§11.6 Asymptotic Expansions

Referenced by:: §11.13(ii)
Permalink:: http://dlmf.nist.gov/11.6

§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 $\nu$

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 $\realpart{\nu}>-\frac{1}{2}$ , 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 $iz$ in (11.6.3) and apply (11.2.5), (11.2.6), (10.27.11).
Keywords:: Struve functions and modified Struve functions
Referenced by:: §11.13(iv)
Permalink:: http://dlmf.nist.gov/11.6.i

11.6.1 $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)\sim\frac{1}{\pi}\sum _{{k=0}}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(k+\tfrac{1}{2}\right)(\tfrac{1}{2}z)^{{\nu-2k-1}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}-k\right)},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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
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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 $\mathop{O\/}\nolimits\!\left(z^{{\nu-2m-1}}\right)$ . 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 $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)\sim\frac{1}{\pi}\sum _{{k=0}}^{\infty}(-1)^{{k+1}}\frac{\mathop{\Gamma\/}\nolimits\!\left(k+\tfrac{1}{2}\right)(\tfrac{1}{2}z)^{{\nu-2k-1}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}-k\right)},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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
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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 $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ and $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ combine (11.6.1), (11.6.2) with (11.2.5), (11.2.6), (10.17.4), and (10.40.1).

11.6.3 $\int _{0}^{z}\mathop{\mathbf{K}_{{0}}\/}\nolimits\!\left(t\right)dt-\frac{2}{\pi}(\mathop{\ln\/}\nolimits\!\left(2z\right)+\EulerConstant)\sim\frac{2}{\pi}\sum _{{k=1}}^{\infty}(-1)^{{k+1}}\frac{(2k)!(2k-1)!}{(k!)^{2}(2z)^{{2k}}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ,

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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

11.6.4 $\int _{0}^{z}\mathop{\mathbf{M}_{{0}}\/}\nolimits\!\left(t\right)dt+\frac{2}{\pi}(\mathop{\ln\/}\nolimits\!\left(2z\right)+\EulerConstant)\sim\frac{2}{\pi}\sum _{{k=1}}^{\infty}\frac{(2k)!(2k-1)!}{(k!)^{2}(2z)^{{2k}}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta$ ,

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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
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where $\EulerConstant$ is Euler’s constant (§5.2(ii)).

§11.6(ii) Large $|\nu|$ , Fixed $z$

Notes:: Apply (5.11.7) to (11.2.1), (11.2.2).
Keywords:: Struve functions and modified Struve functions
Permalink:: http://dlmf.nist.gov/11.6.ii

11.6.5 $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right),\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)\sim\frac{z}{\pi\nu\sqrt{2}}\left(\frac{ez}{2\nu}\right)^{\nu},$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\pi-\delta$ .

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $e$ : base of exponential function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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

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 $z/\nu$

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).
Permalink:: http://dlmf.nist.gov/11.6.iii

For fixed $\lambda(>1)$

11.6.6 $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(\lambda\nu\right)\sim\frac{(\tfrac{1}{2}\lambda\nu)^{{\nu-1}}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\sum _{{k=0}}^{\infty}\frac{k!c_{k}(\lambda)}{\nu^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\tfrac{1}{2}\pi-\delta$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $!$ : $n!$ : factorial, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\nu$ : 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

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

11.6.7 $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(\lambda\nu\right)\sim-\frac{(\frac{1}{2}\lambda\nu)^{{\nu-1}}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)}\sum _{{k=0}}^{\infty}\frac{k!c_{k}(i\lambda)}{\nu^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\frac{1}{2}\pi-\delta$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $!$ : $n!$ : factorial, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\nu$ : 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

Here

11.6.8

$c_{0}(\lambda)=1,$

$c_{1}(\lambda)=2\lambda^{{-2}},$

$c_{2}(\lambda)=6\lambda^{{-4}}-\tfrac{1}{2}\lambda^{{-2}},$

$c_{3}(\lambda)=20\lambda^{{-6}}-4\lambda^{{-4}},$

$c_{4}(\lambda)=70\lambda^{{-8}}-\tfrac{45}{2}\lambda^{{-6}}+\tfrac{3}{8}\lambda^{{-4}},$

Symbols:: $\lambda$ : parameter and $c_{k}(\lambda)$ : expansion function
Permalink:: http://dlmf.nist.gov/11.6.E8
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and for higher coefficients $c_{k}(\lambda)$ see Dingle (1973, p. 203).

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

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

11.6.9 $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(\lambda\nu\right)\sim\mathop{I_{{\nu}}\/}\nolimits\!\left(\lambda\nu\right),$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\tfrac{1}{2}\pi-\delta$ ,

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\nu$ : 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
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and for an estimate of the relative error in this approximation see Watson (1944, p. 336).

§11.6 Asymptotic Expansions

Contents

§11.6(i) Large , Fixed

§11.6(ii) Large , Fixed

§11.6(iii) Large , Fixed

§11.6(i) Large $|z|$ , Fixed $\nu$

§11.6(ii) Large $|\nu|$ , Fixed $z$

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