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

Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.40

§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

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 $\mathop{I_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{K_{{\nu}}\/}\nolimits\!% \left(x\right)$ and ${\mathop{I_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right){\mathop{K_{{\nu}}\/% }\nolimits^{{\prime}}}\!\left(x\right)$ satisfy the same differential equations as ${\mathop{M_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right)=|\mathop{{H^{{(1)}}_{{% \nu}}}\/}\nolimits\!\left(x\right)|^{2}=\mathop{{H^{{(1)}}_{{\nu}}}\/}% \nolimits\!\left(x\right)\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(x\right)$ and ${\mathop{N_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right)=|{\mathop{{H^{{(1)}}_{{% \nu}}}\/}\nolimits^{{\prime}}}\!\left(x\right)|^{2}={\mathop{{H^{{(1)}}_{{\nu}% }}\/}\nolimits^{{\prime}}}\!\left(x\right){\mathop{{H^{{(2)}}_{{\nu}}}\/}% \nolimits^{{\prime}}}\!\left(x\right)$ , 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).
Keywords:: Hankel’s expansions, derivatives, modified Bessel functions
Referenced by:: §10.74(i)
Permalink:: http://dlmf.nist.gov/10.40.i

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

10.40.1 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{{\frac% {1}{2}}}}\sum_{{k=0}}^{\infty}(-1)^{k}\frac{a_{k}(\nu)}{z^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta$ ,

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : nonnegative integer, : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant and $a_{k}(\nu)$ : expansion
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

10.40.2 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)\sim\left(\frac{\pi}{2z}\right)^{% {\frac{1}{2}}}e^{{-z}}\sum_{{k=0}}^{\infty}\frac{a_{k}(\nu)}{z^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{3}{2}\pi-\delta$ ,

Symbols:: : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : nonnegative integer, : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant and $a_{k}(\nu)$ : expansion
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

10.40.3 ${\mathop{I_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)\sim\frac{e^{z}}{(2% \pi z)^{\frac{1}{2}}}\sum_{{k=0}}^{\infty}(-1)^{k}\frac{b_{k}(\nu)}{z^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta$ ,

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : nonnegative integer, : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant and $b_{k}(\nu)$ : expansion
A&S Ref:: 9.7.3
Referenced by:: §10.40(i)
Permalink:: http://dlmf.nist.gov/10.40.E3
Encodings:: TeX, pMML, png

10.40.4 ${\mathop{K_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)\sim-\left(\frac{\pi% }{2z}\right)^{{\frac{1}{2}}}e^{{-z}}\sum_{{k=0}}^{\infty}\frac{b_{k}(\nu)}{z^{% k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{3}{2}\pi-\delta$ .

Symbols:: : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : nonnegative integer, : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant and $b_{k}(\nu)$ : expansion
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

Corresponding expansions for $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ , $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ , ${\mathop{I_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , and ${\mathop{K_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ for other ranges of $\mathop{\mathrm{ph}\/}\nolimits z$ 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 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{\frac{% 1}{2}}}\sum_{{k=0}}^{\infty}(-1)^{k}\frac{a_{k}(\nu)}{z^{k}}\pm ie^{{\pm\nu\pi i% }}\frac{e^{{-z}}}{(2\pi z)^{\frac{1}{2}}}\sum_{{k=0}}^{\infty}\frac{a_{k}(\nu)% }{z^{k}},$ $-\tfrac{1}{2}\pi+\delta\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{2% }\pi-\delta$ .

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : nonnegative integer, : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant and $a_{k}(\nu)$ : expansion
Referenced by:: §10.40(i)
Permalink:: http://dlmf.nist.gov/10.40.E5
Encodings:: TeX, pMML, png

¶ Products

Keywords:: derivatives, modified Bessel functions

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

10.40.6 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{K_{{\nu}}\/}\nolimits\!% \left(z\right)\sim\frac{1}{2z}\left(1-\frac{1}{2}\frac{\mu-1}{(2z)^{2}}+\frac{% 1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2z)^{4}}-\cdots\right),$

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\sim$ : asymptotic equality, : 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

10.40.7 ${\mathop{I_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right){\mathop{K_{{\nu}}\/% }\nolimits^{{\prime}}}\!\left(z\right)\sim-\frac{1}{2z}\left(1+\frac{1}{2}% \frac{\mu-3}{(2z)^{2}}-\frac{1}{2\cdot 4}\frac{(\mu-1)(\mu-45)}{(2z)^{4}}+% \cdots\right),$

