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Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents.

It is important to note that these asymptotic expansions are not uniform with respect to ν; we have to require ν 2 = o(|z|) in order to satisfy Poincaré's definition. There exist other types of large-z expansions which are valid under the weaker condition ν = o(|z|) (see, for instance, [3,12] or [16]); however these expansions do not lend themselves to treatment with our methods.
The main aim of the present paper is to derive new error bounds for the asymptotic expansions (1)- (13) and their re-expanded versions. Due to the various relations between the Hankel and Bessel functions, it is enough to study the remainder terms of the asymptotic expansions (5), (6), (10) and (11) (see Appendix A for more details). Thus, for any nonnegative integers N and M, we write K ν (z) = π 2z 1 2 e −z N−1 n=0 a n (ν) z n + R (K) N (z, ν) , J ν (z) = 2 πz 1 2 cos ω Throughout this paper, if not stated otherwise, empty sums are taken to be zero. The derivations of the estimates for R (K) N (z, ν), R (K ) N (z, ν), R (J ) 2N (z, ν), R (J ) 2M+1 (z, ν), R (J ) 2N (z, ν) and R (J ) 2M+1 (z, ν) are based on new integral representations of these remainder terms. In expressing these new integral representations, we shall employ the following notation (w = 0): is the incomplete gamma function. The functions Λ p (w) and Π p (w) were originally introduced by Dingle [8], [10, pp. 406-407 and p. 415] and, following his convention, we refer to them as basic terminants (but note that Dingle's notation slightly differs from ours, e.g., Λ p−1 (w) is used for our Λ p (w)). These basic terminants are multivalued functions of their argument w and, when the argument is fixed, are entire functions of their order p. We shall also use the concept of the regularized hypergeometric function F(a, b; c; w) which is defined by the power series expansion w n for |w| < 1 and by analytic continuation elsewhere [24, §15.2]. The parameters a, b and c of this function can take arbitrary complex values.
Analogous expressions for the remainder terms R (K ) N (z, ν) and R (J ) N (z, ν) can be written down by applying Theorems 1.1 and 1.2 together with the functional equations and These functional equations follow directly from the connection formulae −2K ν (z) = K ν+1 (z) + K ν−1 (z) and −2J ν (z) = J ν+1 (z) − J ν−1 (z) (see, e.g., [24, 10.6.E1 and 10.29.E1]). It was shown by Boyd [4] that for any non-negative integer N , the remainder term R (K) N (z, ν) can be expressed as provided that |arg z| < π and | (ν)| < N + 1 2 . This representation of R (K) N (z, ν) is the central tool in Boyd's asymptotic analysis of the modified Bessel function K ν (z), in particular, in the derivation of his bounds for R (K) N (z, ν). In the following two theorems, we present results for the remainders R (J ) N (z, ν), R (K ) N (z, ν) and R (J ) N (z, ν) analogous to (22). In our analysis, the purpose of these representations is not to find bounds for the remainders themselves but to prove error bounds for the re-expansions of these remainders.

Theorem 1.3 Let N be a non-negative integer and let ν be an arbitrary complex number such that
provided |arg z| < π 2 .
The subsequent two theorems provide bounds for the remainders R (K) N (z, ν), R (J ) N (z, ν), R (K ) N (z, ν) and R (J ) N (z, ν) when ν is real. These error bounds may be further simplified by employing the various estimates for sup r≥1 |Λ p (2zr)| and sup r≥1 |Π p (2zr)| given in Appendix B. Note that those estimates in Appendix B that depend on the (positive) order p are monotonically increasing functions of p. Therefore, in the following theorems, we minimize, as much as our methods allow, the order of the basic terminants with respect to the following natural requirement: the form of each bound is directly related to the first omitted term of the corresponding asymptotic expansion. We remark that the bounds (26)-(29) remain true even if the orders of the basic terminants are taken to be any positive quantity at least N + max(0, 1 2 − |ν|).

