Asymptotic formulae for the Lommel and Bessel functions and their derivatives

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.


Summary
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.
Inverting the Laplace transform [15], we obtain the following solutions: and v F (t) = 1 2c sin 2ct sin q 2 . (3.5) Inverting the discrete Fourier transform in formulae (3.4) and (3.5), we get where J 2k is the Bessel function of the first kind.
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: This formula is valid for n 1 and is known as the Nicholson-type formula (see [8, p. 142] or [14, pp. 190 and 249]). Here, is the Airy function. We define Observe that it follows from (3.8) that the amplitude of J n (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 t → ∞ (or n → ∞). Note also that the size of the zone, where J n (ct) increases from zero to the first maximum, increases as t 1/3 (or n 1/3 ).
Return to our mechanical problem. Substituting (3.8) into (3.6), we obtain the desired asymptotic representation for the function u k (t)

Derivation of formula (2.1)
Using the Slepyan method [16] of combined asymptotic (t → ∞) 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 v k (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 v LF (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): is the Scorer function. Comparing (3.7) and (4.1), we get the following approximate representation of the Lommel function s 0,n for n 1 in terms of the Scorer function Gi that is similar to (3.8): Observe that the Lommel function s 0,n is defined for even values of n only (i.e. for n = 2k). It follows from (4.2) that the amplitude of s 0,n (ct) in the neighbourhood of the point ct = n decreases as t −1/3 (or n −1/3 ) as t → ∞ (or n → ∞). Note also the size of the zone, where s 0,n (ct) decreases from zero to the first minimum, increases as t 1/3 (or n 1/3 ). 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 s 0,n for n 1 in terms of the Scorer function Gi is similar to the following formula (11.11.17) in [5]: which gives an asymptotic expansion of the associated Anger-Weber function

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 t → ∞: Differentiating (4.2) with respect to time, we get Using (5.1) and assuming t → ∞, we obtain the following asymptotic representation for the first derivative s 0,n for n 1:  Similarly, we derive the following asymptotic representation for the first derivative J n for n 1: 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 t 1/3 (or n 1/3 ).     Let us find the values of n, for which the accuracy of the asymptotic representations is reasonable. Let max |J n (t)| denote the maximum of the modulus of the function J n (t) calculated in a neighbourhood of the point circled in figure 1. The expressions max |J n (t)|, max |s 0,n (t)|, max |s 0,n (t)|, max |F 1 (n, t)|, max |F 2 (n, t)|, max |F 3 (n, t)| and max |F 4 (n, t)| are defined similarly with the help of figures 2-4. Introduce the notation

Numerical experiments
In tables 1 and 2, we give the values of the relative errors δ 1 , δ 2 , δ 3 and δ 4 , calculated for the values of n, specified in figures 1-4. From tables 1 and 2 it follows that the relative errors δ 1 , δ 2 , δ 3 and δ 4 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.
8. Comparison of the results for the function J ν (ct) described in [6] and this paper Both [6] and this paper focus on the study of the behaviour of the Bessel functions J ν (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 [6], where the authors are looking for the values of J ν (ct) for large values of ν and ct. In particular, Jentschura & Lötstedt [6] present, via apparently heroic numerical efforts, the following value J ν (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 J ν (ct) = 0.002614463961695188. Hence, eight significant figures are in agreement with the exact numerical result given in [6].
In figure 5, we plot the graph of the function F 1 (ν,t) for ν = 2000000.2, which, according to formula (3.8), is asymptotically equivalent to the function J ν (ct). From fig. 4 in [6] 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 J n (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 [6] do not appear in our paper, because we use another method.
9. Comparison of the results for Bessel functions described in [9,12] and this paper