We derive new approximate representations of the Lommel functions in terms of the Scorer function and approximate representations of the first derivative of the Lommel functions in terms of the derivative of the Scorer function. Using the same method, we obtain previously known approximate representations of the Nicholson type for Bessel functions and their first derivatives. We study also for what values of the parameters our representations have reasonable accuracy.
2. The main results
Solving problems in mechanics of discrete media [1,2], we derived the following asymptotic formulae that are not available in the literature [3–14], but probably are of general interest 2.1 and 2.2 where s0,n is the Lommel function, Gi is the Scorer function, c and t are positive real numbers, and n and k are positive integers. Prime denotes the derivative with respect to the argument. Each of formulae (2.1) and (2.2) holds true provided that n≫1 or ct≫1.
3. Motivation of our research
In [1,2], the method of integral transformations was used to solve two-dimensional problems of wave propagation in discrete periodic media. In the process of solving those problems, it was necessary to find the functions u and v, provided that their Laplace–Fourier transforms are given by the formulae 3.1 and 3.2 where c is the velocity of the propagation of disturbances, the superscript L denotes the Laplace transform (of parameter p) with respect to time t and the superscript F denotes the discrete Fourier transform (of parameter q) with respect to k:
Formally, the solution to the problem can be written as follows: 3.3 A similar formula holds true for uk(t).
Inverting the Laplace transform , we obtain the following solutions: 3.4 and 3.5
In the problems of mechanics [1,2], it is important to be able to evaluate the behaviour of perturbations in the vicinity of the quasi-front k=ct (quasi-front is a zone, where perturbations change from zero to maximum). Being motivated by this problem, we look for asymptotic representations of the Bessel and Lommel functions for k≫1.
In order to evaluate the behaviour of function (3.6), we use the following asymptotic representation of the Bessel function: 3.8 This formula is valid for n≫1 and is known as the Nicholson-type formula (see , p. 142 or , pp. 190 and 249). Here, is the Airy function.
We define 3.9
Observe that it follows from (3.8) that the amplitude of Jn(ct) in the neighbourhood of the point ct=n (according to (3.9), this point can also be written as z=0) decreases as t−1/3 (or n−1/3) as (or ). Note also that the size of the zone, where Jn(ct) increases from zero to the first maximum, increases as t1/3 (or n1/3).
4. Derivation of formula (2.1)
Using the Slepyan method  of combined asymptotic () inversion of the integral Laplace–Fourier transforms of long-wave disturbances in the vicinity of the ray x=ct, we can find the asymptotic behaviour for vk(t) that is similar to (3.10).
Applying the Slepyan method, we make the substitution p=s+iq(c+c′) and k=(c+c′)t, where c′→0 and defines the vicinity of the ray k=ct, in the inner integral (3.3). This yields We expand the numerator and denominator of the function vLF(s+iq(c+c′),q) in the Taylor series in a small neighbourhood of the point q=0 as s→0 and c′→0: where ε>0 is small enough. Successively integrating and taking into account, that c′=(k−ct)/t, we obtain the following asymptotic formula that is similar to (3.10): 4.1 where is the Scorer function.
Comparing (3.7) and (4.1), we get the following approximate representation of the Lommel function s0,n for n≫1 in terms of the Scorer function Gi that is similar to (3.8): 4.2 Observe that the Lommel function s0,n is defined for even values of n only (i.e. for n=2k). It follows from (4.2) that the amplitude of s0,n(ct) in the neighbourhood of the point ct=n decreases as t−1/3 (or n−1/3) as (or ). Note also the size of the zone, where s0,n(ct) decreases from zero to the first minimum, increases as t1/3 (or n1/3).
Finally, note that above we derived formula (3.10) from formula (3.8) of the Nicholson type solely for the sake of brevity. In fact, (3.10) can be obtained by using the Slepyan method of combined asymptotic inversion of the integral Laplace–Fourier transforms, just as we got above formula (4.1).
The approximate representation (4.2) of the Lommel function s0,n for n≫1 in terms of the Scorer function Gi is similar to the following formula (11.11.17) in : which gives an asymptotic expansion of the associated Anger–Weber function for in terms of the Scorer function .
5. Derivation of formula (2.2)
Observe that, in the following formula, the term (n−ct)/(3t) can be neglected in a neighbourhood of the point n=ct as : 5.1
Similarly, we derive the following asymptotic representation for the first derivative J′n for n≫1: 5.3 From (5.2) and (5.3), we conclude that, in a neighbourhood of the point n=ct, the functions J′n(ct) and s′0,n(ct) decrease as t−2/3 (or n−2/3) when t (or n) increases. Note also the size of the zone, where J′n(ct) and s′0,n(ct) varies from zero to the first extremum, increases as t1/3 (or n1/3).
6. Numerical experiments
In order to determine the accuracy of the asymptotic representations (2.1), (2.2), (3.8) and (5.3), we plot the graphs of the functions that appeared in (2.1), (2.2), (3.8) and (5.3). All figures given below are plotted for the case c=1.
In figures 1–4, we present the plots of the functions of the variable t, which appear in the left- and right-hand sides of formulae (2.1), (2.2), (3.8) and (5.3) for various values of n. The step of the variable t is equal to 0.1. The dashed vertical lines correspond to the coordinates t*=n or z=0.