Theorem 1.5
Let N be a non-negative integer and let ν be an arbitrary real number such that |ν| < N + 1 2 . Then provided |arg z| < 3π 2 , and provided |arg z| < π. Theorem 1.6 Let N be a positive integer and let ν be an arbitrary real number such that |ν| < N − 1 2 . Then provided |arg z| < 3π 2 , and provided |arg z| < π.
It is well known that in the special case when z is positive and ν is real, |ν| < N + 1 2 , we have Here 0 < θ (K) N (z, ν), θ (J ) N (z, ν) < 1 are appropriate numbers that depend on z, ν and N (see, for instance, [24, §10.40(ii)] or [40, pp. 207 and 209]). Said differently, the remainder terms R (K) N (z, ν) and R (J ) N (z, ν) do not exceed the corresponding first neglected terms in absolute value and have the same sign provided that z > 0 and |ν| < N + 1 2 . In the theorem below, we present the analogous results for the remainder terms R (K ) N (z, ν) and R (J ) N (z, ν).

Theorem 1.7
Let N be a positive integer, z be a positive real number, and let ν be an arbitrary real number such that |ν| < N − 1 2 . Then where 0 < θ (K ) N (z, ν), θ (J ) N (z, ν) < 1 are suitable numbers that depend on z, ν and N .
We now consider the case when ν is complex. In the following theorems, we have chosen N + 1 2 as the order of the basic terminants because this is the value that allows us to express the bounds in a form closely related to the first omitted term in the corresponding asymptotic expansion. The precise relation to the first omitted term is examined in Sect. 5. Theorem 1.8 Let N be a non-negative integer and let ν be an arbitrary complex number such that | (ν)| < N + 1 2 . Then provided |arg z| < 3π 2 , and provided |arg z| < π. If 2 (ν) is an odd integer, then the limiting values have to be taken in these bounds. Theorem 1.9 Let N be a positive integer and let ν be an arbitrary complex number such that | (ν)| < N − 1 2 . Then provided |arg z| < 3π 2 , and provided |arg z| < π. If 2 (ν) is an odd integer, then the limiting values have to be taken in these bounds.
We would like to emphasize that the requirement | (ν)| < N + 1 2 (respectively, | (ν)| < N − 1 2 ) in the above theorems is not a serious restriction. Indeed, the index of the numerically least term of the asymptotic expansion (10), for example, is n ≈ 2|z|. Therefore, it is reasonable to choose the optimal N ≈ 2|z|, whereas the condition ν 2 = o(|z|) has to be fulfilled in order to obtain proper approximations from (14).
A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given in Sect. 5.
In the following, we consider re-expansions for the remainders of the asymptotic expansions of the Bessel and modified Bessel function as well as of their derivatives. Reexpansions for the remainder terms of the asymptotic expansions of the functions J 0 (z), Y 0 (z) and K 0 (z), in order to improve their numerical efficacy, were first derived by Stieltjes [37] in 1886 (see also Watson [40,). His results were extended to arbitrary real order ν by Koshliakov [20] and Burnett [6] in 1929 and in 1930, respectively. More general expansions were established later, using formal methods, by Airey [1] in 1937 and by Dingle [9], [10, pp. 441 and 450] in 1959 (see also [7]). Their work was placed on rigorous mathematical foundations by Boyd [4] and Olver [30] in 1990, who derived an exponentially improved asymptotic expansion for the modified Bessel function K ν (z) valid when | arg z| ≤ π . Here, we shall reconsider the result of Boyd and Olver and obtain explicit bounds for the error term of their expansion. Analogous results for the remainders For any non-negative integers N and M, M < N, we write and The re-expansion (36) for | arg z| ≤ π , and for | arg z| ≤ π 2 . If 2 (ν) is an odd integer, then the limiting values are taken in these bounds.
for | arg z| ≤ π , and for | arg z| ≤ π 2 . If 2 (ν) is an odd integer, then the limiting values are taken in these bounds.
