The resurgence properties of the Hankel and Bessel functions of nearly equal order and argument

Abstract

The aim of this paper is to derive new representations for the Hankel functions, the Bessel functions and their derivatives, exploiting the reformulation of the method of steepest descents by Berry and Howls (Proc Roy Soc London Ser A 434:657–675, 1991). Using these representations, we obtain a number of properties of the asymptotic expansions of the Hankel and Bessel functions and their derivatives of nearly equal order and argument, including explicit and numerically computable error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.

Appendices

Appendix A: Computation of the coefficients $B_n\left( \kappa \right)$

In 1952, Lauwerier [11] showed that the coefficients in asymptotic expansions of Laplace-type integrals can be calculated by means of linear recurrence relations. Simple application of his method provides the formula

$$\begin{aligned} B_n \left( \kappa \right) = \frac{{6^{ - \frac{{n + 1}}{3}} }}{{\Gamma \left( {\frac{{n + 1}}{3}} \right) }}\int _0^{ + \infty } {t^{\frac{{n - 2}}{3}} e^{ - \frac{t}{6}} P_n \left( {t,\kappa } \right) dt} , \end{aligned}$$

where the two-variable polynomials $P_0 \left( x,\kappa \right) , P_1 \left( x,\kappa \right) , P_2 \left( x,\kappa \right) ,\ldots$ are given by the recurrence relation

$$\begin{aligned} P_n \left( {x,\kappa } \right) = \frac{{\kappa ^n }}{{n!}} - \sum \limits _{k = 1}^{\left\lfloor {\frac{n}{2}} \right\rfloor } {\frac{1}{{\left( {2k + 3} \right) !}}\int _0^x {P_{n - 2k} \left( {t,\kappa } \right) dt} } \; \text { for } \; n\ge 2, \end{aligned}$$

with $P_0 \left( {x,\kappa } \right) = 1$ and $P_1 \left( {x,\kappa } \right) = \kappa$.

Expanding the higher derivative in (1.9) using Leibniz’s rule and noting that $t^3/\left( \sinh t -t\right)$ is an even function of $t$, we deduce

$$\begin{aligned} B_n \left( \kappa \right) = \sum \limits _{k = 0}^{\left\lfloor {\frac{n}{2}} \right\rfloor } {\frac{{\kappa ^{n - 2k} }}{{\left( {n - 2k} \right) !}}\frac{1}{{\left( {2k} \right) !}}\left[ {\frac{{d^{2k} }}{{dt^{2k} }}\left( {\frac{1}{6}\frac{{t^3 }}{{\sinh t - t}}} \right) ^{\frac{{n + 1}}{3}} } \right] _{t = 0} } , \end{aligned}$$

(8.1)

which is Airey’s [2] representation for the coefficients $B_n\left( \kappa \right)$.

There has been a recent interest in finding explicit formulas for the coefficients in asymptotic expansions of Laplace-type integrals (see [12, 15, 27, 28]). There are two general formulas for these coefficients, one containing Potential polynomials and one containing Bell polynomials. We derive them here for the special case of the coefficients $B_n\left( \kappa \right)$. Let

$$\begin{aligned} \sinh t - t = \sum \limits _{j= 0}^\infty {a_j t^{j + 3} } , \end{aligned}$$

so that

$$\begin{aligned} a_{2j} = \frac{1}{{\left( {2j + 3} \right) !}},\; a_{2j + 1} = 0 \; \text { for } \; j \ge 0. \end{aligned}$$

Let $0 \le i \le j$ be integers and $\rho$ be a complex number. We define the Potential polynomials

$$\begin{aligned} \mathsf {A}_{\rho ,j} = \mathsf {A}_{\rho ,j} \left( {\frac{{a_1 }}{{a_0 }},\frac{{a_2 }}{{a_0 }}, \ldots ,\frac{{a_j }}{{a_0 }}} \right) \end{aligned}$$

and the Bell polynomials

$$\begin{aligned} \mathsf {B}_{j,i} = \mathsf {B}_{j,i} \left( {a_1 ,a_2 , \ldots ,a_{j - i + 1} } \right) \end{aligned}$$

via the expansions

$$\begin{aligned} \left( {1 + \sum \limits _{j = 1}^\infty {\frac{{a_j }}{{a_0 }}t^j } } \right) ^\rho = \sum \limits _{j = 0}^\infty {\mathsf {A}_{\rho ,j} t^j } \; \text { and } \; \mathsf {A}_{\rho ,j} = \sum \limits _{i = 0}^j {\left( {\begin{array}{c}\rho \\ i\end{array}}\right) \frac{1}{a_0^i}\mathsf {B}_{j,i}} . \end{aligned}$$

