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

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Abstract

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.

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Acknowledgements

The author’s research was supported by a research grant (GRANT11863412/ 70NANB15H221) from the National Institute of Standards and Technology. The author greatly appreciates the help of Dorottya Sziráki in improving the presentation of the paper. The author thanks the anonymous referees for their helpful comments and suggestions on the manuscript.

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Correspondence to Gergő Nemes.

Appendices

Appendix A: Relations Between the Various Remainder Terms

In this appendix, we show how the remainder terms of the asymptotic expansions of the functions \(H_{\nu }^{ ( 1,2 ) } ( z ) \), \(H_{\nu }^{ ( 1,2 ) \prime } ( z ) \), \(Y_{\nu } ( z ) \), \(Y'_{\nu } ( z ) \), \(I_{\nu } ( z ) \) and \(I'_{\nu } ( z ) \) may be expressed in terms of the remainders \(R_{N}^{ ( K ) } ( {z,\nu } ) \), \(R_{N}^{ ( J ) } ( {z,\nu } ) \), \(R_{N}^{(K')} ( {z,\nu } ) \) and \(R_{N}^{(J')} ( {z, \nu } ) \) of the asymptotic expansions of \(K_{\nu } ( z ) \), \(J_{\nu } ( z ) \), \(K'_{\nu } ( z ) \) and \(J'_{\nu } ( z ) \).

Let us consider first the Hankel functions. These functions are directly related to the modified Bessel function \(K_{\nu } ( z ) \) through the connection formulae

$$ H_{\nu }^{ ( 1 ) } ( z ) = \frac{2}{{\pi i}}e^{ - \frac{\pi }{2}i\nu } K_{\nu }\bigl( ze^{ - \frac{\pi }{2}i} \bigr),\quad - \frac{ \pi }{2} \le \arg z \le \pi $$

and

$$ H_{\nu }^{ ( 2 ) } ( z ) = - \frac{2}{{\pi i}}e ^{\frac{\pi }{2}i\nu } K_{\nu }\bigl( ze^{\frac{\pi }{2}i} \bigr),\quad - \pi \le \arg z \le \frac{\pi }{2} $$

(see, e.g., [24, 10.27.E8]). We substitute (14) into the right-hand sides and match the notation with those of (1) and (3) in order to obtain

$$ H_{\nu }^{ ( 1 ) } ( z ) = \biggl( { \frac{2}{{\pi z}}} \biggr) ^{\frac{1}{2}} e^{i\omega } \Biggl( {\sum _{n = 0}^{N - 1} {i^{n} \frac{{a_{n} ( \nu ) }}{ {z^{n} }}} + R_{N}^{ ( K ) } \bigl(ze^{ - \frac{\pi }{2}i} , \nu \bigr)} \Biggr) $$
(91)

and

$$ H_{\nu }^{ ( 2 ) } ( z ) = \biggl( { \frac{2}{{\pi z}}} \biggr) ^{\frac{1}{2}} e^{ - i\omega } \Biggl( {\sum _{n = 0}^{N - 1} { ( { - i} ) ^{n} \frac{{a_{n} ( \nu ) }}{{z^{n} }}} + R_{N}^{ ( K ) } \bigl(ze^{\frac{ \pi }{2}i} , \nu \bigr)} \Biggr) . $$
(92)

The restrictions on \(\arg z\) may now be removed by appealing to analytic continuation.

The corresponding expressions for the derivatives can most readily be obtained by first substituting (91) and (92) into the right-hand sides of the connection formulae \(-2H_{\nu }^{ ( 1 ) \prime } ( z ) = H_{\nu +1}^{ ( 1 ) } ( z ) -H _{\nu -1}^{ ( 1 ) } ( z ) \) and \(-2H_{\nu }^{ ( 2 ) \prime } ( z ) = H_{\nu +1}^{ ( 2 ) } ( z ) -H _{\nu -1}^{ ( 2 ) } ( z ) \) (cf. [24, 10.6.E1]). Then employing the relations (20) and (66), and taking into account the notation of (2) and (4), we deduce

$$ H_{\nu }^{ ( 1 ) \prime } ( z ) = i \biggl( {\frac{2}{ {\pi z}}} \biggr) ^{\frac{1}{2}} e^{i\omega } \Biggl( {\sum _{n = 0}^{N - 1} {i^{n} \frac{{b_{n} ( \nu ) }}{{z^{n} }}} + R _{N}^{(K')} \bigl( ze^{ - \frac{\pi }{2}i} ,\nu \bigr)} \Biggr) $$

and

$$ H_{\nu }^{ ( 2 ) \prime } ( z ) = - i \biggl( {\frac{2}{ {\pi z}}} \biggr) ^{\frac{1}{2}} e^{ - i\omega } \Biggl( {\sum _{n = 0}^{N - 1} { ( { - i} ) ^{n} \frac{{b_{n} ( \nu ) }}{{z^{n} }}} + R_{N}^{(K')} \bigl( ze^{\frac{\pi }{2}i} , \nu \bigr)} \Biggr) . $$

