10 Bessel Functions Kelvin Functions 10.66 Expansions in Series of Bessel Functions 10.68 Modulus and Phase Functions

§10.67 Asymptotic Expansions for Large Argument

Permalink:: http://dlmf.nist.gov/10.67

§10.67(i) $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{bei}_{{\nu}}\/}% \nolimits x,\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{kei}_{{% \nu}}\/}\nolimits x$ , and Derivatives
§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu=0$

§10.67(i) $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{bei}_{{\nu}}\/}% \nolimits x,\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{kei}_{{% \nu}}\/}\nolimits x$ , and Derivatives

Notes:: Combine (10.61.1), (10.61.2), and their differentiated forms with (10.40.1)–(10.40.4). To obtain the exponentially-small terms in (10.67.3), (10.67.4), (10.67.7), and (10.67.8), use the identity $\pi i\mathop{I_{{\nu}}\/}\nolimits\!\left(xe^{{\pi i/4}}\right)=\mathop{K_{{% \nu}}\/}\nolimits\!\left(xe^{{-3\pi i/4}}\right)-e^{{\nu\pi i}}\mathop{K_{{\nu% }}\/}\nolimits\!\left(xe^{{\pi i/4}}\right)$ , obtained from (10.27.6) and (10.27.9). The final sentence in §10.67(i) is justified by error bounds obtained as in §10.40(iii).
Keywords:: Kelvin functions
Referenced by:: §10.67(i)
Permalink:: http://dlmf.nist.gov/10.67.i

Define $a_{k}(\nu)$ and $b_{k}(\nu)$ as in §§10.17(i) and 10.17(ii). Then as $x\to\infty$ with $\nu$ fixed,

10.67.1 $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x\sim e^{{-x/\sqrt{2}}}\left(\frac{% \pi}{2x}\right)^{{\frac{1}{2}}}\*\sum_{{k=0}}^{\infty}\frac{a_{k}(\nu)}{x^{k}}% \mathop{\cos\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{k% }{4}+\frac{1}{8}\right)\pi\right),$

Symbols:: $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\sim$ : asymptotic equality, : nonnegative integer, : real variable, $\nu$ : complex parameter and $a_{k}(\nu)$ : expansion
A&S Ref:: 9.10.3, 9.10.5--9.10.7 (modified)
Referenced by:: §10.61(v), §10.67(ii), §10.68(iii)
Permalink:: http://dlmf.nist.gov/10.67.E1
Encodings:: TeX, pMML, png

10.67.2 $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x\sim-e^{{-x/\sqrt{2}}}\left(\frac{% \pi}{2x}\right)^{{\frac{1}{2}}}\*\sum_{{k=0}}^{\infty}\frac{a_{k}(\nu)}{x^{k}}% \*\mathop{\sin\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac% {k}{4}+\frac{1}{8}\right)\pi\right).$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : base of exponential function, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : real variable, $\nu$ : complex parameter and $a_{k}(\nu)$ : expansion
A&S Ref:: 9.10.4, 9.10.5--9.10.7 (modified)
Referenced by:: §10.61(v)
Permalink:: http://dlmf.nist.gov/10.67.E2
Encodings:: TeX, pMML, png

10.67.3 $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x\sim\frac{e^{{x/\sqrt{2}}}}{(2\pi x)% ^{{\frac{1}{2}}}}\*\sum_{{k=0}}^{\infty}\frac{a_{k}(\nu)}{x^{k}}\mathop{\cos\/% }\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{3k}{4}-\frac{1}% {8}\right)\pi\right)-\frac{1}{\pi}(\mathop{\sin\/}\nolimits\!\left(2\nu\pi% \right)\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x+\mathop{\cos\/}\nolimits\!% \left(2\nu\pi\right)\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x),$

