11 Struve and Related Functions Struve and Modified Struve Functions 11.5 Integral Representations 11.7 Integrals and Sums

§11.6 Asymptotic Expansions

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Referenced by:: §11.13(ii)
Permalink:: http://dlmf.nist.gov/11.6
See also:: Annotations for Ch.11

§11.6(i) Large $|z|$ , Fixed $\nu$
§11.6(ii) Large $|\nu|$ , Fixed $z$
§11.6(iii) Large $|\nu|$ , Fixed $z/\nu$

§11.6(i) Large $|z|$ , Fixed $\nu$

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Keywords:: Struve functions and modified Struve functions, asymptotic expansions, large argument, remainder terms
Notes:: For (11.6.1) apply Watson’s lemma (§2.4(i)) to (11.5.2), or combine Watson (1944, p. 333, Eq. (2)) with (11.2.5). See also the subsequent text in this reference, and Olver (1997b, p. 277, Ex. 15.5). For (11.6.2) convert (11.5.4) to a loop integral $\int_{0}^{(1+)}$ to remove the restriction $\Re\nu>-\frac{1}{2}$ , extend the loop to pass through the point $t=\infty$ on the positive real axis, then apply Laplace’s method (§2.4(iii)) to each of the two integrals with paths from $t=0$ to $t=\infty$ , one passing below $t=1$ and the other passing above $t=1$ . For (11.6.3) write the integrals over the intervals $[0,\infty)$ and $[z,\infty)$ ; use (11.6.1) with the first term extracted, and a limiting procedure on the integral over $[0,\infty)$ . For (11.6.4) replace $z$ by $i z$ in (11.6.3) and apply (11.2.5), (11.2.6), (10.27.11).
Referenced by:: §11.13(iv)
Permalink:: http://dlmf.nist.gov/11.6.i
See also:: Annotations for §11.6 and Ch.11

11.6.1		$\mathbf{K}_{\nu}\left(z\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{\Gamma% \left(k+\tfrac{1}{2}\right)(\tfrac{1}{2}z)^{\nu-2k-1}}{\Gamma\left(\nu+\tfrac{% 1}{2}-k\right)},$
$\|\operatorname{ph}z\|\leq\pi-\delta$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : 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 $m(\geq 0)$ terms, then the remainder term $R_{m}(z)$ is $O\left(z^{\nu-2m-1}\right)$ . If $\nu$ is real, $z$ is positive, and $m+\tfrac{1}{2}-\nu\geq 0$ , then $R_{m}(z)$ is of the same sign and numerically less than the first neglected term.

11.6.2		$\mathbf{M}_{\nu}\left(z\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}(-1)^{k+1}% \frac{\Gamma\left(k+\tfrac{1}{2}\right)(\tfrac{1}{2}z)^{\nu-2k-1}}{\Gamma\left% (\nu+\tfrac{1}{2}-k\right)},$
$\|\operatorname{ph}z\|\leq\tfrac{1}{2}\pi-\delta$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : 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		$\int_{0}^{z}\mathbf{K}_{0}\left(t\right)\,\mathrm{d}t-\frac{2}{\pi}(\ln\left(2% z\right)+\gamma)\sim\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(2k)!(2k-1% )!}{(k!)^{2}(2z)^{2k}},$
$\|\operatorname{ph}z\|\leq\pi-\delta$ ,
ⓘ Symbols: $\gamma$ : Euler’s constant, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : 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		$\int_{0}^{z}\mathbf{M}_{0}\left(t\right)\,\mathrm{d}t+\frac{2}{\pi}(\ln\left(2% z\right)+\gamma)\sim\frac{2}{\pi}\sum_{k=1}^{\infty}\frac{(2k)!(2k-1)!}{(k!)^{% 2}(2z)^{2k}},$
$\|\operatorname{ph}z\|\leq\tfrac{1}{2}\pi-\delta$ ,
ⓘ Symbols: $\gamma$ : Euler’s constant, $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : 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(ii) Large $|\nu|$ , Fixed $z$