We remark that Boyd also considered the problem of bounding the remainder R (K) N,M (z, ν) in the special case when ν is real and |ν| < 1 2 . His result can be recovered from (40) by using (30)). A brief discussion on the sharpness of the error bound (40) is given in Sect. 5.
In the case when z is positive and ν is real, we shall show that the remainder terms do not exceed the corresponding first neglected terms in absolute value and have the same sign provided that |ν| < M + 1 2 (respectively |ν| < M − 1 2 ). More precisely, we will prove that the following theorems hold.

Theorem 1.12 Let N and M be arbitrary non-negative integers satisfying M < N. Let z be a positive real number and ν be an arbitrary real number such that
and and The best accuracy available from the re-expansions (36)- (39) can be obtained by setting N = 4|z| + ρ and M = 2|z| + σ , with ρ and σ being appropriate (fixed) real numbers. With these choices of N and M, Theorems 1.10 and 1.11, combined with Stirling's approximation for the gamma function and the bounds (96) and (97), imply that as z → ∞ in the sector |arg z| ≤ π , and as z → ∞ in the sector |arg z| ≤ π 2 . (Throughout this paper, we use subscripts in the O notations to indicate the dependence of the implied constant on certain parameters.) The estimates (48) and (49) can alternatively be deduced from the hyperasymptotic theory of second order linear differential equations, discussed in the papers [26] and [27].
Results analogous to those given in this paper for other ranges of arg z can be obtained by making use of the analytic continuation formulae for the Hankel and Bessel functions (see, e.g., [24, §10.11 and §10.34]). Also, by setting ν = 1 3 or ν = 2 3 , one can derive error bounds for the asymptotic expansions of the Airy functions and their derivatives, respectively, but we shall not pursue the details here.
As a closing remark, we would like to point out that the techniques used in this paper should generalize to integrals over steepest descent contours. Indeed, the theory developed by Berry and Howls ([2], especially equations (12) and (15)) and Howls ([17], especially equations (17) and (35)) enables us to derive representations analogous to (16) for the remainder terms of asymptotic expansions arising from an application of the method of steepest descents. The estimation of these new representations would then provide error bounds for the method of steepest descents which are presumably more general and sharper than those that exist in the literature (cf., in particular, [5]).
The remaining part of the paper is structured as follows. In Sect. 2, we prove the representations for the remainder terms stated in Theorems 1.1-1.4. In Sect. 3, we prove the error bounds given in Theorems 1.5-1.9. Section 4 discusses the proof of the bounds for the remainders of the re-expansions stated in Theorems 1.10-1.13. The paper concludes with a discussion in Sect. 5.

Proof of the Representations for the Remainder Terms
In this section, we prove the representations for the remainder terms stated in Theorems 1.1-1.4. In order to prove Theorems 1.1 and 1.2, we require certain estimates for the regularized hypergeometric function given in the following lemma.

Lemma 2.1 Let ν and λ be arbitrary fixed complex numbers such that
as t → 0+, and as t → +∞.
Proof The estimate (50) is a consequence of the power series expansion of the regularized hypergeometric function. To prove the estimate (51), we use the known linear transformation formula for the regularized hypergeometric function (see, e.g., [24, 15.8.E1]) and obtain for t > 0. Now, by employing the known behavior of the regularized hypergeometric function near one (see, e.g., [24, §15.4(ii)]), we find that as t → +∞. Combining these estimates with (52), the result claimed in (51) follows.