To identify the remainder terms in the asymptotic expansion (7) of the Bessel function \(Y_{\nu } ( z ) \), we may proceed as follows. This function is related to the modified Bessel function \(K_{\nu } ( z ) \) via the connection formula

$$\begin{aligned} -\pi Y_{\nu } ( z ) = & e^{ - \frac{\pi }{2}i\nu } K_{ \nu } \bigl( {ze^{ - \frac{\pi }{2}i} } \bigr) + e^{\frac{\pi }{2}i \nu } K_{\nu } \bigl( {ze^{\frac{\pi }{2}i} } \bigr) \\ = & i\sin \omega \bigl( {e^{\frac{\pi }{4}i} e^{iz} K_{\nu } \bigl( {ze^{\frac{\pi }{2}i} } \bigr) + e^{ - \frac{\pi }{4}i} e^{ - iz} K _{\nu } \bigl( {ze^{ - \frac{\pi }{2}i} } \bigr) } \bigr) \\ &{} - \cos \omega \bigl( {e^{\frac{\pi }{4}i} e^{iz} K_{\nu } \bigl( {ze^{\frac{\pi }{2}i} } \bigr) - e^{ - \frac{\pi }{4}i} e^{ - iz} K _{\nu } \bigl( {ze^{ - \frac{\pi }{2}i} } \bigr) } \bigr) , \end{aligned}$$

with \(\vert {\arg z} \vert \leq \frac{\pi }{2}\) [24, 10.27.E10]. If we now compare this relation with (55) and (56), we readily establish that

$$\begin{aligned} Y_{\nu } ( z ) =& \biggl( { \frac{2}{{\pi z}}} \biggr) ^{ \frac{1}{2}} \Biggl( \sin \omega \Biggl( {\sum _{n = 0}^{N - 1} { ( { - 1} ) ^{n} \frac{{a_{2n} ( \nu ) }}{{z^{2n} }}} + R_{2N}^{ ( J ) } ( {z,\nu } ) } \Biggr) \\ &{} + \cos \omega \Biggl( {\sum _{m = 0}^{M - 1} { ( { - 1} ) ^{m} \frac{{a_{2m + 1} ( \nu ) }}{{z^{2m + 1} }}} - R _{2M + 1}^{ ( J ) } ( {z,\nu } ) } \Biggr) \Biggr) . \end{aligned}$$
(93)

We can now remove the restriction on \(\arg z\) using analytic continuation.

Consider now the derivative \(Y'_{\nu } ( z ) \). The simplest way to derive the required expression is by substituting the formula (93) into the connection formula \(-2Y'_{\nu } ( z ) = Y_{\nu +1} ( z ) -Y_{\nu -1} ( z ) \) (cf. [24, 10.6.E1]). Employing the relations (21) and (66), and matching our notation with that of (8), we find

$$\begin{aligned} Y'_{\nu } ( z ) =& \biggl( {\frac{2}{{\pi z}}} \biggr) ^{ \frac{1}{2}} \Biggl( \cos \omega \Biggl( {\sum _{n = 0}^{N - 1} { ( { - 1} ) ^{n} \frac{{b_{2n} ( \nu ) }}{{z^{2n} }}} + R_{2N}^{(J')} ( {z,\nu } ) } \Biggr) \\ &{} -\sin \omega \Biggl( {\sum _{m = 0}^{M - 1} { ( { - 1} ) ^{m} \frac{{b_{2m + 1} ( \nu ) }}{{z^{2m + 1} }}} - R _{2M + 1}^{(J')} ( {z,\nu } ) } \Biggr) \Biggr) . \end{aligned}$$

The modified Bessel function \(I_{\nu } ( z ) \) can be treated in the following manner. This function is related to the modified Bessel function \(K_{\nu } ( z ) \) through the connection formula

$$ I_{\nu } ( z ) = \mp \frac{i}{\pi }K_{\nu } \bigl( {ze^{ \mp \pi i} } \bigr) \pm \frac{i}{\pi }e^{ \pm \pi i\nu } K_{\nu } ( z ) $$
(94)

(see, for instance, [24, 10.34.E3]). Let \(N\) and \(M\) be arbitrary non-negative integers. We express the functions \(K_{\nu } ( {ze ^{ \mp \pi i} } ) \) and \(K_{\nu } ( z ) \) as

$$ K_{\nu } \bigl( {ze^{ \mp \pi i} } \bigr) = \pm i \biggl( { \frac{\pi }{ {2z}}} \biggr) ^{\frac{1}{2}} e^{z} \Biggl( {\sum _{n = 0}^{N - 1} { ( { - 1} ) ^{n} \frac{{a_{n} ( \nu ) }}{{z^{n} }}} + R_{N}^{ ( K ) } \bigl( {ze^{ \mp \pi i} , \nu } \bigr) } \Biggr) $$

and

$$ K_{\nu } ( z ) = \biggl( {\frac{\pi }{{2z}}} \biggr) ^{ \frac{1}{2}} e^{ - z} \Biggl( {\sum _{m = 0}^{M - 1} { \frac{ {a_{m} ( \nu ) }}{{z^{m} }}} + R_{M}^{ ( K ) } ( {z,\nu } ) } \Biggr) $$