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Keywords:: Struve functions and modified Struve functions, asymptotic expansions, generalized, large order
Notes:: Apply (5.11.7) to (11.2.1), (11.2.2).
Permalink:: http://dlmf.nist.gov/11.6.ii
See also:: Annotations for §11.6 and Ch.11

11.6.5		$\mathbf{H}_{\nu}\left(z\right),\mathbf{L}_{\nu}\left(z\right)\sim\frac{z}{\pi% \nu\sqrt{2}}\left(\frac{ez}{2\nu}\right)^{\nu},$
$\|\operatorname{ph}\nu\|\leq\pi-\delta$ .
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order and $\delta$ : arbitrary small positive constant Referenced by: (11.11.7) 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)).

§11.6(iii) Large $|\nu|$ , Fixed $z/\nu$

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Notes:: For (11.6.6) and (11.6.9) see Watson (1944, §10.43): a similar method can be used for (11.6.7), starting from (11.5.4).
Referenced by:: Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/11.6.iii
Change (effective with 1.0.11):: The phrase “and for higher coefficients $c_{k}(\lambda)$ see Dingle (1973, p. 203)” that appeared formerly below (11.6.8) was replaced by a sentence with reference to Nemes (2015b).
See also:: Annotations for §11.6 and Ch.11

For fixed $\lambda(>1)$

11.6.6		$\mathbf{K}_{\nu}\left(\lambda\nu\right)\sim\frac{(\tfrac{1}{2}\lambda\nu)^{\nu% -1}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}\sum_{k=0}^{\infty}\frac{k!% c_{k}(\lambda)}{\nu^{k}},$
$\|\operatorname{ph}\nu\|\leq\tfrac{1}{2}\pi-\delta$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\operatorname{ph}$ : 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		$\mathbf{M}_{\nu}\left(\lambda\nu\right)\sim-\frac{(\frac{1}{2}\lambda\nu)^{\nu% -1}}{\sqrt{\pi}\Gamma\left(\nu+\frac{1}{2}\right)}\sum_{k=0}^{\infty}\frac{k!c% _{k}(i\lambda)}{\nu^{k}},$
$\|\operatorname{ph}\nu\|\leq\frac{1}{2}\pi-\delta$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, $\operatorname{ph}$ : 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	$\displaystyle c_{0}(\lambda)$	$\displaystyle=1,$
	$\displaystyle c_{1}(\lambda)$	$\displaystyle=2\lambda^{-2},$
	$\displaystyle c_{2}(\lambda)$	$\displaystyle=6\lambda^{-4}-\tfrac{1}{2}\lambda^{-2},$
	$\displaystyle c_{3}(\lambda)$	$\displaystyle=20\lambda^{-6}-4\lambda^{-4},$
	$\displaystyle c_{4}(\lambda)$	$\displaystyle=70\lambda^{-8}-\tfrac{45}{2}\lambda^{-6}+\tfrac{3}{8}\lambda^{-4}.$
ⓘ 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		$\mathbf{L}_{\nu}\left(\lambda\nu\right)\sim I_{\nu}\left(\lambda\nu\right),$
$\|\operatorname{ph}\nu\|\leq\tfrac{1}{2}\pi-\delta$ ,
ⓘ Symbols: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\operatorname{ph}$ : 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).

11.5 Integral Representations 11.7 Integrals and Sums

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§11.6 Asymptotic Expansions

Contents

§11.6(i) Large |z|, Fixed ν

§11.6(ii) Large |ν|, Fixed z

§11.6(iii) Large |ν|, Fixed z/ν

§11.6(i) Large $|z|$ , Fixed $\nu$

§11.6(ii) Large $|\nu|$ , Fixed $z$

§11.6(iii) Large $|\nu|$ , Fixed $z/\nu$