10.67.4 $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x\sim\frac{e^{{x/\sqrt{2}}}}{(2\pi x)% ^{{\frac{1}{2}}}}\*\sum_{{k=0}}^{\infty}\frac{a_{k}(\nu)}{x^{k}}\mathop{\sin\/% }\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{3k}{4}-\frac{1}% {8}\right)\pi\right)+\frac{1}{\pi}(\mathop{\cos\/}\nolimits\!\left(2\nu\pi% \right)\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x-\mathop{\sin\/}\nolimits\!% \left(2\nu\pi\right)\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x).$

10.67.5 ${\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits^{{\prime}}}x\sim-e^{{-x/\sqrt{2}}}% \left(\frac{\pi}{2x}\right)^{{\frac{1}{2}}}\sum_{{k=0}}^{\infty}\frac{b_{k}(% \nu)}{x^{k}}\*\mathop{\cos\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{% \nu}{2}+\frac{k}{4}-\frac{1}{8}\right)\pi\right),$

Symbols:: $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\sim$ : asymptotic equality, : nonnegative integer, : real variable, $\nu$ : complex parameter and $b_{k}(\nu)$ : expansion
A&S Ref:: §9.10 (modified)
Permalink:: http://dlmf.nist.gov/10.67.E5
Encodings:: TeX, pMML, png

10.67.6 ${\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits^{{\prime}}}x\sim e^{{-x/\sqrt{2}}}% \left(\frac{\pi}{2x}\right)^{{\frac{1}{2}}}\sum_{{k=0}}^{\infty}\frac{b_{k}(% \nu)}{x^{k}}\*\mathop{\sin\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{% \nu}{2}+\frac{k}{4}-\frac{1}{8}\right)\pi\right).$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : base of exponential function, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : real variable, $\nu$ : complex parameter and $b_{k}(\nu)$ : expansion
A&S Ref:: §9.10 (modified)
Permalink:: http://dlmf.nist.gov/10.67.E6
Encodings:: TeX, pMML, png

10.67.7 ${\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{\prime}}}x\sim\frac{e^{{x/\sqrt{2}% }}}{(2\pi x)^{{\frac{1}{2}}}}\*\sum_{{k=0}}^{\infty}\frac{b_{k}(\nu)}{x^{k}}% \mathop{\cos\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{3% k}{4}+\frac{1}{8}\right)\pi\right)-\frac{1}{\pi}(\mathop{\sin\/}\nolimits\!% \left(2\nu\pi\right){\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits^{{\prime}}}x+% \mathop{\cos\/}\nolimits\!\left(2\nu\pi\right){\mathop{\mathrm{kei}_{{\nu}}\/}% \nolimits^{{\prime}}}x),$

Symbols:: $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : real variable, $\nu$ : complex parameter and $b_{k}(\nu)$ : expansion
A&S Ref:: §9.10 (modified)
Referenced by:: §10.67(i), §10.67(i), §10.69
Permalink:: http://dlmf.nist.gov/10.67.E7
Encodings:: TeX, pMML, png

10.67.8 ${\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{\prime}}}x\sim\frac{e^{{x/\sqrt{2}% }}}{(2\pi x)^{{\frac{1}{2}}}}\*\sum_{{k=0}}^{\infty}\frac{b_{k}(\nu)}{x^{k}}% \mathop{\sin\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{3% k}{4}+\frac{1}{8}\right)\pi\right)+\frac{1}{\pi}(\mathop{\cos\/}\nolimits\!% \left(2\nu\pi\right){\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits^{{\prime}}}x-% \mathop{\sin\/}\nolimits\!\left(2\nu\pi\right){\mathop{\mathrm{kei}_{{\nu}}\/}% \nolimits^{{\prime}}}x).$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : real variable, $\nu$ : complex parameter and $b_{k}(\nu)$ : expansion
A&S Ref:: §9.10 (modified)
Referenced by:: §10.67(i), §10.67(i), §10.69
Permalink:: http://dlmf.nist.gov/10.67.E8
Encodings:: TeX, pMML, png

The contributions of the terms in $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x$ , $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x$ , ${\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits^{{\prime}}}x$ , and ${\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits^{{\prime}}}x$ on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). However, their inclusion improves numerical accuracy.