(cf. Eq. (14)), substitute these expressions into the right-hand side of the functional relation (94) and match the notation with that of (12) in order to obtain

$$\begin{aligned} I_{\nu } ( z ) =& \frac{{e^{z} }}{{ ( {2\pi z} ) ^{\frac{1}{2}} }} \Biggl( {\sum _{n = 0}^{N - 1} { ( { - 1} ) ^{n} \frac{{a_{n} ( \nu ) }}{{z^{n} }}} + R_{N}^{ ( K ) } \bigl( {ze^{ \mp \pi i} , \nu } \bigr) } \Biggr) \\ &{}\pm ie^{ \pm \pi i\nu } \frac{{e^{ - z} }}{{ ( {2\pi z} ) ^{ \frac{1}{2}} }} \Biggl( {\sum _{m = 0}^{M - 1} {\frac{{a_{m} ( \nu ) }}{{z^{m} }}} + R_{M}^{ ( K ) } ( {z,\nu } ) } \Biggr) . \end{aligned}$$
(95)

The analogous expression for the derivative can be deduced by substituting (95) into the right-hand side of the connection formula \(2I'_{\nu } ( z ) = I_{\nu + 1} ( z ) + I _{\nu - 1} ( z ) \) (see, e.g., [24, 10.29.E1]). By making use of the relations (20) and (66) and taking into account the notation of (13), we obtain

$$\begin{aligned} I'_{\nu } ( z ) =& \frac{{e^{z} }}{{ ( {2\pi z} ) ^{\frac{1}{2}} }} \Biggl( {\sum _{n = 0}^{N - 1} { ( { - 1} ) ^{n} \frac{{b_{n} ( \nu ) }}{{z^{n} }}} + R_{N}^{(K')} \bigl( {ze^{ \mp \pi i} , \nu } \bigr) } \Biggr) \\ &{} \mp ie^{ \pm \pi i \nu } \frac{{e^{ - z} }}{{ ( {2\pi z} ) ^{\frac{1}{2}} }} \Biggl( {\sum _{m = 0}^{M - 1} {\frac{{b_{m} ( \nu ) }}{ {z^{m} }}} + R_{M}^{(K')} ( {z,\nu } ) } \Biggr) . \end{aligned}$$

Appendix B: Bounds for the Basic Terminants

In this appendix, we prove some simple estimates for the absolute value of the basic terminants \(\varLambda_{p} ( w ) \) and \(\varPi_{p} ( w ) \) with \(p>0\). These estimates depend only on \(p\) and the argument of \(w\) and therefore also provide bounds for the quantities \(\sup _{r \geq 1} | {\varLambda_{p} ( {2zr } ) } |\) and \(\sup _{r \geq 1} | {\varPi _{p} ( {2zr } ) } |\) which appear in Theorems 1.51.9.

Proposition B.1

For any \(p>0\), the following holds:

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le \textstyle\begin{cases} 1 & \textit{if } \vert \arg w \vert \leq \frac{\pi }{2}, \\ \min ( { \vert \csc ( {\arg w} ) \vert ,\sqrt{e ( {p + \frac{1}{2}} ) } } ) & \textit{if } \frac{\pi }{2} < \vert \arg w \vert \leq \pi , \end{cases} $$
(96)

and

$$ \bigl\vert {\varPi_{p} ( w ) } \bigr\vert \le \textstyle\begin{cases} 1 & \textit{if } \vert \arg w \vert \leq \frac{\pi }{4}, \\ \min ( { \vert \csc ( {2\arg w} ) \vert ,\sqrt{ \frac{e}{4} ( {p + \frac{3}{2}} ) } } ) & \textit{if } \frac{\pi }{4} < \vert \arg w \vert \leq \frac{\pi }{2}. \end{cases} $$
(97)

Moreover, when \(w\) is positive, we have \(0<\varLambda_{p} ( w ) <1\) and \(0<\varPi_{p} ( w ) <1\).

Proof

To prove the estimate (96), it suffices to consider the range \(0 \le \arg w \le \pi \), because \(\varLambda_{p} ( {\bar{w}} ) = \overline{\varLambda_{p} ( w ) }\). Our starting point is the integral representation

$$ \varLambda_{p} ( w ) = \frac{1}{{\varGamma ( {p } ) }} \int_{0}^{ + \infty } {\frac{{t^{p-1} e^{ - t} }}{{1 + t/w}}dt} , $$
(98)

which is valid when \(\vert \arg w \vert <\pi \) and \(p>0\) (cf. [24, 8.6.E4]). For \(t \geq 0\), we have

$$ \biggl\vert {1 + \frac{t}{w}} \biggr\vert \ge \textstyle\begin{cases} 1 & \mbox{if } 0 \leq \arg w \leq \frac{\pi }{2}, \\ \sin ( \arg w ) & \mbox{if } \frac{\pi }{2} < \arg w < \pi , \end{cases} $$
(99)

and therefore

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le \textstyle\begin{cases} 1 & \mbox{if } 0 \leq \arg w \leq \frac{\pi }{2}, \\ \csc ( {\arg w} ) & \mbox{if } \frac{\pi }{2} < \arg w < \pi . \end{cases} $$

