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 by . 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 $I_{\nu}\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{% \infty}(-1)^{k}\frac{a_{k}(\nu)}{z^{k}},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$ ,

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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, : nonnegative integer, : 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 $K_{\nu}\left(z\right)\sim\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\sum_{% k=0}^{\infty}\frac{a_{k}(\nu)}{z^{k}},$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$ ,

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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, : nonnegative integer, : 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 $I_{\nu}'\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{% \infty}(-1)^{k}\frac{b_{k}(\nu)}{z^{k}},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$ ,

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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, : nonnegative integer, : 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 $K_{\nu}'\left(z\right)\sim-\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\sum% _{k=0}^{\infty}\frac{b_{k}(\nu)}{z^{k}},$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$ .

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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, : nonnegative integer, : 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 yields the following more general (and more accurate) version of (10.40.1):

10.40.5 $I_{\nu}\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{% \infty}(-1)^{k}\frac{a_{k}(\nu)}{z^{k}}\pm ie^{\pm\nu\pi i}\frac{e^{-z}}{(2\pi z% )^{\frac{1}{2}}}\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{z^{k}},$ $-\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{2}\pi-\delta$ .

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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, : nonnegative integer, : 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 $I_{\nu}\left(z\right)K_{\nu}\left(z\right)\sim\frac{1}{2z}\left(1-\frac{1}{2}% \frac{\mu-1}{(2z)^{2}}+\frac{1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2z)^{4}% }-\dotsb\right),$

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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, : 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 $I_{\nu}'\left(z\right)K_{\nu}'\left(z\right)\sim-\frac{1}{2z}\left(1+\frac{1}{% 2}\frac{\mu-3}{(2z)^{2}}-\frac{1}{2\cdot 4}\frac{(\mu-1)(\mu-45)}{(2z)^{4}}+% \dotsb\right),$

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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, : 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 $\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\sim\left(\frac{\pi}{2z}% \right)^{\frac{1}{2}}\frac{\nu e^{-z}}{z}\sum_{k=0}^{\infty}\frac{\alpha_{k}(% \nu)}{(8z)^{k}},$

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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 with respect to , $\,\partial\NVar{x}$ : partial differential of , : nonnegative integer, : 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 $\alpha_{k}(\nu)=\frac{(4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k+1)^{% 2})}{(k+1)!}\*\left(\frac{1}{4\nu^{2}-1^{2}}+\frac{1}{4\nu^{2}-3^{2}}+\cdots+% \frac{1}{4\nu^{2}-(2k+1)^{2}}\right).$

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Defines:: $\alpha_{k}(\nu)$ : polynomial coefficient (locally)
Symbols:: : factorial (as in ), : 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 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 $K_{\nu}\left(z\right)=\left(\frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\left(% \sum_{k=0}^{\ell-1}\frac{a_{k}(\nu)}{z^{k}}+R_{\ell}(\nu,z)\right),$ $\ell=1,2,\dotsc$ .

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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, : nonnegative integer, : 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 $|R_{\ell}(\nu,z)|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\infty}\left(t^{-\ell}% \right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,\infty}\left(t^{-1}% \right)\right),$

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Defines:: $R_{\ell}(\nu,z)$ : remainder (locally)
Symbols:: $\exp\NVar{z}$ : exponential function, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : total variation, : 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 $\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&|% \operatorname{ph}z|\leq\tfrac{1}{2}\pi,\\ \chi(\ell)|z|^{-\ell},&\tfrac{1}{2}\pi\leq|\operatorname{ph}z|\leq\pi,\\ 2\chi(\ell)|\Re z|^{-\ell},&\pi\leq|\operatorname{ph}z|<\tfrac{3}{2}\pi,\end{cases}$

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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 : 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 , 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 $R_{\ell}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi\right)\*\left(\sum_{k=0}^{m-1}% \frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left(2z\right)+R_{m,\ell}(\nu,z)\right),$

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