Let us find the values of n, for which the accuracy of the asymptotic representations is reasonable. Let denote the maximum of the modulus of the function Jn(t) calculated in a neighbourhood of the point circled in figure 1. The expressions , , , , , and are defined similarly with the help of figures 2–4. Introduce the notation
In tables 1 and 2, we give the values of the relative errors δ1, δ2, δ3 and δ4, calculated for the values of n, specified in figures 1–4. From tables 1 and 2 it follows that the relative errors δ1, δ2, δ3 and δ4 monotonically decrease as n increases.
From figures 1–4, for every pair of the functions, we see that, as n increases, the matching of the amplitudes of all local extrema get better, not only of the circled ones. From figures 1–4, we see also that, for every pair of the functions, the matching of the oscillation frequencies get better as n increases. For each pair of functions, the best approximation is achieved in a neighbourhood of the point n=t (or z=0). Note that, in the problems of mechanics, this neighbourhood corresponds to the quasi-front of the propagating wave.
Besides, we compared the plots of the functions (treated as functions of the variable n), which appear in the left- and right-hand sides of formulae (2.1), (2.2), (3.8) and (5.3), for various fixed values of t. That comparison showed that, for every pair of the functions, the plots agree best of all in a neighbourhood of the point n=t, that formulae (2.1), (2.2), (3.8) and (5.3) can be used starting from t≈6, and that the agreement gets better as t increases. For every pair of the functions, the distinction appears at a sufficiently large distance from the point n=t. Moreover, this distinction appears in the frequency of oscillations only. The maximal amplitudes of oscillations do not differ very much even at large distances from the point n=t.
— the amplitudes of s0,n(ct) and Jn(ct) in the neighbourhood of the point ct=n decrease as t−1/3 (or n−1/3) as (or ;
— the amplitudes of s′0,n(ct) and J′n(ct) in the neighbourhood of the point ct=n decrease as t−2/3 (or n−2/3) as (or ;
— the size of the zone, where s0,n(ct), Jn(ct), s′0,n(ct) and J′n(ct) varies from zero to the first extremum, increases as t1/3 (or n1/3) as (or ; and
8. Comparison of the results for the function Jν(ct) described in  and this paper
Both  and this paper focus on the study of the behaviour of the Bessel functions Jν(ct) when the argument ct and order ν are nearly equal. However, in this article, the emphasis is on the smallest values of ν=n and ct, for which the asymptotic formula (3.8) has reasonable accuracy for solving the problems of discrete periodic media [1,2]. This differs in our paper from , where the authors are looking for the values of Jν(ct) for large values of ν and ct. In particular, Jentschura & Lötstedt  present, via apparently heroic numerical efforts, the following value Jν(ct)=0.002614463954691926 for ν=5000000.2 and ct=5000000.1. In this formula, the values of the argument ct and order ν of the Bessel function are the largest ones for which we know the value of the Bessel function from the scientific literature. For ν=5000000.2 and ct=5000000.1, the asymptotic formula (3.8) yields Jν(ct)=0.002614463961695188. Hence, eight significant figures are in agreement with the exact numerical result given in .
In figure 5, we plot the graph of the function F1(ν,t) for ν=2000000.2, which, according to formula (3.8), is asymptotically equivalent to the function Jν(ct). From fig. 4 in  and figure 5, we see that the behaviour of the plots is the same.
Numerical experiments, discussed in this section, show that formula (3.8) is valid not only for the Bessel function Jn(ct) of a positive integer order n, but also for the case where the order is a positive real number.
The problems associated with the confluence of the saddle points in the cusp region explained in  do not appear in our paper, because we use another method.
Note that formulae (3.8) and (5.3) present the leading terms of the following complete asymptotic expansions of Bessel functions and their first derivatives (see pp. 281 and 287 in ): 9.1 and 9.2 Here, ξ=(ν−y)(y/2)−1/3, , |arg y|≤π−ε, y−ν=O(y1/3), ν is a positive real number, the coefficients Pk(ξ), Qk(ξ), , are polynomials in ξ described in  (for example, P0(ξ)=1, Q0(ξ)=0, and .
Besides formulae (3.8) and (5.3), (8.1) and (8.2), there are other asymptotic expansions of Bessel functions and their first derivatives for large values of the order. We mean the following formulae (10.19.8) and (10.19.12) in  (the same formulae are available on p. 414 in ): 9.3 and 9.4 where a is a fixed complex number; ν is a complex number, such that and its argument satisfies the inequality with some δ>0; the coefficients Pk(a), Qk(a), Rk(a) and Sk(a) are polynomials in a, in particular, P0(a)=1, Q0(a)=3/10a2, R0(a)=1 and S0(a)=3/5a3−1/5.
At first glance, formulae (8.1) and (8.2) are similar to formulae (8.3) and (8.4). But in fact, they differ radically, because in (8.3) and (8.4) the expansion is carried out in powers of the order ν of the functions, whereas in formulae (8.1) and (8.2) the expansion is carried out in the powers of the argument of the functions.
The author would like to thank the anonymous referees for their helpful comments.
- Received July 14, 2014.
- Accepted September 10, 2014.
© 2014 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.