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu=0$

Notes:: First, replace the $\mathop{\cos\/}\nolimits$ and $\mathop{\sin\/}\nolimits$ functions in (10.67.1)–(10.67.4) by exponential functions by constructing the corresponding expansions for $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x\pm i\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x$ and $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x\pm i\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x$ and discarding the exponentially-small terms. Then set $\nu=0$ and apply straightforward manipulations.
Keywords:: Kelvin functions
Permalink:: http://dlmf.nist.gov/10.67.ii

As $x\to\infty$

10.67.9 ${\mathop{\mathrm{ber}\/}\nolimits^{{2}}}x+{\mathop{\mathrm{bei}\/}\nolimits^{{% 2}}}x\sim\frac{e^{{x\sqrt{2}}}}{2\pi x}\left(1+\frac{1}{4\sqrt{2}}\frac{1}{x}+% \frac{1}{64}\frac{1}{x^{2}}-\frac{33}{256\sqrt{2}}\frac{1}{x^{3}}-\frac{1797}{% 8192}\frac{1}{x^{4}}+\cdots\right),$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : base of exponential function, $\sim$ : asymptotic equality and : real variable
A&S Ref:: 9.10.27
Permalink:: http://dlmf.nist.gov/10.67.E9
Encodings:: TeX, pMML, png

10.67.10 $\mathop{\mathrm{ber}\/}\nolimits x{\mathop{\mathrm{bei}\/}\nolimits^{{\prime}}% }x-{\mathop{\mathrm{ber}\/}\nolimits^{{\prime}}}x\mathop{\mathrm{bei}\/}% \nolimits x\sim\frac{e^{{x\sqrt{2}}}}{2\pi x}\left(\frac{1}{\sqrt{2}}+\frac{1}% {8}\frac{1}{x}+\frac{9}{64\sqrt{2}}\frac{1}{x^{2}}+\frac{39}{512}\frac{1}{x^{3% }}+\frac{75}{8192\sqrt{2}}\frac{1}{x^{4}}+\cdots\right),$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : base of exponential function, $\sim$ : asymptotic equality and : real variable
A&S Ref:: 9.10.28
Permalink:: http://dlmf.nist.gov/10.67.E10
Encodings:: TeX, pMML, png

10.67.11 $\mathop{\mathrm{ber}\/}\nolimits x{\mathop{\mathrm{ber}\/}\nolimits^{{\prime}}% }x+\mathop{\mathrm{bei}\/}\nolimits x{\mathop{\mathrm{bei}\/}\nolimits^{{% \prime}}}x\sim\frac{e^{{x\sqrt{2}}}}{2\pi x}\left(\frac{1}{\sqrt{2}}-\frac{3}{% 8}\frac{1}{x}-\frac{15}{64\sqrt{2}}\frac{1}{x^{2}}-\frac{45}{512}\frac{1}{x^{3% }}+\frac{315}{8192\sqrt{2}}\frac{1}{x^{4}}+\cdots\right),$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : base of exponential function, $\sim$ : asymptotic equality and : real variable
A&S Ref:: 9.10.29
Permalink:: http://dlmf.nist.gov/10.67.E11
Encodings:: TeX, pMML, png

10.67.12 $\left({\mathop{\mathrm{ber}\/}\nolimits^{{\prime}}}x\right)^{2}+\left({\mathop% {\mathrm{bei}\/}\nolimits^{{\prime}}}x\right)^{2}\sim\frac{e^{{x\sqrt{2}}}}{2% \pi x}\left(1-\frac{3}{4\sqrt{2}}\frac{1}{x}+\frac{9}{64}\frac{1}{x^{2}}+\frac% {75}{256\sqrt{2}}\frac{1}{x^{3}}+\frac{2475}{8192}\frac{1}{x^{4}}+\cdots\right).$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : base of exponential function, $\sim$ : asymptotic equality and : real variable
A&S Ref:: 9.10.30
Permalink:: http://dlmf.nist.gov/10.67.E12
Encodings:: TeX, pMML, png