To complete the proof, one has to show that \(\vert {\varLambda_{p} ( w ) } \vert \le \sqrt{e ( {p + \frac{1}{2}} ) }\) when \(\frac{\pi }{2} < \arg w \le \pi \). For this purpose, we deform the contour of integration in (98) by rotating it through an acute angle \(\varphi \). Thus, by appealing to Cauchy’s theorem and analytic continuation, we have, for arbitrary \(0<\varphi <\frac{\pi }{2}\), that

$$ \varLambda_{p} ( w ) = \frac{1}{{\varGamma ( {p } ) }} \int_{0}^{ + \infty e^{i\varphi } } {\frac{{t^{p-1} e^{ - t} }}{{1 + t/w}}dt} $$
(100)

when \(\frac{\pi }{2} < \arg w \le \pi \). Employing the inequality (99), we find that

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le \frac{1}{{\cos^{p} \varphi }} \times \textstyle\begin{cases} 1 & \mbox{if } \frac{\pi }{2} < \arg w \le \frac{\pi }{2} + \varphi , \\ \csc ( \arg w -\varphi ) & \mbox{if } \frac{\pi }{2} + \varphi < \arg w \le \pi . \end{cases} $$

We now choose the value of \(\varphi \) which minimizes the right-hand side of this inequality when \(\arg w =\pi \), namely \(\varphi = \arctan (p^{ - \frac{1}{2}} )\). We may then claim that

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le \frac{1}{{\cos^{p} ( \arctan (p^{ - \frac{1}{2}} ))}} = \biggl( {1 + \frac{1}{{p }}} \biggr) ^{\frac{{p }}{2}} < \sqrt{e \biggl( {p + \frac{1}{2}} \biggr) } , $$

when \(\frac{\pi }{2} < \arg w \le \frac{\pi }{2} + \arctan (p^{ - \frac{1}{2}} )\), where the last inequality can be obtained by means of elementary analysis. In the remaining case \(\frac{\pi }{2} + \arctan (p^{ - \frac{1}{2}} ) < \arg w \le \pi \), we have

$$\begin{aligned} \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le& \frac{{\csc ( \arg w - \arctan (p^{ - \frac{1}{2}} ))}}{{\cos^{p} (\arctan (p^{ - \frac{1}{2}} ))}} \\ \le& \frac{{\csc ( \pi - \arctan (p^{ - \frac{1}{2}} ))}}{{\cos^{p} (\arctan (p^{ - \frac{1}{2}} ))}} = \biggl( {1 + \frac{1}{ {p }}} \biggr) ^{\frac{p+1}{2}} p^{\frac{1}{2}} < \sqrt{e \biggl( {p + \frac{1}{2}} \biggr) } . \end{aligned}$$

For the last step, note that the quantity \(( {1 + \frac{1}{{p}}} ) ^{\frac{p+1}{2}} \sqrt{\frac{{p }}{{p + a}}}\), as a function of \(p>0\), increases monotonically if and only if \(a\geq \frac{1}{2}\), in which case it has limit \(\sqrt{e}\).

The estimate (97) can be proved in a similar way, starting from the representation

$$ \varPi_{p} ( w ) = \frac{1}{{\varGamma ( {p } ) }} \int _{0}^{ + \infty } {\frac{{t^{p-1} e^{ - t} }}{{1 + ( {t/w} ) ^{2} }}dt} , $$
(101)

which is valid when \(\vert \arg w \vert <\frac{\pi }{2}\) and \(p>0\). This representation can be obtained from (98) and the definition of the basic terminant \(\varPi_{p} ( w ) \).

Finally, in the case of a positive \(w\), notice that \(0 < 1/ ( 1 + t/w ) < 1\) and \(0 < 1/(1 + ( {t/w} ) ^{2}) < 1\) for any \(t>0\). Therefore, the integral representations (98) and (101) combined with the mean value theorem of integration imply that \(0<\varLambda_{p} ( w ) <1\) and \(0<\varPi_{p} ( w ) <1\). □

Proposition B.2

For any \(p>0\) and \(w\) with \(\frac{\pi }{2} < \vert {\arg w} \vert < \frac{3\pi }{2}\), we have

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le \frac{{ \vert {\csc ( {\arg w - \varphi } ) } \vert }}{{\cos^{p} \varphi }}, $$
(102)

where \(\varphi \) is the unique solution of the implicit equation

$$ ( {p + 1} ) \cos ( {\arg w -2\varphi } ) = ( {p - 1} ) \cos ( \arg w ) $$

that satisfies \(0 < \varphi < - \frac{\pi }{2} + \arg w\) if \(\frac{\pi }{2} < \arg w < \pi \), \(- \pi + \arg w < \varphi < \frac{ \pi }{2}\) if \(\pi \le \arg w < \frac{3\pi }{2}\), \(0 < \varphi < \frac{ \pi }{2} + \arg w\) if \(- \pi < \arg w < -\frac{\pi }{2}\) and \(- \frac{\pi }{2} < \varphi < \pi + \arg w\) if \(- \frac{{3\pi }}{2} < \arg w \le - \pi \).