We continue with the proof of (16). The modified Bessel function can be represented in terms of the regularized hypergeometric function as for | arg u| < π 2 and with an arbitrary complex λ, (λ) > 0 [25, ent. 19.2, p. 195]. Inserting this representation into (22) and changing the order of integration, we deduce The change in the order of integration is justified because the infinite double integrals are absolutely convergent, which can be seen by appealing to Lemma 2.1. The inner integral is expressible in terms of the basic terminant Λ p (w) since Eq. (98)), and therefore we obtain Since Λ p (w) = O(1) as w → ∞ in the sector |arg w| < 3π 2 , by analytic continuation, this representation is valid in a wider range than (22), namely in |arg z| < 3π 2 .
Let us now turn our attention to the proof of (18). It is enough to show that for any positive integer N and fixed complex numbers z and ν satisfying |arg z| < 3π 2 and | (ν)| < N + 1 2 . Note that the convergence of the integral in the second line is guaranteed by Lemma 2.1 and the boundedness of the basic terminant. By a simple algebraic manipulation and an application of the beta integral (cf. [24, 5.12.E3]), we can assert that By continuity, we have and thus, it remains to prove that as t → 0+. Consequently, the integrand in the first line of (53 as t → 0+ and, appealing to Lemma 2.1 and to the boundedness of the basic terminant, it as t → +∞. Therefore, there exists a positive constant C z,ν,N , independent of t and λ such that the absolute value of the integrand in the first line of (53) is bounded from above by for any t > 0. Since N ≥ 1 and | (ν)| < N + 1 2 , the function (54) is absolutely integrable on the positive t -axis and the proof is complete.

Proof of the Error Bounds
In this section, we prove the bounds for the remainder terms R (K) N (z, ν), R (J ) N (z, ν), R (K ) N (z, ν) and R (J ) N (z, ν) given in Theorems 1.5-1.9. To this end, we shall state and prove a series of lemmata.
Proof From (14), we can assert that which itself is a consequence of the connection formula −2K ν (z) = K ν+1 (z) + K ν−1 (z). Formula (64) follows from (18) by an argument similar to the derivation of (63) from (16); the details are left to the reader.
The bound (28) can be proved as follows. Assume that N is a positive integer and ν is an arbitrary real number such that |ν| < N − 1 2 . By substituting (18) into the functional relation (20), we find By Lemma 3.2, we can then assert that provided |arg z| < 3π 2 . The right-hand side of this inequality can be simplified using Lemma 3.2 and the representation (64) for the coefficients b N (ν) leading to the required estimate (28). The bound (29) can be deduced similarly from (19) and (21).
To prove formula (31), we first note that 0 < Λ p (w) < 1 whenever w > 0 and p > 0 (see Proposition B.1). Employing Lemma 3.2, the mean value theorem of integration and the fact that the summands in the large parentheses in (67) have the same sign, we can infer that for some 0 < θ (K ) N (z, ν) < 1 depending on z, ν and N . Comparing this expression with the representation (64) for the coefficients b N (ν) yields the required formula (31) for R (K ) N (z, ν). The corresponding result for R (J ) N (z, ν) can be obtained using the analogue of the representation (67) and the fact that 0 < Π p (w) < 1 whenever w > 0 and p > 0 (see Proposition B.1); we leave the details to the reader.
We now turn to the proof of the bound (32). Assume that N is a non-negative integer and ν is an arbitrary complex number satisfying | (ν)| < N + 1 2 . First note that since 1 2 for any t > 0 (see, for instance, [24, 15.4.E13]), the following inequality holds: Taking λ = 1 2 in (16) and employing (68), we find that as long as |arg z| < 3π 2 . Now the required bound (32) can be obtained by simplifying the right-hand side of the above inequality using the representation (62) (with λ = 1 2 ) for the coefficients a N (ν) (with (ν) in place of ν). The estimate (33) can be deduced in essentially the same way from (17) with the specific choice of λ = 1 2 . We close this section by proving the estimate (34). For this purpose, suppose that N is a positive integer and ν is an arbitrary complex number such that | (ν)| < N − 1 2 . By taking λ = 1 2 in (16) and substituting the result into the functional relation (20), we obtain An application of the inequality (68) yields for |arg z| < 3π 2 . We then simplify the right-hand side of this inequality using the representation (63) (with λ = 1 2 ) for the coefficients b N (ν) (with (ν) in place of ν) to arrive at the required estimate (34). The bound (35) can be deduced in a similar manner from (17) and (21) with the value λ = 1 2 .

