, is the second positive zeroof the derivative of the hyper-Bessel function, which interlaces with the zeros j ~σ, and j ~σ, [1,Theorem 5].Then from (4.5) and (4.6), we obtain J n ∼ pµ /p ∞ X k =0 ( − ) k A k p k ( kp + 1) Z ∞ e − nτ p τ kp dτ = 1( nµ ) /p ∞ X k =0 ( − ) k A k ( np ) k ( kp + 1) Z ∞ e − w w k +1 /p − dw. R. B. Paris
Evaluation of the integral as a gamma function then produces
Theorem 2 . With p = m + 1 and µ = Q mj =1 ( σ j + 1) − , we have the expansion J n ∼ p ( nµ ) /p ∞ X k =0 ( − ) k A k ( np ) k Γ( k + 1 p + 1) (4.8) as n → ∞ , where the first five coefficients A k are listed in (4.7). In Table 4 we show values of the absolute relative error against truncation index k in theevaluation of J n in the case m = 2 using (4.8). The value of J n was obtained by a high-precision numerical integration procedure with the first zero when σ = , σ = given by j ~σ, . = 4 . Values of the absolute relative error in the computation of J n against truncation index k when m = 2and σ = 1 / σ = 3 /
4. . k n = 20 n = 50 n = 1000 6 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − Appendix: The asymptotic behaviour of J σ ,...,σ m ( x ) for x → + ∞ The hypergeometric-type function f ( z ) = ∞ X k =0 z k Q mj =1 Γ( σ j + 1 + k ) k !is associated with the parameters (see [7, § κ = m + 1 , h = 1 , ϑ = − m + m X j =1 σ j . (A.1)Define the formal exponential asymptotic sum E ( z ) := Z ϑ e Z ∞ X k =0 A k Z − k , Z := κ ( hz ) /κ , Here we follow the notation of [7, § κ the quantity m + 1, although in Section 4 thisquantity was denoted by p symptotic expansion of an integral A k are constants independent of z with A = (2 π ) − m/ κ − − ϑ . Then, when κ > m ≥
2) the asymptotic expansion of f ( z ) is given by [7, § f ( z ) ∼ P X r = − P E ( ze πir ) ( | z | → ∞ , | arg z | ≤ π ) , where P is chosen such that 2 P + 1 is the smallest odd integer satisfying 2 P + 1 > κ .For the hypergeometric function appearing in (4.1) we have Z = xe πi/κ . Then, when m ≥ F m ( − ( x/κ ) κ ) ∼ , m Y j =1 Γ( σ j + 1) P X r = − P E ( xe (2 r +1) πi ) ( x → + ∞ ) . The dominant exponential sums correspond to r = 0 and r = −
1, whence we obtain o F m ( − ( x/κ ) κ ) ∼ m Y j =1 Γ( σ j + 1) { E ( xe πi ) + E ( xe − πi ) }∼ A m Y j =1 Γ( σ j + 1) x ϑ e x cos π/κ cos (cid:26) x sin πκ + πϑκ (cid:27) . Hence the laeding behaviour of J σ ,...,σ m ( x ) is given by J σ ,...,σ m ( x ) ∼ π ) − m/ ( m + 1) / (cid:18) xm + 1 (cid:19) − m/ e x cos π/ ( m +1) cos (cid:26) x sin πm + 1 + πϑm + 1 (cid:27) (A.2)as x → + ∞ ; see also [8].When m = 1, σ = ν , the approximation (A.2) reduces to the well-known leading behaviourof the classical Bessel function [5, (10.17.3)] J ν ( x ) ∼ r πx cos (cid:26) x − πν − π (cid:27) ( x → + ∞ ) . However, when m ≥ J σ ,...,σ m ( x ) is oscillatory with an exponentiallygrowing amplitude as x → + ∞ , and so is of a completely different asymptotic structure to thatof J ν ( x ). References [1] I. Aktas, A. Baricz and S. Singh, Geometric and monotonic properties of hyper-Besselfunction, The Ramanujan Journal DOI.org/10.1007/s11139-018-9195-9, 2019.[2] K. Ball, Cube slicing in R n , Proc. Amer. Math. Soc. (1986) 465–473.[3] A.L. Jones, An extension of an inequality involving modified Bessel functions, J. Math.Phys. (1968) 220–221.2 R. B. Paris [4] R. Kerman, R. Ol’Hava and S. Spektor, An asymptotically sharp form of Ball’s integralinequality, Proc. Amer. Math. Soc. (2015) 3839–3846.[5] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark (eds.),
NIST Handbook of Math-ematical Functions , Cambridge University Press, Cambridge, 2010.[6] R.B. Paris, Asymptotics of some generalised sine-integrals, arXiv:2011.05156 (2020).[7] R.B. Paris and D. Kaminski,
Asymptotics and Mellin-Barnes Integrals , Cambridge Uni-versity Press, Cambridge, 2011.[8] R.B. Paris and A.D. Wood, Results old and new on the hyper-Bessel equation, Proc. Roy.Soc. Edinburgh106A