Similarly, for any \(p>0\) and \(w\) with \(\frac{\pi }{4} < \vert {\arg w} \vert < \pi \), we have

$$ \bigl\vert {\varPi_{p} ( w ) } \bigr\vert \le \frac{{ \vert {\csc ( {2 ( {\arg w - \varphi ' } ) } ) } \vert }}{ {\cos^{p} \varphi ' }}, $$
(103)

where \(\varphi '\) is the unique solution of the implicit equation

$$ ( {p + 2} ) \cos \bigl( {2\arg w -3\varphi '} \bigr) = ( {p - 2} ) \cos \bigl( 2\arg w-\varphi ' \bigr) $$

that satisfies \(0 < \varphi ' < - \frac{\pi }{4} + \arg w\) if \(\frac{\pi }{4} < \arg w < \frac{\pi }{2}\), \(- \frac{\pi }{2} + \arg w < \varphi ' < -\frac{\pi }{4}+\arg w\) if \(\frac{\pi }{2} \le \arg w < \frac{3\pi }{4}\), \(- \frac{\pi }{2} + \arg w < \varphi ' < \frac{\pi }{2}\) if \(\frac{3\pi }{4} \le \arg w < \pi \), \(\frac{\pi }{4} + \arg w < \varphi ' < 0\) if \(-\frac{\pi }{2} < \arg w < -\frac{\pi }{4}\), \(\frac{\pi }{4} + \arg w < \varphi ' < \frac{ \pi }{2}+\arg w\) if \(-\frac{3\pi }{4} < \arg w \le -\frac{\pi }{2}\) and \(- \frac{\pi }{2} < \varphi ' < \frac{\pi }{2}+ \arg w\) if \(-\pi < \arg w \le -\frac{3\pi }{4}\).

We remark that the values of \(\varphi \) and \(\varphi '\) in this proposition are chosen so as to minimize the right-hand sides of the inequalities (102) and (103), respectively.

Proof

It is enough to prove (102) when \(\frac{\pi }{2}<\arg w< \frac{3 \pi }{2}\), because this inequality for the case \(-\frac{3\pi }{2}< \arg w< -\frac{\pi }{2}\) can be derived by taking complex conjugates. Note that the representation (100) is actually valid in the wider range \(\frac{\pi }{2} < \arg w < \pi + \varphi \). Thus, we can infer that

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le \frac{{\csc ( {\arg w - \varphi } ) }}{{\cos^{p} \varphi }}, $$

provided \(\frac{\pi }{2}+\varphi < \arg w < \pi + \varphi \). We would like to choose \(\varphi \) in a way that the right-hand side of this inequality is minimized. A lemma of Meijer [23, pp. 953–954] shows that this minimization problem has a unique solution with the properties given in the statement of Proposition B.2.

The bound for \(\varPi_{p} ( w ) \) can be proved similarly: we deform the contour of integration in (101) by rotating it through an arbitrary acute angle \(\varphi '\), and then we employ the inequality (99) and use the corresponding lemma of Meijer [23, p. 956] to minimize the resulting estimate. □

Proposition B.3

For any \(p>0\), we have

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le 1 + \varGamma \biggl( {\frac{{p }}{2} + 1} \biggr) \mathbf{F} \biggl( { \frac{1}{2},\frac{p}{2};\frac{ {p}}{2} + 1;\cos^{2} ( {\arg w} ) } \biggr) \le 1 + \chi ( {p } ) , $$
(104)

for \(\frac{\pi }{2} < \vert {\arg w} \vert \le \pi \), and

$$ \bigl\vert {\varPi_{p} ( w ) } \bigr\vert \le 1 + \frac{1}{2}\varGamma \biggl( {\frac{{p }}{2} + 1} \biggr) \mathbf{F} \biggl( {\frac{1}{2}, \frac{p}{2};\frac{{p }}{2} + 1; \sin^{2} ( {\arg w} ) } \biggr) \le 1 + \frac{{\chi ( {p } ) }}{2}, $$
(105)

for \(\frac{\pi }{4} < \vert {\arg w} \vert \le \frac{\pi }{2}\). Here \(\chi ( {p } ) \) is defined by (77).

Proof

It is sufficient to prove (104) for \(\frac{\pi }{2}<\arg w \leq \pi \), as the estimates for \(-\pi \leq \arg w<-\frac{\pi }{2}\) can be derived by taking complex conjugates. To do so, we consider the integral representation

$$ \varLambda_{p} ( w ) = \int_{0}^{ + \infty } {\frac{{e^{ - t} }}{{ ( {1 + t/w} ) ^{p} }}dt} , $$

which is valid when \(\vert \arg w \vert <\pi \) and \(p>0\) (cf. [24, 8.6.E5]). Integrating once by parts, we obtain

$$ \varLambda_{p} ( w ) = 1 - \frac{p}{w} \int_{0}^{ + \infty } {\frac{{e^{ - t} }}{{ ( {1 + t/w} ) ^{p + 1} }}dt} . $$
(106)