(8.2)

Naturally, these polynomials can be defined for arbitrary power series with $a_0\ne 0$. It is possible to express the Potential polynomials with complex parameter in terms of Potential polynomials with integer parameter using the following formula of Comtet [7, p.142]

$$\begin{aligned} \mathsf {A}_{\rho ,j} = \frac{\Gamma \left( { - \rho + j + 1} \right) }{j!\Gamma \left( { - \rho } \right) }\sum \limits _{i = 0}^j {\frac{\left( { - 1} \right) ^i}{- \rho + i}\left( {\begin{array}{c}j\\ i\end{array}}\right) \mathsf {A}_{i,j} } . \end{aligned}$$

(8.3)

With these notations we can write (8.1) as

$$\begin{aligned} B_n \left( \kappa \right) = \sum \limits _{k = 0}^{\left\lfloor {\frac{n}{2}} \right\rfloor } {\frac{{\kappa ^{n - 2k} }}{{\left( {n - 2k} \right) !}}\mathsf {A}_{ - \frac{{n + 1}}{3},2k} } . \end{aligned}$$

Using (8.2) and (8.3) we find

$$\begin{aligned} B_n \left( \kappa \right) = \sum \limits _{k = 0}^{\left\lfloor {\frac{n}{2}} \right\rfloor } {\frac{{\kappa ^{n - 2k} }}{{\left( {n - 2k} \right) !}}\sum \limits _{j = 0}^{2k} {\left( { - 1} \right) ^j 6^j \frac{{\Gamma \left( {\frac{{n + 1}}{3} + j} \right) }}{{j!\Gamma \left( {\frac{{n + 1}}{3}} \right) }}\mathsf {B}_{2k,j}} } \end{aligned}$$

and

$$\begin{aligned} B_n \left( \kappa \right) = \sum \limits _{k = 0}^{\left\lfloor {\frac{n}{2}} \right\rfloor } {\frac{{\kappa ^{n - 2k} }}{{\left( {n - 2k} \right) !}}\frac{{3\Gamma \left( {\frac{{n + 1}}{3} + 2k + 1} \right) }}{{\left( {2k} \right) !\Gamma \left( {\frac{{n + 1}}{3}} \right) }}\sum \limits _{j = 0}^{2k} {\frac{{\left( { - 1} \right) ^j }}{{n + 3j + 1}} \left( {\begin{array}{c}2k\\ j\end{array}}\right) \mathsf {A}_{j,2k} } } . \end{aligned}$$

(8.4)

The quantities $\mathsf {B}_{k,j}$ and $\mathsf {A}_{j,k}$ appearing in these formulas may be computed from the recurrence relations

$$\begin{aligned} \mathsf {B}_{k,j} = \sum \limits _{i= 1}^{k - j + 1} {a_i \mathsf {B}_{k - i,j - 1} } \; \text { and } \; \mathsf {A}_{j,k} = \sum \limits _{i = 0}^{k} {\frac{a_i}{a_0}\mathsf {A}_{j - 1,k - i} } \end{aligned}$$

with $\mathsf {B}_{0,0} = \mathsf {A}_{0,0} = 1$, $\mathsf {B}_{i,0} = \mathsf {A}_{0,i} = 0$ $\left( i \ge 1\right)$, $\mathsf {B}_{i,1} = a_0 \mathsf {A}_{1,i} = a_i$ (see Nemes [15]). In the paper [16], it was shown that the Potential polynomials $\mathsf {A}_{j,2k}$ in (8.4) can be written as

$$\begin{aligned} \mathsf {A}_{j,2k} = \sum \limits _{i = 0}^j {\left( { - 1} \right) ^{j + i} \left( {\begin{array}{c}j\\ i\end{array}}\right) \frac{{2^{2k + 2j} 6^j }}{{\left( {2k + 2j} \right) !}}B_{2k + 2j}^{\left( { - i} \right) } \left( { - \frac{i}{2}} \right) } , \end{aligned}$$

where $B_n^{\left( \ell \right) } \left( x\right)$ stands for the generalised Bernoulli polynomials, which are defined by the exponential generating function

$$\begin{aligned} \left( \frac{t}{e^t - 1} \right) ^\ell e^{x t} = \sum \limits _{n = 0}^\infty {B_n^{\left( \ell \right) } \left( x\right) \frac{t^n}{n!}} \; \text { for } \; \left| t\right| < 2\pi . \end{aligned}$$

For basic properties of these polynomials, see Milne-Thomson [14] or Nörlund [18].

