§10.40 Asymptotic Expansions for Large Argument

With the notation of §§10.17(i) and 10.17(ii), as $z\to \mathrm{\infty }$ with $\nu$ fixed,

10.40.1		$I_{\nu }\left(z\right)$	$\sim {\displaystyle \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}}{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{\displaystyle \frac{a_k(\nu )}{z^k}},$
$|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient A&S Ref: 9.7.1 Referenced by: §10.30(ii), §10.40(i), §10.40(i), §10.40(ii), §10.40(iii), §10.45, §10.67(i), §11.6(i) Permalink: http://dlmf.nist.gov/10.40.E1 Encodings: TeX, pMML, png See also: Annotations for §10.40(i), §10.40 and Ch.10
10.40.2		$K_{\nu }\left(z\right)$	$\sim {\left({\displaystyle \frac{\pi }{2z}}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{a_k(\nu )}{z^k}},$
$|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient A&S Ref: 9.7.2 Referenced by: §10.40(i), §10.40(i), §10.40(ii), §10.40(iii), §10.41(iv), §10.45 Permalink: http://dlmf.nist.gov/10.40.E2 Encodings: TeX, pMML, png See also: Annotations for §10.40(i), §10.40 and Ch.10

10.40.3		$$I_{\nu }^{\prime }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_k(\nu )}{z^k},$$
$|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $b_k(\nu )$: polynomial coefficient A&S Ref: 9.7.3 Referenced by: §10.40(i) Permalink: http://dlmf.nist.gov/10.40.E3 Encodings: TeX, pMML, png See also: Annotations for §10.40(i), §10.40 and Ch.10

10.40.4		$$K_{\nu }^{\prime }\left(z\right)\sim -{\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{z^k},$$
$|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $b_k(\nu )$: polynomial coefficient A&S Ref: 9.7.4 Referenced by: §10.40(i), §10.40(i), §10.67(i) Permalink: http://dlmf.nist.gov/10.40.E4 Encodings: TeX, pMML, png See also: Annotations for §10.40(i), §10.40 and Ch.10

Corresponding expansions for $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$, $I_{\nu }^{\prime }\left(z\right)$, and $K_{\nu }^{\prime }\left(z\right)$ for other ranges of $\mathrm{ph}z$ are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). In particular, use of (10.34.3) with $m=0$ yields the following more general (and more accurate) version of (10.40.1):

10.40.5		$$I_{\nu }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_k(\nu )}{z^k}\pm \mathrm{i}{\mathrm{e}}^{\pm \nu \pi \mathrm{i}}\frac{{\mathrm{e}}^{-z}}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$
$-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$.
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient Referenced by: §10.40(i) Permalink: http://dlmf.nist.gov/10.40.E5 Encodings: TeX, pMML, png See also: Annotations for §10.40(i), §10.40 and Ch.10

In the expansion (10.40.2) assume that $z>0$ and the sum is truncated when $k=\mathrm{\ell }-1$. Then the remainder term does not exceed the first neglected term in absolute value and has the same sign provided that $\mathrm{\ell }\ge \max (|\nu |-\frac{1}{2},1)$.

For the error term in (10.40.1) see §10.40(iii).

For (10.40.2) write

10.40.10		$$K_{\nu }\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\left(\sum _{k=0}^{\mathrm{\ell }-1}\frac{a_k(\nu )}{z^k}+R_{\mathrm{\ell }}(\nu ,z)\right),$$
$\mathrm{\ell }=1,2,\mathrm{\dots }$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient and $R_{\mathrm{\ell }}(\nu ,z)$: remainder Referenced by: §10.40(i), §10.40(iii), §10.40(iv) Permalink: http://dlmf.nist.gov/10.40.E10 Encodings: TeX, pMML, png See also: Annotations for §10.40(iii), §10.40 and Ch.10

Then

10.40.11		$$|R_{\mathrm{\ell }}(\nu ,z)|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\mathrm{\infty }}\left(t^{-1}\right)\right),$$
ⓘ Defines: $R_{\mathrm{\ell }}(\nu ,z)$: remainder (locally) Symbols: $\exp z$: exponential function, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient Permalink: http://dlmf.nist.gov/10.40.E11 Encodings: TeX, pMML, png See also: Annotations for §10.40(iii), §10.40 and Ch.10

where $𝒱$ denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that $|\mathrm{\Re }t|$ changes monotonically. Bounds for ${𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ are given by

10.40.12
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, ${𝒱}_{a,b}\left(f\right)$: total variation and $z$: complex variable Referenced by: §10.40(i), §10.40(iii), Erratum (V1.0.11) for Equation (10.40.12) Permalink: http://dlmf.nist.gov/10.40.E12 Encodings: TeX, pMML, png Correction (effective with 1.0.11): Originally the third constraint was written incorrectly as $\pi \le |\mathrm{ph}z|\le \frac{3}{2}\pi$. Suggested 2014-11-05 by Gergő Nemes See also: Annotations for §10.40(iii), §10.40 and Ch.10

where $\chi (\mathrm{\ell })={\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}\mathrm{\ell }+1\right)/\mathrm{\Gamma }\left(\frac{1}{2}\mathrm{\ell }+\frac{1}{2}\right)$; see §9.7(i).

A similar result for (10.40.1) is obtained by combining (10.34.3), with $m=0$, and (10.40.10)–(10.40.12); see Olver (1997b, p. 269).

In (10.40.10)

10.40.13		$$R_{\mathrm{\ell }}(\nu ,z)={(-1)}^{\mathrm{\ell }}2\cos \left(\nu \pi \right)\left(\sum _{k=0}^{m-1}\frac{a_k(\nu )}{z^k}{G}_{\mathrm{\ell }-k}\left(2z\right)+R_{m,\mathrm{\ell }}(\nu ,z)\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $G_p\left(z\right)$: rescaled terminant function, $m$: integer, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient and $R_{m,\mathrm{\ell }}(\nu ,z)$: remainder Permalink: http://dlmf.nist.gov/10.40.E13 Encodings: TeX, pMML, png See also: Annotations for §10.40(iv), §10.40 and Ch.10

where $G_p\left(z\right)$ is given by (10.17.16). If $z\to \mathrm{\infty }$ with $|\mathrm{\ell }-2|z||$ bounded and $m$ $(\ge 0)$ fixed, then

10.40.14		$$R_{m,\mathrm{\ell }}(\nu ,z)=O\left({\mathrm{e}}^{-2|z|}{z}^{-m}\right),$$
$|\mathrm{ph}z|\le \pi$.
ⓘ Defines: $R_{m,\mathrm{\ell }}(\nu ,z)$: remainder (locally) Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $m$: integer, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.40.E14 Encodings: TeX, pMML, png See also: Annotations for §10.40(iv), §10.40 and Ch.10

For higher re-expansions of the remainder term see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Paris (2001a, b).