10 Bessel Functions Modified Bessel Functions 10.39 Relations to Other Functions 10.41 Asymptotic Expansions for Large Order

§10.40 Asymptotic Expansions for Large Argument

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Keywords:: asymptotic expansions for large argument, modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.40
See also:: Annotations for Ch.10

§10.40(i) Hankel’s Expansions
§10.40(ii) Error Bounds for Real Argument and Order
§10.40(iii) Error Bounds for Complex Argument and Order
§10.40(iv) Exponentially-Improved Expansions

§10.40(i) Hankel’s Expansions

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Keywords:: Hankel’s expansions, asymptotic expansions for large argument, derivatives, for modified Bessel functions, modified Bessel functions
Notes:: See Watson (1944, pp. 202–203) and Olver (1997b, pp. 250–251 and 325). (10.40.3) and (10.40.4) are obtained by differentiation of (10.40.1) and (10.40.2); compare §2.1(iii). (10.40.6) and (10.40.7) are obtained by multiplication of (10.40.1)–(10.40.4): that the coefficients are the same as in (10.18.17) and (10.18.19) is a consequence of the fact that $I_{\nu}\left(x\right)K_{\nu}\left(x\right)$ and $I_{\nu}'\left(x\right)K_{\nu}'\left(x\right)$ satisfy the same differential equations as ${M_{\nu}}^{2}\left(x\right)=|{H^{(1)}_{\nu}}\left(x\right)|^{2}={H^{(1)}_{\nu}% }\left(x\right){H^{(2)}_{\nu}}\left(x\right)$ and ${N_{\nu}}^{2}\left(x\right)=|{H^{(1)}_{\nu}}'\left(x\right)|^{2}={H^{(1)}_{\nu% }}'\left(x\right){H^{(2)}_{\nu}}'\left(x\right)$ , respectively, except for replacement of $x$ by $i x$ . For the statement concerning the accuracy of (10.40.5) use the error bounds given by (10.40.10)–(10.40.12).
Referenced by:: §10.74(i)
Permalink:: http://dlmf.nist.gov/10.40.i
See also:: Annotations for §10.40 and Ch.10

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

10.40.1	$\displaystyle I_{\nu}\left(z\right)$	$\displaystyle\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{\infty}(-1)^{% k}\frac{a_{k}(\nu)}{z^{k}},$
$\|\operatorname{ph}z\|\leq\tfrac{1}{2}\pi-\delta$ ,
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{ph}$ : 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	$\displaystyle K_{\nu}\left(z\right)$	$\displaystyle\sim\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\sum_{k=0}^{% \infty}\frac{a_{k}(\nu)}{z^{k}},$
$\|\operatorname{ph}z\|\leq\tfrac{3}{2}\pi-\delta$ ,
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : 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		$I_{\nu}'\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{% \infty}(-1)^{k}\frac{b_{k}(\nu)}{z^{k}},$
$\|\operatorname{ph}z\|\leq\tfrac{1}{2}\pi-\delta$ ,
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{ph}$ : 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		$K_{\nu}'\left(z\right)\sim-\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\sum% _{k=0}^{\infty}\frac{b_{k}(\nu)}{z^{k}},$
$\|\operatorname{ph}z\|\leq\tfrac{3}{2}\pi-\delta$ .
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : 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 $I_{\nu}\left(z\right)$ , $K_{\nu}\left(z\right)$ , $I_{\nu}'\left(z\right)$ , and $K_{\nu}'\left(z\right)$ for other ranges of $\operatorname{ph}z$ 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		$I_{\nu}\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{% \infty}(-1)^{k}\frac{a_{k}(\nu)}{z^{k}}\pm ie^{\pm\nu\pi i}\frac{e^{-z}}{(2\pi z% )^{\frac{1}{2}}}\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{z^{k}},$
$-\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{2}\pi-\delta$ .
ⓘ 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, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{ph}$ : 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

Products

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Keywords:: asymptotic expansions for large argument, derivatives, for products, modified Bessel functions
See also:: Annotations for §10.40(i), §10.40 and Ch.10

With $\mu=4\nu^{2}$ and fixed,

10.40.6	$\displaystyle I_{\nu}\left(z\right)K_{\nu}\left(z\right)$	$\displaystyle\sim\frac{1}{2z}\left(1-\frac{1}{2}\frac{\mu-1}{(2z)^{2}}+\frac{1% \cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2z)^{4}}-\dotsb\right),$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.7.5 Referenced by: §10.40(i), §10.40(i) Permalink: http://dlmf.nist.gov/10.40.E6 Encodings: TeX, pMML, png See also: Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10
10.40.7	$\displaystyle I_{\nu}'\left(z\right)K_{\nu}'\left(z\right)$	$\displaystyle\sim-\frac{1}{2z}\left(1+\frac{1}{2}\frac{\mu-3}{(2z)^{2}}-\frac{% 1}{2\cdot 4}\frac{(\mu-1)(\mu-45)}{(2z)^{4}}+\dotsb\right),$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.7.6 Referenced by: §10.40(i), §10.40(i) Permalink: http://dlmf.nist.gov/10.40.E7 Encodings: TeX, pMML, png See also: Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

as $z\to\infty$ in $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$ . The general terms in (10.40.6) and (10.40.7) can be written down by analogy with (10.18.17), (10.18.19), and (10.18.20).