Proof of the Error Bounds for the Re-expansions
In this section, we prove the bounds for the remainder terms R (K) N,M (z, ν), R (J ) N,M (z, ν), R (K ) N,M (z, ν) and R (J ) N,M (z, ν) that are given in Theorems 1.10-1.13. We begin by stating and proving the following lemma.

Lemma 4.1 Let N be a non-negative integer and let ν be an arbitrary complex number. Then
provided that | (ν)| < N − 1 2 and N ≥ 1.
Proof Formula (69) can be derived by substituting the integral representation (22) into the right-hand side of the equality (65). To prove (70), we substitute (69) into (66) and use the functional relation −2K We continue with the proof of the estimate (40). Let N and M be arbitrary non-negative integers satisfying M < N. Suppose that | (ν)| < M + 1 2 and | arg z| < π. We begin by replacing the function K ν (t) in the integral formula (22) for R (K) N (z, ν) by its truncated asymptotic expansion and using the representation (98) of the basic terminant Λ p (w). In this way, we obtain (36) with When passing to the second equality, we have made a change of integration variable from t to u by t = |z|u. The remainder R (K) M (|z|u, ν) is given by the integral formula (22), which can be re-expressed in the form Noting that for positive u and t , the substitution of (73) into (72) and trivial estimation yield the upper bound In arriving at this bound, we have made use of the fact that |K ν (t)| < K (ν) (t) for any t > 0 and of the representation (69) for the coefficients a N (ν) (with M and (ν) in place of N and ν). Since |(u − 1)/(u + e i arg z )| ≤ 1 for positive u, we find, after simplification, that By continuity, this bound holds in the closed sector | arg z| ≤ π , and therefore the proof of the estimate (40) is complete.
To prove the bound (41), we proceed in a similar manner. Let N and M be arbitrary nonnegative integers satisfying M < N. Assume further that | (ν)| < M + 1 2 and | arg z| < π 2 . We replace the function K ν (t) in the integral formula (23) for R (J ) N (z, ν) by its truncated asymptotic expansion (71) and employ the representation (101) of the basic terminant Π p (w), to obtain (38) with By an argument similar to that which led to (74), we deduce the bound Now, using the fact that |(u − 1)/(u 2 + e 2i arg z )| ≤ 1 for any positive u, after simplification we arrive at By continuity, this bound holds in the closed sector | arg z| ≤ π 2 , and thus the proof of the estimate (41) is complete.
The corresponding bounds (42) and (43) for the remainder terms R (K ) N,M (z, ν) and R (J ) N,M (z, ν) can be proved in an analogous manner using the representations (15), (24), (25) and (70). We leave the details to the reader.
We now turn to the proof of the expression (44). Let N and M be arbitrary non-negative integers satisfying M < N. Suppose that z is positive and ν is a real number such that |ν| < M + 1 2 . Applying the first formula in (30) (with M and t in place of N and z) in the representation (72) for R (K) N,M (z, ν), and then using the mean value theorem of integration, we find that Here

Discussion
In this paper, we have derived new integral representations and estimates for the remainder terms of the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions, and their derivatives. We have also constructed error bounds for the reexpansions of these remainders. In this section, we shall discuss the sharpness of our error bounds and their relation to other results in the literature.
First, we show that the error bounds proved in this paper are reasonably sharp. For the sake of brevity, we consider only the bounds for R (K) N (z, ν) and R (K) N,M (z, ν); the other remainder terms can be treated in a similar manner.
