§11.6 Asymptotic Expansions

11.6.1		$${\mathbf{K}}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$
$|\mathrm{ph}z|\le \pi -\delta$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{K}}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant A&S Ref: 12.1.29 Referenced by: §11.6(i), §11.6(i), §11.6(i), §11.6(i) Permalink: http://dlmf.nist.gov/11.6.E1 Encodings: TeX, pMML, png See also: Annotations for §11.6(i), §11.6 and Ch.11

where $\delta$ is an arbitrary small positive constant. If the series on the right-hand side of (11.6.1) is truncated after $m(\ge 0)$ terms, then the remainder term $R_m(z)$ is $O\left(z^{\nu -2m-1}\right)$. If $\nu$ is real, $z$ is positive, and $m+\frac{1}{2}-\nu \ge 0$, then $R_m(z)$ is of the same sign and numerically less than the first neglected term.

11.6.2		$${\mathbf{M}}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$
$|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{M}}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant A&S Ref: 12.2.6 Referenced by: §11.6(i), §11.6(i), §11.6(i) Permalink: http://dlmf.nist.gov/11.6.E2 Encodings: TeX, pMML, png See also: Annotations for §11.6(i), §11.6 and Ch.11

For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445).

For the corresponding expansions for ${\mathbf{H}}_{\nu }\left(z\right)$ and ${\mathbf{L}}_{\nu }\left(z\right)$ combine (11.6.1), (11.6.2) with (11.2.5), (11.2.6), (10.17.4), and (10.40.1).

11.6.3		$${\int }_0^z{\mathbf{K}}_0\left(t\right)dt-\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$
$|\mathrm{ph}z|\le \pi -\delta$,
ⓘ Symbols: $\gamma$: Euler’s constant, ${\mathbf{K}}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant A&S Ref: 12.1.32 Referenced by: §11.6(i) Permalink: http://dlmf.nist.gov/11.6.E3 Encodings: TeX, pMML, png See also: Annotations for §11.6(i), §11.6 and Ch.11

11.6.4		$${\int }_0^z{\mathbf{M}}_0\left(t\right)dt+\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$
$|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,
ⓘ Symbols: $\gamma$: Euler’s constant, ${\mathbf{M}}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant A&S Ref: 12.2.8 Referenced by: §11.6(i) Permalink: http://dlmf.nist.gov/11.6.E4 Encodings: TeX, pMML, png See also: Annotations for §11.6(i), §11.6 and Ch.11

where $\gamma$ is Euler’s constant (§5.2(ii)).

11.6.5		$${\mathbf{H}}_{\nu }\left(z\right),{\mathbf{L}}_{\nu }\left(z\right)\sim \frac{z}{\pi \nu \sqrt{2}}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$
$|\mathrm{ph}\nu |\le \pi -\delta$.
ⓘ Symbols: ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${\mathbf{L}}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order and $\delta$: arbitrary small positive constant Referenced by: §11.11(ii) Permalink: http://dlmf.nist.gov/11.6.E5 Encodings: TeX, pMML, png See also: Annotations for §11.6(ii), §11.6 and Ch.11

More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)).

For fixed $\lambda (>1)$

11.6.6		$${\mathbf{K}}_{\nu }\left(\lambda \nu \right)\sim \frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\lambda )}{{\nu }^k},$$
$|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{K}}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function A&S Ref: 12.1.34 Referenced by: §11.6(iii) Permalink: http://dlmf.nist.gov/11.6.E6 Encodings: TeX, pMML, png See also: Annotations for §11.6(iii), §11.6 and Ch.11

and for fixed $\lambda$ $(>0)$

11.6.7		$${\mathbf{M}}_{\nu }\left(\lambda \nu \right)\sim -\frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\mathrm{i}\lambda )}{{\nu }^k},$$
$|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{M}}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function Referenced by: §11.6(iii) Permalink: http://dlmf.nist.gov/11.6.E7 Encodings: TeX, pMML, png See also: Annotations for §11.6(iii), §11.6 and Ch.11

Here

11.6.8		$c_0(\lambda )$	$=1,$
		$c_1(\lambda )$	$=2{\lambda }^{-2},$
		$c_2(\lambda )$	$=6{\lambda }^{-4}-\frac{1}{2}{\lambda }^{-2},$
		$c_3(\lambda )$	$=20{\lambda }^{-6}-4{\lambda }^{-4},$
		$c_4(\lambda )$	$=70{\lambda }^{-8}-\frac{45}{2}{\lambda }^{-6}+\frac{3}{8}{\lambda }^{-4}.$
ⓘ Symbols: $\lambda$: parameter and $c_k(\lambda )$: expansion function Referenced by: §11.6(iii) Permalink: http://dlmf.nist.gov/11.6.E8 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §11.6(iii), §11.6 and Ch.11

These and higher coefficients $c_k(\lambda )$ can be computed via the representations in Nemes (2015b).

For the corresponding result for ${\mathbf{H}}_{\nu }\left(\lambda \nu \right)$ use (11.2.5) and (10.19.6). See also Watson (1944, p. 336).

For fixed $\lambda$ $(>0)$

11.6.9		$${\mathbf{L}}_{\nu }\left(\lambda \nu \right)\sim I_{\nu }\left(\lambda \nu \right),$$
$|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,
ⓘ Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${\mathbf{L}}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter Referenced by: §11.6(iii) Permalink: http://dlmf.nist.gov/11.6.E9 Encodings: TeX, pMML, png See also: Annotations for §11.6(iii), §11.6 and Ch.11

and for an estimate of the relative error in this approximation see Watson (1944, p. 336).