10.67.13 ${\mathop{\mathrm{ker}\/}\nolimits^{{2}}}x+{\mathop{\mathrm{kei}\/}\nolimits^{{% 2}}}x\sim\frac{\pi}{2x}e^{{-x\sqrt{2}}}\left(1-\frac{1}{4\sqrt{2}}\frac{1}{x}+% \frac{1}{64}\frac{1}{x^{2}}+\frac{33}{256\sqrt{2}}\frac{1}{x^{3}}-\frac{1797}{% 8192}\frac{1}{x^{4}}+\cdots\right),$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : base of exponential function, $\sim$ : asymptotic equality and : real variable
A&S Ref:: 9.10.31
Permalink:: http://dlmf.nist.gov/10.67.E13
Encodings:: TeX, pMML, png

10.67.14 $\mathop{\mathrm{ker}\/}\nolimits x{\mathop{\mathrm{kei}\/}\nolimits^{{\prime}}% }x-{\mathop{\mathrm{ker}\/}\nolimits^{{\prime}}}x\mathop{\mathrm{kei}\/}% \nolimits x\sim-\frac{\pi}{2x}e^{{-x\sqrt{2}}}\left(\frac{1}{\sqrt{2}}-\frac{1% }{8}\frac{1}{x}+\frac{9}{64\sqrt{2}}\frac{1}{x^{2}}-\frac{39}{512}\frac{1}{x^{% 3}}+\frac{75}{8192\sqrt{2}}\frac{1}{x^{4}}+\cdots\right),$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : base of exponential function, $\sim$ : asymptotic equality and : real variable
A&S Ref:: 9.10.32
Permalink:: http://dlmf.nist.gov/10.67.E14
Encodings:: TeX, pMML, png

10.67.15 $\mathop{\mathrm{ker}\/}\nolimits x{\mathop{\mathrm{ker}\/}\nolimits^{{\prime}}% }x+\mathop{\mathrm{kei}\/}\nolimits x{\mathop{\mathrm{kei}\/}\nolimits^{{% \prime}}}x\sim-\frac{\pi}{2x}e^{{-x\sqrt{2}}}\left(\frac{1}{\sqrt{2}}+\frac{3}% {8}\frac{1}{x}-\frac{15}{64\sqrt{2}}\frac{1}{x^{2}}+\frac{45}{512}\frac{1}{x^{% 3}}+\frac{315}{8192\sqrt{2}}\frac{1}{x^{4}}+\cdots\right),$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : base of exponential function, $\sim$ : asymptotic equality and : real variable
A&S Ref:: 9.10.33
Permalink:: http://dlmf.nist.gov/10.67.E15
Encodings:: TeX, pMML, png

10.67.16 $\left({\mathop{\mathrm{ker}\/}\nolimits^{{\prime}}}x\right)^{2}+\left({\mathop% {\mathrm{kei}\/}\nolimits^{{\prime}}}x\right)^{2}\sim\frac{\pi}{2x}e^{{-x\sqrt% {2}}}\left(1+\frac{3}{4\sqrt{2}}\frac{1}{x}+\frac{9}{64}\frac{1}{x^{2}}-\frac{% 75}{256\sqrt{2}}\frac{1}{x^{3}}+\frac{2475}{8192}\frac{1}{x^{4}}+\cdots\right).$

Symbols:: $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : base of exponential function, $\sim$ : asymptotic equality and : real variable
A&S Ref:: 9.10.34
Permalink:: http://dlmf.nist.gov/10.67.E16
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.66 Expansions in Series of Bessel Functions 10.68 Modulus and Phase Functions

§10.67 Asymptotic Expansions for Large Argument

Contents

§10.67(i) , and Derivatives

§10.67(ii) Cross-Products and Sums of Squares in the Case

§10.67(i) $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{bei}_{{\nu}}\/}% \nolimits x,\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x,\mathop{\mathrm{kei}_{{% \nu}}\/}\nolimits x$ , and Derivatives

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu=0$