Let N be any non-negative integer, ν a real number and z a complex number. Suppose that |ν| < N + 1 2 and | arg z| ≤ π 2 . Under these assumptions, it follows from Theorem 1.5 and Proposition B.1 that By the definition of an asymptotic expansion, lim z→∞ |z| N |R (K) N (z, ν)| = |a N (ν)| for any fixed N ≥ 0. Therefore, when | arg z| ≤ π 2 , the estimate (75) and hence our error bound (26) cannot be improved in general.
Consider now the case when π 2 < | arg z| ≤ π (ν is still real and satisfies |ν| < N + 1 2 ). Combining Theorem 1.5 (or Theorem 1.8) with Propositions B.1 and B.3, we deduce that Here, following Olver [28], we use the notation for any p > 0. The bound (76) is reasonably sharp as long as |csc(arg z)| is not very large, i.e., when | arg z| is bounded away from π . As | arg z| approaches π , the factor |csc(arg z)| grows indefinitely and therefore it has to be replaced by 1+χ(N + 1 2 ). By Stirling's formula, 1 2 as N → +∞, and therefore the appearance of this factor in the bound (76) may give the impression that this estimate is unrealistic for large N . However, this is not the case, as the following argument shows. We may suppose, without loss of generality, that 2ν is not equal to an odd integer, because otherwise, the remainder term R (K) N (z, ν) becomes identically zero, since |ν| < N + 1 2 . Recently, Paris [34] showed that if a and a − b + 1 are fixed and neither of them is a non-positive integer, then the confluent hypergeometric function satisfies as x → +∞, provided that |N − x| remains bounded. In the following, we assume that ν is fixed, |N − 2x| is bounded and x is large and positive. Employing the expansion (78) for the right-hand side of the functional relation (see, e.g., [24, 13.6.E10]) and comparing the result with (14), we obtain From (76), we have Now, using our assumptions on N , ν and z, we have, by Stirling's formula, Similarly, Combining (82) and (83), we see that the upper bound in (81) is asymptotically equal to the right-hand side of the equality (80). Consequently, when | arg z| is equal or close to π , the estimate (76) and thus the error bound (26) cannot be improved in general. Finally, assume that π < |arg z| < 3π 2 , ν is real and |ν| < N + 1 2 . If we combine Theorem 1.5 (or Theorem 1.8) with Propositions B.3 and B.4, we obtain the estimate Elementary analysis shows that the second factor on the right-hand side of (84), as a function of N , remains bounded, provided that |arg z| − π = O(N − 1 2 ). This gives a reasonable estimate for the remainder term R (K) N (z, ν). Otherwise, |cos(arg z)| N+ 1 2 can take very small values when N is large, making the bound (84) completely unrealistic in most of the sectors π < |arg z| < 3π 2 . This deficiency of the bound (84) (and (26)) is necessary and is due to the omission of certain exponentially small terms arising from the Stokes phenomenon related to the asymptotic expansion (10) of the modified Bessel function (for a detailed discussion, see [29]). Thus, the use of the asymptotic expansion (10) should be confined to the sector | arg z| ≤ π . For other ranges of arg z, one should use the analytic continuation formulae for the modified Bessel function [24, 10.34.E3].
The bound (26) and the bound (32) which we discuss below and which covers the case of complex ν differ significantly only in the component that depends on ν. Therefore, when analyzing the estimate (32) we can focus mainly on understanding the relationship between the quantities |cos(π ν)| |cos(π (ν))| |a N ( (ν))| and |a N (ν)|. With the aid of the definition (9), one readily infers that Now, we make the assumptions that N − | (ν)| → +∞ and | (ν)| = o(N 1 2 ) as N → +∞. With these provisos, it can easily be shown, using for example Stirling's formula, that the quotient of gamma functions in (85) is asymptotically 1 for large N . Consequently, the right-hand side of the inequality (32) is asymptotic to if N is large. Now, using the same argument as in the case of real ν above, we can conclude that the bound (32) is sharp in general (at least when z is restricted to the sector | arg z| ≤ π ). If the assumption | (ν)| = o(N 1 2 ) is replaced by the weaker condition | (ν)| = O(N 1 2 ), the quotient in (85) is still bounded and hence, the estimate (32) is still reasonably sharp. Otherwise, if N − | (ν)| is small and | (ν)| is much larger than N 1 2 , this quotient may grow exponentially fast in | (ν)|, which can make the bound (32) completely unrealistic. Note, however, that in practical applications, it is reasonable to truncate the asymptotic expansion (10) just before its numerically least term, i.e., when N ≈ 2|z|. And so, since the expansion (10) is valid only when ν 2 = o(|z|), the conditions N − | (ν)| → +∞ and | (ν)| = o(N 1 2 ) are automatically satisfied in this case.
