10 Bessel Functions Kelvin Functions 10.70 Zeros 10.72 Mathematical Applications

§10.71 Integrals

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Referenced by:: §10.1, §10.61(i)
Permalink:: http://dlmf.nist.gov/10.71
See also:: Annotations for Ch.10

§10.71(i) Indefinite Integrals
§10.71(ii) Definite Integrals
§10.71(iii) Compendia

§10.71(i) Indefinite Integrals

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Keywords:: Kelvin functions, indefinite, integrals
Notes:: Differentiate and use (10.63.2) and (10.68.5). See also Young and Kirk (1964, pp. xvi–xvii).
Permalink:: http://dlmf.nist.gov/10.71.i
See also:: Annotations for §10.71 and Ch.10

In the following equations $f_{\nu},g_{\nu}$ is any one of the four ordered pairs given in (10.63.1), and $\widehat{f}_{\nu},\widehat{g}_{\nu}$ is either the same ordered pair or any other ordered pair in (10.63.1).

10.71.1 $\int x^{1+\nu}f_{\nu}\,\mathrm{d}x=-\frac{x^{1+\nu}}{\sqrt{2}}(f_{\nu+1}-g_{% \nu+1})=-x^{1+\nu}\left(\frac{\nu}{x}g_{\nu}-g_{\nu}^{\prime}\right),$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function
A&S Ref:: 9.9.21
Permalink:: http://dlmf.nist.gov/10.71.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.71(i), §10.71 and Ch.10

10.71.2 $\int x^{1-\nu}f_{\nu}\,\mathrm{d}x=\frac{x^{1-\nu}}{\sqrt{2}}(f_{\nu-1}-g_{\nu% -1})=x^{1-\nu}\left(\frac{\nu}{x}g_{\nu}+g_{\nu}^{\prime}\right).$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function
A&S Ref:: 9.9.22
Permalink:: http://dlmf.nist.gov/10.71.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §10.71(i), §10.71 and Ch.10

10.71.3 $\int x(f_{\nu}\widehat{g}_{\nu}-g_{\nu}\widehat{f}_{\nu})\,\mathrm{d}x=\frac{x% }{2\sqrt{2}}\left(\widehat{f}_{\nu}(f_{\nu+1}+g_{\nu+1})-\widehat{g}_{\nu}(f_{% \nu+1}-g_{\nu+1})-f_{\nu}(\widehat{f}_{\nu+1}+\widehat{g}_{\nu+1})+g_{\nu}(% \widehat{f}_{\nu+1}-\widehat{g}_{\nu+1})\right)=\tfrac{1}{2}x(f_{\nu}^{\prime}% \widehat{f}_{\nu}-f_{\nu}\widehat{f}_{\nu}^{\prime}+g_{\nu}^{\prime}\widehat{g% }_{\nu}-g_{\nu}\widehat{g}^{\prime}_{\nu}),$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function, $g_{\nu}$ a Kelvin function, $\widehat{f}_{\nu}$ a Kelvin function and $\widehat{g}_{\nu}$ a Kelvin function
A&S Ref:: 9.9.23
Permalink:: http://dlmf.nist.gov/10.71.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §10.71(i), §10.71 and Ch.10

10.71.4 $\int x(f_{\nu}\widehat{g}_{\nu}+g_{\nu}\widehat{f}_{\nu})\,\mathrm{d}x=\tfrac{% 1}{4}x^{2}(2f_{\nu}\widehat{g}_{\nu}-f_{\nu-1}\widehat{g}_{\nu+1}-f_{\nu+1}% \widehat{g}_{\nu-1}+2g_{\nu}\widehat{f}_{\nu}-g_{\nu-1}\widehat{f}_{\nu+1}-g_{% \nu+1}\widehat{f}_{\nu-1}).$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function, $g_{\nu}$ a Kelvin function, $\widehat{f}_{\nu}$ a Kelvin function and $\widehat{g}_{\nu}$ a Kelvin function
A&S Ref:: 9.9.24
Permalink:: http://dlmf.nist.gov/10.71.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §10.71(i), §10.71 and Ch.10

10.71.5 $\int x(f_{\nu}^{2}+g_{\nu}^{2})\,\mathrm{d}x=x(f_{\nu}g_{\nu}^{\prime}-f_{\nu}% ^{\prime}g_{\nu})=-\frac{x}{\sqrt{2}}(f_{\nu}f_{\nu+1}+g_{\nu}g_{\nu+1}-f_{\nu% }g_{\nu+1}+f_{\nu+1}g_{\nu}),$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function
A&S Ref:: 9.9.25
Permalink:: http://dlmf.nist.gov/10.71.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §10.71(i), §10.71 and Ch.10

10.71.6 $\int xf_{\nu}g_{\nu}\,\mathrm{d}x=\tfrac{1}{4}x^{2}\left(2f_{\nu}g_{\nu}-f_{% \nu-1}g_{\nu+1}-f_{\nu+1}g_{\nu-1}\right),$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function
A&S Ref:: 9.9.26
Permalink:: http://dlmf.nist.gov/10.71.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §10.71(i), §10.71 and Ch.10

10.71.7 $\int x(f_{\nu}^{2}-g_{\nu}^{2})\,\mathrm{d}x=\tfrac{1}{2}x^{2}\left(f_{\nu}^{2% }-f_{\nu-1}f_{\nu+1}-g_{\nu}^{2}+g_{\nu-1}g_{\nu+1}\right).$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function
A&S Ref:: 9.9.27
Permalink:: http://dlmf.nist.gov/10.71.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §10.71(i), §10.71 and Ch.10

Examples

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See also:: Annotations for §10.71(i), §10.71 and Ch.10

10.71.8

$\int x{M_{\nu}}^{2}\left(x\right)\,\mathrm{d}x=x(\operatorname{ber}_{\nu}x% \operatorname{bei}_{\nu}'x-\operatorname{ber}_{\nu}'x\operatorname{bei}_{\nu}x),$

$\int x{N_{\nu}}^{2}\left(x\right)\,\mathrm{d}x=x(\operatorname{ker}_{\nu}x% \operatorname{kei}_{\nu}'x-\operatorname{ker}_{\nu}'x\operatorname{kei}_{\nu}x),$