Some other properties of the polynomials $B_n \left( \kappa \right)$ can be found in the paper of Schöbe [25].

Appendix B: Auxiliary estimates

In this appendix, we prove some estimates for the remainder $R_N^{\left( {H} \right) } \left( {z,\kappa } \right)$. First, we prove the estimate (5.6). Suppose that $0 \le \arg z \le \pi$ and $\left| \mathfrak {R}\left( \kappa \right) \right| < \frac{N + 1}{3}$ with $N\ge 0$. Trivial estimation of (1.10) yields

$$\begin{aligned}&\left| {R_N^{\left( {H} \right) } \left( {z,\kappa } \right) } \right| \\&\ \le \frac{{\left| {e^{2\pi i\kappa } } \right| }}{{6\pi \left| z \right| ^{\frac{{N + 1}}{3}} }}\int _0^{ + \infty } {t^{\frac{{N - 2}}{3}} e^{ - 2\pi t} \left| {\frac{{e^{\frac{{\left( {N + 1} \right) \pi i}}{3}} }}{{1 + i\left( {t/z} \right) ^{\frac{1}{3}} e^{\frac{\pi }{3}i} }} - \frac{{e^{\left( {N + 1} \right) \pi i} }}{{1 - i\left( {t/z} \right) ^{\frac{1}{3}} }}} \right| \left| {H_{it + \kappa }^{\left( 1 \right) } \left( {it} \right) } \right| dt}\\&\ \quad +\, \frac{{\left| {e^{ - 2\pi i\kappa } } \right| }}{{6\pi \left| z \right| ^{\frac{{N + 1}}{3}} }}\int _0^{ + \infty } {t^{\frac{{N - 2}}{3}} e^{ - 2\pi t} \left| {\frac{{e^{\frac{{\left( {N + 1} \right) \pi i}}{3}} }}{{1 - i\left( {t/z} \right) ^{\frac{1}{3}} e^{\frac{\pi }{3}i} }} - \frac{{e^{\left( {N + 1} \right) \pi i} }}{{1 + i\left( {t/z} \right) ^{\frac{1}{3}} }}} \right| \left| {H_{it - \kappa }^{\left( 1 \right) } \left( {it} \right) } \right| dt} . \end{aligned}$$

Employing the elementary inequality

$$\begin{aligned} \frac{1}{{\left| {1 - re^{i\vartheta } } \right| }} \le {\left\{ \begin{array}{ll} \left| \csc \vartheta \right| &{} \; \text { if } \; 0 < \left| \vartheta \text { mod } 2\pi \right| <\frac{\pi }{2}, \\ 1 &{} \; \text { if } \; \frac{\pi }{2} \le \left| \vartheta \text { mod } 2\pi \right| \le \pi , \end{array}\right. } \end{aligned}$$

(9.1)

for $r>0$, we obtain

$$\begin{aligned} \left| {\frac{{e^{\frac{{\left( {N + 1} \right) \pi i}}{3}} }}{{1 + i\left( {t/z} \right) ^{\frac{1}{3}} e^{\frac{\pi }{3}i} }} - \frac{{e^{\left( {N + 1} \right) \pi i} }}{{1 - i\left( {t/z} \right) ^{\frac{1}{3}} }}} \right| \le \frac{1}{{\left| {1 + i\left( {t/z} \right) ^{\frac{1}{3}} e^{\frac{\pi }{3}i} } \right| }} + \frac{1}{{\left| {1 - i\left( {t/z} \right) ^{\frac{1}{3}} } \right| }}\\ \le 2 + 2 = 4 \end{aligned}$$

and

$$\begin{aligned} \left| {\frac{{e^{\frac{{\left( {N + 1} \right) \pi i}}{3}} }}{{1 - i\left( {t/z} \right) ^{\frac{1}{3}} e^{\frac{\pi }{3}i} }} - \frac{{e^{\left( {N + 1} \right) \pi i} }}{{1 + i\left( {t/z} \right) ^{\frac{1}{3}} }}} \right| \le \frac{1}{{\left| {1 - i\left( {t/z} \right) ^{\frac{1}{3}} e^{\frac{\pi }{3}i} } \right| }} + \frac{1}{{\left| {1 + i\left( {t/z} \right) ^{\frac{1}{3}} } \right| }} \le 1 + 1 = 2. \end{aligned}$$