Assuming that \(\frac{\pi }{2}<\arg w<\pi \), we can deform the contour of integration in (106) by rotating it through a right angle. And therefore, by Cauchy’s theorem and analytic continuation, we have

$$ \varLambda_{p} ( w ) = 1 - \frac{{p }}{w} \int_{0}^{ + \infty e^{\frac{\pi }{2}i} } {\frac{{e^{ - t} }}{{ ( {1 + t/w} ) ^{p + 1} }}dt} = 1 - i \frac{p}{w} \int_{0}^{ + \infty } {\frac{{e^{ - it} }}{{ ( {1 + it/w} ) ^{p + 1} }}dt} $$
(107)

when \(\frac{\pi }{2}<\arg w\leq \pi \). Hence, we may assert that

$$\begin{aligned} \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert & \le 1 + \frac{{p }}{ { \vert w \vert }} \int_{0}^{ + \infty } {\frac{{dt}}{{ \vert {1 + it/w} \vert ^{p + 1} }}} \\ &= 1 + \frac{{p }}{{ \vert w \vert }} \int_{0}^{ + \infty } {\frac{{dt}}{{(1 + 2t/ \vert w \vert \sin ( {\arg w} ) + t^{2} / \vert w \vert ^{2} )^{\frac{p+1}{2}} }}} \\ & = 1 + p \int_{0}^{ + \infty } {\frac{{du}}{{(1 + 2u\sin ( {\arg w} ) + u^{2} )^{\frac{p+1}{2}} }}} . \end{aligned}$$

To simplify this result, we can proceed as follows. When \(\arg w= \pi \), we perform a change of integration variable from \(u\) to \(t\) by \(t=u^{2}\), and, using the beta integral together with the known evaluation of the regularized hypergeometric function with argument 1 (see, e.g., [24, 5.12.E3 and 15.4.E20]), we obtain

$$\begin{aligned} \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le& 1 + \frac{p}{2} \int _{0}^{ + \infty } {\frac{{t^{ - \frac{1}{2}} }}{{ ( {1 + t} ) ^{\frac{{p + 1}}{2}} }}dt} = 1 + \frac{{\varGamma ( {\frac{1}{2}} ) \varGamma ( {\frac{p}{2} + 1} ) }}{{\varGamma ( 1 ) \varGamma ( {\frac{p}{2} + \frac{1}{2}} ) }} \\ =& 1 + \varGamma \biggl( {\frac{p}{2} + 1} \biggr) \mathbf{F} \biggl( {\frac{1}{2}, \frac{p}{2};\frac{p}{2} + 1;1} \biggr) . \end{aligned}$$

Since \(\cos^{2} ( \pi ) =1\), this result is in agreement with (104). In the case that \(\frac{\pi }{2}<\arg w < \pi \), a change of variable from \(u\) to \(t\) via \(t = 1 - \sin^{2} (\arg w)/(u + \sin (\arg w))^{2}\) and the well-known integral representation of the regularized hypergeometric function (see [24, 15.6.E1]) yield

$$\begin{aligned} \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le& 1 + \frac{p}{2} \sin ( {\arg w} ) \int_{0}^{1} {\frac{{ ( {1 - t} ) ^{\frac{p}{2} - 1} }}{{(1 - t\cos^{2} ( {\arg w} ) )^{\frac{ {p + 1}}{2}} }}dt} \\ = & 1 + \sin ( {\arg w} ) \varGamma \biggl( {\frac{p}{2} + 1} \biggr) \mathbf{F} \biggl( {\frac{p}{2}+\frac{1}{2},1; \frac{p}{2} + 1;\cos^{2} ( {\arg w} ) } \biggr) . \end{aligned}$$

The linear transformation \(\sin (\arg w)\mathbf{F} ( {\frac{p}{2}+ \frac{1}{2},1;\frac{p}{2} + 1;\cos^{2} (\arg w)} ) = \mathbf{F} ( \frac{1}{2},\frac{p}{2};\frac{p}{2} + 1; \cos^{2} (\arg w) ) \) (cf. [24, 15.8.E1]) then shows that this bound is equivalent to the required one in (104).

To obtain the second inequality in (104), note that

$$\begin{aligned} \varGamma \biggl( {\frac{{p }}{2} + 1} \biggr) \mathbf{F} \biggl( { \frac{{1 }}{2}, \frac{p}{2};\frac{{p }}{2} + 1; \cos^{2} ( {\arg w} ) } \biggr) & \le \varGamma \biggl( { \frac{{p }}{2} + 1} \biggr) \mathbf{F} \biggl( {\frac{ {1 }}{2}, \frac{p}{2};\frac{{p}}{2} + 1;1} \biggr) \\ & = \frac{{\varGamma ( {\frac{1}{2}} ) \varGamma ( {\frac{ {p }}{2} + 1} ) }}{{\varGamma ( 1 ) \varGamma ( {\frac{p}{2} + \frac{1}{2}} ) }} = \pi^{\frac{1}{2}} \frac{{\varGamma ( {\frac{{p }}{2} + 1} ) }}{{\varGamma ( {\frac{{p }}{2} + \frac{1}{2}} ) }} = \chi ( {p } ) . \end{aligned}$$