$\nu$ -Derivative

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Keywords:: asymptotic expansion for large argument, asymptotic expansions for large argument, derivatives with respect to order, for derivatives with respect to order, modified Bessel functions
See also:: Annotations for §10.40(i), §10.40 and Ch.10

For fixed $\nu$ ,

10.40.8		$\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\sim\left(\frac{\pi}{2z}% \right)^{\frac{1}{2}}\frac{\nu e^{-z}}{z}\sum_{k=0}^{\infty}\frac{\alpha_{k}(% \nu)}{(8z)^{k}},$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter and $\alpha_{k}(\nu)$ : polynomial coefficient Permalink: http://dlmf.nist.gov/10.40.E8 Encodings: TeX, pMML, png See also: Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

as $z\to\infty$ in $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$ . Here $\alpha_{0}(\nu)=1$ and

10.40.9		$\alpha_{k}(\nu)=\frac{(4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k+1)^{% 2})}{(k+1)!}\*\left(\frac{1}{4\nu^{2}-1^{2}}+\frac{1}{4\nu^{2}-3^{2}}+\cdots+% \frac{1}{4\nu^{2}-(2k+1)^{2}}\right).$
ⓘ Defines: $\alpha_{k}(\nu)$ : polynomial coefficient (locally) Symbols: $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.40.E9 Encodings: TeX, pMML, png See also: Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

§10.40(ii) Error Bounds for Real Argument and Order

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Keywords:: asymptotic expansions for large argument, error bounds, modified Bessel functions
Notes:: See Watson (1944, pp.206–207) or Olver (1997b, pp. 268–269) and (10.27.8).
Permalink:: http://dlmf.nist.gov/10.40.ii
See also:: Annotations for §10.40 and Ch.10

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

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

§10.40(iii) Error Bounds for Complex Argument and Order

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Keywords:: asymptotic expansions for large argument, error bounds, modified Bessel functions
Notes:: See Olver (1997b, pp. 266–267) and (10.27.8).
Referenced by:: §10.40(ii), §10.67(i)
Permalink:: http://dlmf.nist.gov/10.40.iii
See also:: Annotations for §10.40 and Ch.10

For (10.40.2) write

10.40.10		$K_{\nu}\left(z\right)=\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\left(% \sum_{k=0}^{\ell-1}\frac{a_{k}(\nu)}{z^{k}}+R_{\ell}(\nu,z)\right),$
$\ell=1,2,\dotsc$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $a_{k}(\nu)$ : polynomial coefficient and $R_{\ell}(\nu,z)$ : 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		$\|R_{\ell}(\nu,z)\|\leq 2\|a_{\ell}(\nu)\|\mathcal{V}_{z,\infty}\left(t^{-\ell}% \right)\*\exp\left(\|\nu^{2}-\tfrac{1}{4}\|\mathcal{V}_{z,\infty}\left(t^{-1}% \right)\right),$
ⓘ Defines: $R_{\ell}(\nu,z)$ : remainder (locally) Symbols: $\exp\NVar{z}$ : exponential function, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : 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 $\mathcal{V}$ denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that $|\Re t|$ changes monotonically. Bounds for $\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)$ are given by

10.40.12		$\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)\leq\begin{cases}\|z\|^{-\ell},&\|% \operatorname{ph}z\|\leq\tfrac{1}{2}\pi,\\ \chi(\ell)\|z\|^{-\ell},&\tfrac{1}{2}\pi\leq\|\operatorname{ph}z\|\leq\pi,\\ 2\chi(\ell)\|\Re z\|^{-\ell},&\pi\leq\|\operatorname{ph}z\|<\tfrac{3}{2}\pi,\end{cases}$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : 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 $\pi\leq\|\operatorname{ph}z\|<\frac{3}{2}\pi$ was written incorrectly as $\pi\leq\|\operatorname{ph}z\|\leq\frac{3}{2}\pi$ . Suggested 2014-11-05 by Gergő Nemes See also: Annotations for §10.40(iii), §10.40 and Ch.10

where $\chi(\ell)=\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}\ell+1\right)/\Gamma\left(% \tfrac{1}{2}\ell+\tfrac{1}{2}\right)$ ; 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).

§10.40(iv) Exponentially-Improved Expansions

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Keywords:: asymptotic expansions for large argument, exponentially-improved, modified Bessel functions
Notes:: See Olver (1991b), together with (10.39.6).
Permalink:: http://dlmf.nist.gov/10.40.iv
See also:: Annotations for §10.40 and Ch.10

In (10.40.10)

10.40.13		$R_{\ell}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi\right)\*\left(\sum_{k=0}^{m-1}% \frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left(2z\right)+R_{m,\ell}(\nu,z)\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $G_{\NVar{p}}\left(\NVar{z}\right)$ : rescaled terminant function, $m$ : integer, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $a_{k}(\nu)$ : polynomial coefficient and $R_{m,\ell}(\nu,z)$ : 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 $z\to\infty$ with $|\ell-2|z||$ bounded and $m$ $(\geq 0)$ fixed, then

10.40.14		$R_{m,\ell}(\nu,z)=O\left(e^{-2\|z\|}z^{-m}\right),$
$\|\operatorname{ph}z\|\leq\pi$ .
ⓘ Defines: $R_{m,\ell}(\nu,z)$ : remainder (locally) Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : 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).

10.39 Relations to Other Functions 10.41 Asymptotic Expansions for Large Order

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