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

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Permalink:: http://dlmf.nist.gov/10.67
See also:: Annotations for Ch.10

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

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

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Keywords:: Kelvin functions, asymptotic expansions for large argument, exponentially-small contributions
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 $\pi iI_{\nu}\left(xe^{\pi i/4}\right)=K_{\nu}\left(xe^{-3\pi i/4}\right)-e^{% \nu\pi i}K_{\nu}\left(xe^{\pi i/4}\right)$ , 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).
Referenced by:: §10.67(i)
Permalink:: http://dlmf.nist.gov/10.67.i
See also:: Annotations for §10.67 and Ch.10

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

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Symbols:: $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, : nonnegative integer, : real variable, $\nu$ : complex parameter and $a_{k}(\nu)$ : polynomial coefficient
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
See also:: Annotations for §10.67(i), §10.67 and Ch.10

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

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, : nonnegative integer, : real variable, $\nu$ : complex parameter and $a_{k}(\nu)$ : polynomial coefficient
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
See also:: Annotations for §10.67(i), §10.67 and Ch.10

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

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Symbols:: $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, : nonnegative integer, : real variable, $\nu$ : complex parameter and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 9.10.1, 9.10.5--9.10.7 (modified)
Referenced by:: §10.61(v), §10.67(i), §10.67(i), §10.68(iii), §10.69
Permalink:: http://dlmf.nist.gov/10.67.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(i), §10.67 and Ch.10

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

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, : nonnegative integer, : real variable, $\nu$ : complex parameter and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 9.10.2, 9.10.5--9.10.7 (modified)
Referenced by:: §10.61(v), §10.67(i), §10.67(i), §10.67(ii), §10.68(iii), §10.69
Permalink:: http://dlmf.nist.gov/10.67.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(i), §10.67 and Ch.10

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

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Symbols:: $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, : nonnegative integer, : real variable, $\nu$ : complex parameter and $b_{k}(\nu)$ : polynomial coefficient
A&S Ref:: §9.10 (modified)
Permalink:: http://dlmf.nist.gov/10.67.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(i), §10.67 and Ch.10

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

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, : nonnegative integer, : real variable, $\nu$ : complex parameter and $b_{k}(\nu)$ : polynomial coefficient
A&S Ref:: §9.10 (modified)
Permalink:: http://dlmf.nist.gov/10.67.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(i), §10.67 and Ch.10

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

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Symbols:: $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, : nonnegative integer, : real variable, $\nu$ : complex parameter and $b_{k}(\nu)$ : polynomial coefficient
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
See also:: Annotations for §10.67(i), §10.67 and Ch.10

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

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, : nonnegative integer, : real variable, $\nu$ : complex parameter and $b_{k}(\nu)$ : polynomial coefficient
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
See also:: Annotations for §10.67(i), §10.67 and Ch.10

The contributions of the terms in $\operatorname{ker}_{\nu}x$ , $\operatorname{kei}_{\nu}x$ , $\operatorname{ker}_{\nu}'x$ , and $\operatorname{kei}_{\nu}'x$ 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 $\nu=0$

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Keywords:: Kelvin functions, asymptotic expansions for large argument, cross-products and sums of squares
Notes:: First, replace the $\cos$ and $\sin$ functions in (10.67.1)–(10.67.4) by exponential functions by constructing the corresponding expansions for $\operatorname{ber}_{\nu}x\pm i\operatorname{bei}_{\nu}x$ and $\operatorname{ker}_{\nu}x\pm i\operatorname{kei}_{\nu}x$ and discarding the exponentially-small terms. Then set $\nu=0$ and apply straightforward manipulations.
Permalink:: http://dlmf.nist.gov/10.67.ii
See also:: Annotations for §10.67 and Ch.10

As $x\to\infty$

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

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : real variable
A&S Ref:: 9.10.27
Permalink:: http://dlmf.nist.gov/10.67.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