Hence, we have the estimate

$$\begin{aligned} \left| {R_N^{\left( {H} \right) } \left( {z,\kappa } \right) } \right|\le & {} \frac{{2\left| {e^{2\pi i\kappa } } \right| }}{{3\pi \left| z \right| ^{\frac{{N + 1}}{3}} }}\int _0^{ + \infty } {t^{\frac{{N - 2}}{3}} e^{ - 2\pi t} \left| {H_{it + \kappa }^{\left( 1 \right) } \left( {it} \right) } \right| dt}\\&+ \frac{{\left| {e^{ - 2\pi i\kappa } } \right| }}{{3\pi \left| z \right| ^{\frac{{N + 1}}{3}} }}\int _0^{ + \infty } {t^{\frac{{N - 2}}{3}} e^{ - 2\pi t} \left| {H_{it - \kappa }^{\left( 1 \right) } \left( {it} \right) } \right| dt} . \end{aligned}$$

It remains to show that the integrals on the right-hand side are convergent. The integrands are continuous functions of $t>0$. The asymptotic expansion (1.1) shows that for large positive $t$ and fixed $\kappa$, we have

$$\begin{aligned} H_{it \pm \kappa }^{\left( 1 \right) } \left( {it} \right) \sim \frac{2}{{3\pi }}6^{\frac{1}{3}} \frac{{\sqrt{3} }}{2}\frac{{\Gamma \left( {\frac{1}{3}} \right) }}{{t^{\frac{1}{3}} }}. \end{aligned}$$

We also have

$$\begin{aligned} \left| { H_{it \pm \kappa }^{\left( 1 \right) } \left( {it} \right) } \right| = {\left\{ \begin{array}{ll} \mathcal {O}_\kappa \left( {t^{ - \left| {\mathfrak {R}\left( \kappa \right) } \right| } } \right) &{} \; \text { if } \; \kappa \ne 0, \\ \mathcal {O}\left( {\log t} \right) &{} \; \text { if } \; \kappa = 0, \end{array}\right. } \end{aligned}$$

as $t\rightarrow 0$ along the positive real axis (see, e.g., [22, Sect. 10.7(i)]). Therefore the integrals are convergent and we conclude that there is a constant $C_N \left( \kappa \right) >0$ depending only on $N$ and $\kappa$, such that if $0 \le \arg z \le \pi$ and $\left| \mathfrak {R}\left( \kappa \right) \right| < \frac{N + 1}{3}$, then

$$\begin{aligned} \left| {R_N^{\left( {H} \right) } \left( {z,\kappa } \right) } \right| \le \frac{{C_N \left( \kappa \right) }}{{\left| z \right| ^{\frac{{N + 1}}{3}} }}. \end{aligned}$$

Finally, we extend the region of validity of the expansions (1.1) and (1.2). It was given by Watson that the $N$th remainders in the asymptotic expansions (1.1) and (1.2) satisfy

$$\begin{aligned} R_N^{\left( H \right) } \left( {z,\kappa } \right) = \mathcal {O}_{\kappa ,N,\delta } \left( {\frac{1}{{\left| z \right| ^{\frac{{N + 1}}{3}} }}} \right) \; \text { and } \; - R_N^{\left( H \right) } \left( {ze^{\pi i} , - \kappa } \right) = \mathcal {O}_{\kappa ,N,\delta } \left( {\frac{1}{{\left| z \right| ^{\frac{{N + 1}}{3}} }}} \right) \end{aligned}$$

as $z\rightarrow \infty$ in $\left| {\arg z} \right| \le \pi - \delta < \pi$, with $0<\delta \le \pi$ being fixed. However, from the connection between the two remainders it is seen that the first estimate is valid in the larger sector $- \pi + \delta \le \arg z \le 2\pi - \delta$, and the second estimate is valid in the larger sector $- 2\pi + \delta \le \arg z \le \pi - \delta$.