Let us now turn to the analysis of the estimate (40) for R (K) N,M (z, ν). When z is large, |z| M |R (K) M (|z|, ν)| |a M (ν)| holds, and therefore the first term on the right-hand side of the inequality (40) is of the same order of magnitude as the first neglected term in the expansion (36). It can be shown that, when N − M is large, the second term is comparable with, or less than, the first term (except near the zeros of Λ N−M (2z)). The proof of this fact is identical to the proof given by Boyd [4] for the case when ν is real and |ν| < 1 2 and it is therefore not pursued here. In summary, the bound (40) is comparable with the first neglected term in the expansion (36), unless z is close to a zero of Λ N−M (2z), and thus, this bound is reasonably sharp. We would like to remark that an estimate for R (K) N,M (z, ν), different from (40), can be derived using the functional relation (79) and an error bound for the re-expansion of the remainder term in the large-z asymptotic expansion of the confluent hypergeometric function U(a, b, z) which is due to Olver [32]. Since the resulting estimate is significantly more complicated than (40), we omit the details.
There are two major methods in the literature for obtaining bounds for the remainder terms R (K) N (z, ν) and R (J ) N (z, ν): one is based on integral representations, while the other uses differential equations. The former approach was taken by Schläfli [36], Watson [40, pp. 209-210] and Meijer [23]. The bounds proved by Schläfli and Meijer were later examined by Döring [11], in the case of real ν, who effected simplifications to make them more easily computable. These results of the authors can all be deduced as direct consequences of Theorem 1.8 and Propositions B.1 and B.2.
Weber [41] and Olver [28] used differential equation methods to derive different types of estimates for the remainder R (K) N (z, ν). Since Olver's bounds are more general and sharper than those found by Weber, we consider only Olver's estimates below. First, we state his results and then compare them with our estimates.
Let z and ν be arbitrary complex numbers such that | arg z| < 3π 2 . Then Olver's bounds read as follows:   (87), V denotes the variational operator, in particular V z,∞ (t −N ) = N ∞ z |t| −N−1 |dt| and the path of variation is subject to the condition that | (t)| changes monotonically. The following are simple bounds for V z,∞ (t −N ): where χ(N) is given by (77) (84) that if ν is real, our bound (26) (in its region of validity, i.e., when |ν| < N + 1 2 ) is about twice as sharp as (89). When ν is complex and satisfies the assumptions made after (85), our bound (32) is again better than (89) for large N . In particular, if the asymptotic expansion (10) is truncated at or near its numerically least term (i.e., when N ≈ 2|z|) and ν satisfies ν 2 = o(|z|), then the estimate (32) is sharper than (89) for large z (or, equivalently, for large N ). In any other situations, the estimates (89), and hence Olver's (86) and (87), are preferable over (26) or (32).
In Figs. 1 and 2, we compare numerically Olver's estimate (89) with the following bound which follows by combining Theorem 1.8 with Propositions B.1 and B.3:
The estimate (97) can be proved in a similar way, starting from the representation which is valid when | arg w| < π 2 and p > 0. This representation can be obtained from (98) and the definition of the basic terminant Π p (w).