10.67.10 $\operatorname{ber}x\operatorname{bei}'x-\operatorname{ber}'x\operatorname{bei}% x\sim\frac{e^{x\sqrt{2}}}{2\pi x}\left(\frac{1}{\sqrt{2}}+\frac{1}{8}\frac{1}{% x}+\frac{9}{64\sqrt{2}}\frac{1}{x^{2}}+\frac{39}{512}\frac{1}{x^{3}}+\frac{75}% {8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),$

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : real variable
A&S Ref:: 9.10.28
Permalink:: http://dlmf.nist.gov/10.67.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

10.67.11 $\operatorname{ber}x\operatorname{ber}'x+\operatorname{bei}x\operatorname{bei}'% x\sim\frac{e^{x\sqrt{2}}}{2\pi x}\left(\frac{1}{\sqrt{2}}-\frac{3}{8}\frac{1}{% x}-\frac{15}{64\sqrt{2}}\frac{1}{x^{2}}-\frac{45}{512}\frac{1}{x^{3}}+\frac{31% 5}{8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),$

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : real variable
A&S Ref:: 9.10.29
Permalink:: http://dlmf.nist.gov/10.67.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

10.67.12 $\left(\operatorname{ber}'x\right)^{2}+\left(\operatorname{bei}'x\right)^{2}% \sim\frac{e^{x\sqrt{2}}}{2\pi x}\left(1-\frac{3}{4\sqrt{2}}\frac{1}{x}+\frac{9% }{64}\frac{1}{x^{2}}+\frac{75}{256\sqrt{2}}\frac{1}{x^{3}}+\frac{2475}{8192}% \frac{1}{x^{4}}+\dotsi\right).$

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : real variable
A&S Ref:: 9.10.30
Permalink:: http://dlmf.nist.gov/10.67.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

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

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : real variable
A&S Ref:: 9.10.31
Permalink:: http://dlmf.nist.gov/10.67.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

10.67.14 $\operatorname{ker}x\operatorname{kei}'x-\operatorname{ker}'x\operatorname{kei}% x\sim-\frac{\pi}{2x}e^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}-\frac{1}{8}\frac{1}% {x}+\frac{9}{64\sqrt{2}}\frac{1}{x^{2}}-\frac{39}{512}\frac{1}{x^{3}}+\frac{75% }{8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),$

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : real variable
A&S Ref:: 9.10.32
Permalink:: http://dlmf.nist.gov/10.67.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

10.67.15 $\operatorname{ker}x\operatorname{ker}'x+\operatorname{kei}x\operatorname{kei}'% x\sim-\frac{\pi}{2x}e^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}+\frac{3}{8}\frac{1}% {x}-\frac{15}{64\sqrt{2}}\frac{1}{x^{2}}+\frac{45}{512}\frac{1}{x^{3}}+\frac{3% 15}{8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),$

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : real variable
A&S Ref:: 9.10.33
Permalink:: http://dlmf.nist.gov/10.67.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

10.67.16 $\left(\operatorname{ker}'x\right)^{2}+\left(\operatorname{kei}'x\right)^{2}% \sim\frac{\pi}{2x}e^{-x\sqrt{2}}\left(1+\frac{3}{4\sqrt{2}}\frac{1}{x}+\frac{9% }{64}\frac{1}{x^{2}}-\frac{75}{256\sqrt{2}}\frac{1}{x^{3}}+\frac{2475}{8192}% \frac{1}{x^{4}}+\dotsi\right).$

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Symbols:: $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : real variable
A&S Ref:: 9.10.34
Permalink:: http://dlmf.nist.gov/10.67.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §10.67(ii), §10.67 and Ch.10

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

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§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) $\operatorname{ber}_{\nu}x,\operatorname{bei}_{\nu}x,\operatorname{ker}_{\nu}x,% \operatorname{kei}_{\nu}x$ , and Derivatives

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