The bounds (105) for \(\varPi_{p} ( w ) \) may be proved as follows. From the definition of the basic terminant \(\varPi_{p} ( w ) \), we can infer that

$$ \bigl\vert {\varPi_{p} ( w ) } \bigr\vert \le \frac{1}{2} \bigl( { | {\varLambda_{p} \bigl(we^{\frac{\pi }{2}i} \bigr)} | + | { \varLambda_{p} \bigl(we ^{ - \frac{\pi }{2}i} \bigr)} |} \bigr) . $$

When \(\frac{\pi }{4} < \pm \arg w \le \frac{\pi }{2}\), the term \(| {\varLambda_{p} (we^{\pm \frac{\pi }{2}i} )} |\) is bounded by (104), and we can see that \(| {\varLambda_{p} (we^{ \mp \frac{ \pi }{2}i} )} | \leq 1\) from (96). □

Before we proceed to our last set of bounds, we would like to compare the estimates given in Propositions B.1 and B.3. For the purpose of brevity, we consider only the bounds for \(\varLambda_{p} ( w ) \); the other basic terminant \(\varPi_{p} ( w ) \) can be treated in a similar way.

First, assume that \(\frac{\pi }{2}< \vert \arg w \vert <\pi \), \(\vert \arg w \vert \) is bounded away from \(\pi \) and \(p\) is large. Employing a linear transformation formula and the large-\(c\) asymptotics of the regularized hypergeometric function \(\mathbf{F} ( {a,b;c;w} ) \) (see, e.g., [24, 15.8.E1 and 15.12.E2]), we find that

$$\begin{aligned} {\mathbf{F}} \biggl( {\frac{1}{2},\frac{p}{2};\frac{p}{2} + 1;\cos^{2} ( {\arg w} ) } \biggr) & = \bigl\vert {\csc ( {\arg w} ) } \bigr\vert {\mathbf{F}} \biggl( {\frac{1}{2},1;\frac{p}{2} + 1; - \cot^{2} ( {\arg w} ) } \biggr) \\ & = \frac{{ \vert {\csc ( {\arg w} ) } \vert }}{{\varGamma ( {\frac{p}{2} + 1} ) }} \biggl( {1 + \mathcal{O}_{\arg w} \biggl( { \frac{1}{p}} \biggr) } \biggr) . \end{aligned}$$

Consequently, from (104), we have

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le 1 + \bigl\vert {\csc ( {\arg w} ) } \bigr\vert \biggl( {1 + \mathcal{O}_{\arg w} \biggl( {\frac{1}{p}} \biggr) } \biggr) . $$

By comparing this inequality with (96), one sees that, under the above circumstances, the estimate (96) is sharper than (104).

Consider now the case when \(\frac{\pi }{2}< \vert \arg w \vert <\pi \), \(\vert \arg w \vert \) is close to \(\pi \) and \(p\) is bounded. A linear transformation formula [24, 15.8.E4] and the definition of the regularized hypergeometric function yield

$$\begin{aligned} {\mathbf{F}} \biggl( {\frac{1}{2},\frac{p}{2};\frac{p}{2} + 1;\cos^{2} ( {\arg w} ) } \biggr) = & \frac{\pi }{{\varGamma ( {\frac{p}{2} + \frac{1}{2}} ) }}{ \mathbf{F}} \biggl( {\frac{1}{2},\frac{p}{2}; \frac{1}{2}; \sin^{2} ( {\arg w} ) } \biggr) \\ &{} - \pi^{\frac{1}{2}} \frac{{ \vert {\sin ( {\arg w} ) } \vert }}{ {\varGamma ( {\frac{p}{2}} ) }}{\mathbf{F}} \biggl( { \frac{p}{2} + \frac{1}{2},1;\frac{3}{2};\sin^{2} ( {\arg w} ) } \biggr) \\ = & \frac{{\pi^{\frac{1}{2}} }}{{\varGamma ( {\frac{p}{2} + \frac{1}{2}} ) }} - \frac{{2 \vert {\sin ( {\arg w} ) } \vert }}{{\varGamma ( {\frac{p}{2}} ) }} + \mathcal{O}_{p} \bigl( {\sin^{2} ( {\arg w} ) } \bigr) . \end{aligned}$$

Whence, by (104), we deduce

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le 1 + \chi ( p ) - \bigl\vert {\sin ( {\arg w} ) } \bigr\vert p + \mathcal{O}_{p} \bigl( {\sin^{2} ( {\arg w} ) } \bigr) . $$

Comparison with (96) then shows that, under the above circumstances, the bound (104) is sharper and behaves better than (96), unless perhaps when \(\sqrt{e ( {p + \frac{1}{2}} ) } < \vert {\csc ( {\arg w} ) } \vert \).

Finally, we discuss the case when \(\vert \arg w \vert =\pi \). It can readily be shown that

$$ \sqrt{e \biggl( {p + \frac{1}{2}} \biggr) } < 1 + \chi ( {p } ) \quad \mbox{for } 0 < p < 5.9564 \ldots , $$

whence (96) is sharper than (104) when \(\vert \arg w \vert \) is equal or close to \(\pi \) and \(p\) is small. On the other hand, the saddle point method applied to (107) shows that when \(\vert \arg w \vert = \pi \) and \(\vert p - \vert w \vert \vert \) is bounded, the asymptotics

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \sim \biggl( { \frac{ {\pi p}}{2}} \biggr) ^{\frac{1}{2}} $$

holds as \(p\to +\infty \). Thus, using the large-\(p\) asymptotics \(\chi ( p ) \sim ( {\frac{{\pi p}}{2}} ) ^{ \frac{1}{2}}\), we can infer that the inequality \(\vert {\varLambda_{p} ( w ) } \vert \le 1 + \chi ( p ) \) is asymptotically sharp.