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\sim$ : asymptotic equality, : 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

as $z\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta$ . 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

Keywords:: modified Bessel functions

For fixed $\nu$ ,

10.40.8 $\frac{\partial\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}\sim% \left(\frac{\pi}{2z}\right)^{{\frac{1}{2}}}\frac{\nu e^{{-z}}}{z}\sum_{{k=0}}^% {\infty}\frac{\alpha_{k}(\nu)}{(8z)^{k}},$

Symbols:: : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\sim$ : asymptotic equality, : nonnegative integer, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.40.E8
Encodings:: TeX, pMML, png

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

10.40.9 $\alpha_{k}(\nu)=\frac{(4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k+1)^{% 2})}{(k+1)!}\*\left(\frac{1}{4\nu^{2}-1^{2}}+\frac{1}{4\nu^{2}-3^{2}}+\cdots+% \frac{1}{4\nu^{2}-(2k+1)^{2}}\right).$

Symbols:: : : factorial, : nonnegative integer and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.40.E9
Encodings:: TeX, pMML, png

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

Notes:: See Watson (1944, pp.206–207) or Olver (1997b, pp. 268–269) and (10.27.8).
Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.40.ii

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

Notes:: See Olver (1997b, pp. 266–267) and (10.27.8).
Keywords:: modified Bessel functions
Referenced by:: §10.40(ii), §10.67(i)
Permalink:: http://dlmf.nist.gov/10.40.iii

For (10.40.2) write

10.40.10 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=\left(\frac{\pi}{2z}\right)^{{% \frac{1}{2}}}e^{{-z}}\left(\sum_{{k=0}}^{{\ell-1}}\frac{a_{k}(\nu)}{z^{k}}+R_{% \ell}(\nu,z)\right),$ $\ell=1,2,\ldots$ .

Symbols:: : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : nonnegative integer, : complex variable, $\nu$ : complex parameter, $a_{k}(\nu)$ : expansion 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

Then

10.40.11 $|R_{\ell}(\nu,z)|\leq 2|a_{\ell}(\nu)|\mathop{\mathcal{V}_{{z,\infty}}\/}% \nolimits\!\left(t^{{-\ell}}\right)\*\mathop{\exp\/}\nolimits\left(|\nu^{2}-% \tfrac{1}{4}|\mathop{\mathcal{V}_{{z,\infty}}\/}\nolimits\!\left(t^{{-1}}% \right)\right),$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\mathcal{V}_{{a,b}}\/}\nolimits\!\left(f\right)$ : total variation, : complex variable, $\nu$ : complex parameter, $a_{k}(\nu)$ : expansion and $R_{\ell}(\nu,z)$ : remainder
Permalink:: http://dlmf.nist.gov/10.40.E11
Encodings:: TeX, pMML, png

where $\mathop{\mathcal{V}\/}\nolimits$ denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that $|\realpart{t}|$ changes monotonically. Bounds for $\mathop{\mathcal{V}_{{z,\infty}}\/}\nolimits\!\left(t^{{-\ell}}\right)$ are given by

10.40.12 $\mathop{\mathcal{V}_{{z,\infty}}\/}\nolimits\!\left(t^{{-\ell}}\right)\leq% \begin{cases}|z|^{{-\ell}},&|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{2}% \pi,\\ \chi(\ell)|z|^{{-\ell}},&\frac{1}{2}\pi\leq|\mathop{\mathrm{ph}\/}\nolimits z|% \leq\pi,\\ 2\chi(\ell)|\realpart{z}|^{{-\ell}},&\pi\leq|\mathop{\mathrm{ph}\/}\nolimits z% |\leq\frac{3}{2}\pi,\end{cases}$

Symbols:: $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $\mathop{\mathcal{V}_{{a,b}}\/}\nolimits\!\left(f\right)$ : total variation and : complex variable
Referenced by:: §10.40(i), §10.40(iii)
Permalink:: http://dlmf.nist.gov/10.40.E12
Encodings:: TeX, pMML, png

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

Notes:: See Olver (1991b), together with (10.39.6).
Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.40.iv

In (10.40.10)

10.40.13 $R_{\ell}(\nu,z)=(-1)^{\ell}2\mathop{\cos\/}\nolimits(\nu\pi)\*\left(\sum_{{k=0% }}^{{m-1}}\frac{a_{k}(\nu)}{z^{k}}\mathop{G_{{\ell-k}}\/}\nolimits\!\left(2z% \right)+R_{{m,\ell}}(\nu,z)\right),$