Proposition B.4

For any \(p>0\), we have

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le \frac{{\sqrt{2 \pi p} }}{{ \vert {\cos ( {\arg w} ) } \vert ^{p} }} + \bigl\vert {\varLambda_{p} \bigl(we^{ \mp 2\pi i} \bigr)} \bigr\vert \le \frac{{2\chi ( {p } ) }}{{ \vert {\cos ( {\arg w} ) } \vert ^{p} }} + \bigl\vert {\varLambda_{p} \bigl(we^{ \mp 2\pi i} \bigr)} \bigr\vert , $$
(108)

for \(\pi < \pm \arg w < \frac{3\pi }{2}\), and

$$ \bigl\vert {\varPi_{p} (w)} \bigr\vert \le \frac{{\sqrt{2\pi p} }}{{2 \vert {\sin ( {\arg w} ) } \vert ^{p} }} + \bigl\vert {\varPi_{p} \bigl(we ^{ \mp \pi i} \bigr)} \bigr\vert \le \frac{{\chi ( {p } ) }}{{ \vert {\sin ( {\arg w} ) } \vert ^{p } }} + \bigl\vert { \varPi_{p} \bigl(we ^{ \mp \pi i} \bigr)} \bigr\vert , $$
(109)

for \(\frac{\pi }{2} < \pm \arg w <\pi \).

The dependence on \(\vert w \vert \) in these estimates may be eliminated by employing the bounds for \(\vert {\varLambda_{p} (we^{ \mp 2\pi i} )} \vert \) and \(\vert {\varPi_{p} (we^{ \mp \pi i} )} \vert \) that were derived previously.

Proof

The proof of (108) is based on the functional relation

$$ \varLambda_{p} ( w ) = \pm 2\pi i \frac{{e^{ \mp \pi ip} }}{ {\varGamma ( {p } ) }}w^{p} e^{w} + \varLambda_{p} \bigl(we^{ \mp 2 \pi i} \bigr) $$
(110)

(cf. [24, 8.2.E10]). We take the upper or lower sign in (110) according as \(\pi < \arg w < \frac{3\pi }{2}\) or \(-\frac{3\pi }{2} < \arg w < -\pi \). Now, from (110) we can infer that

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le 2\pi \frac{1}{{\varGamma ( {p } ) }} \vert w \vert ^{p } e^{ - \vert w \vert \vert {\cos ( {\arg w} ) } \vert } + \bigl\vert {\varLambda_{p} \bigl(we ^{ \mp 2\pi i} \bigr)} \bigr\vert . $$

Notice that the quantity \(r^{p} e^{ - r\alpha }\), as a function of \(r>0\), takes its maximum value at \(r=p/\alpha \) when \(\alpha >0\) and \(p>0\). We therefore find that

$$ \bigl\vert {\varLambda_{p} ( w ) } \bigr\vert \le 2\pi \frac{1}{{\varGamma ( {p } ) }}\frac{{p^{p} e^{ - p} }}{{ \vert {\cos ( {\arg w} ) } \vert ^{p } }} + \bigl\vert {\varLambda_{p} \bigl(we^{ \mp 2 \pi i} \bigr)} \bigr\vert \le \frac{{\sqrt{2\pi p} }}{{ \vert {\cos ( {\arg w} ) } \vert ^{p } }} + \bigl\vert {\varLambda_{p} \bigl(we^{ \mp 2 \pi i} \bigr)} \bigr\vert . $$

The second inequality can be obtained from the well known fact that \(\sqrt{2\pi } p^{p - \frac{1}{2}} e^{ - p} \le \varGamma ( p ) \) for any \(p>0\) (see, for instance, [24, 5.6.E1]). We derive the second bound in (108) from the result that

$$ \biggl( \frac{p}{2} \biggr) ^{\frac{1}{2}} \le \frac{{\varGamma ( {\frac{p}{2} + 1} ) }}{{\varGamma ( {\frac{p}{2} + \frac{1}{2}} ) }} $$

for any \(p>0\) (see, e.g., [24, 5.6.E4]) and the definition of \(\chi ( p ) \).

The estimate (109) can be deduced in an analogous manner, starting from the functional relation

$$ \varPi_{p} (w) = \pm \pi i \frac{{e^{ \mp \frac{\pi }{2}ip} }}{{\varGamma ( {p } ) }}w^{p } e^{ \pm iw} + \varPi_{p} \bigl(we^{ \mp \pi i} \bigr), $$

which can be obtained from (110) and the definition of the basic terminant \(\varPi_{p} ( w ) \). □

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Nemes, G. Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl Math 150, 141–177 (2017). https://doi.org/10.1007/s10440-017-0099-0

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