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DLMF: §10.71 Integrals ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions

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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

Contents

§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		$\displaystyle\int x^{1+\nu}f_{\nu}\,\mathrm{d}x$	$\displaystyle=-\frac{x^{1+\nu}}{\sqrt{2}}(f_{\nu+1}-g_{\nu+1})=-x^{1+\nu}\left% (\frac{\nu}{x}g_{\nu}-g_{\nu}^{\prime}\right),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : 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		$\displaystyle\int x^{1-\nu}f_{\nu}\,\mathrm{d}x$	$\displaystyle=\frac{x^{1-\nu}}{\sqrt{2}}(f_{\nu-1}-g_{\nu-1})=x^{1-\nu}\left(% \frac{\nu}{x}g_{\nu}+g_{\nu}^{\prime}\right).$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : 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		$\displaystyle\int x(f_{\nu}\widehat{g}_{\nu}-g_{\nu}\widehat{f}_{\nu})\,% \mathrm{d}x$	$\displaystyle=\frac{x}{2\sqrt{2}}\left(\widehat{f}_{\nu}(f_{\nu+1}+g_{\nu+1})-% \widehat{g}_{\nu}(f_{\nu+1}-g_{\nu+1})-f_{\nu}(\widehat{f}_{\nu+1}+\widehat{g}% _{\nu+1})+g_{\nu}(\widehat{f}_{\nu+1}-\widehat{g}_{\nu+1})\right)=\tfrac{1}{2}% x(f_{\nu}^{\prime}\widehat{f}_{\nu}-f_{\nu}\widehat{f}_{\nu}^{\prime}+g_{\nu}^% {\prime}\widehat{g}_{\nu}-g_{\nu}\widehat{g}^{\prime}_{\nu}),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : 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		$\displaystyle\int x(f_{\nu}\widehat{g}_{\nu}+g_{\nu}\widehat{f}_{\nu})\,% \mathrm{d}x$	$\displaystyle=\tfrac{1}{4}x^{2}(2f_{\nu}\widehat{g}_{\nu}-f_{\nu-1}\widehat{g}% _{\nu+1}-f_{\nu+1}\widehat{g}_{\nu-1}+2g_{\nu}\widehat{f}_{\nu}-g_{\nu-1}% \widehat{f}_{\nu+1}-g_{\nu+1}\widehat{f}_{\nu-1}).$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : 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		$\int x(f_{\nu}^{2}+g_{\nu}^{2})\,\mathrm{d}x=x(f_{\nu}g_{\nu}^{\prime}-f_{\nu}% ^{\prime}g_{\nu})=-\frac{x}{\sqrt{2}}(f_{\nu}f_{\nu+1}+g_{\nu}g_{\nu+1}-f_{\nu% }g_{\nu+1}+f_{\nu+1}g_{\nu}),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : 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		$\displaystyle\int xf_{\nu}g_{\nu}\,\mathrm{d}x$	$\displaystyle=\tfrac{1}{4}x^{2}\left(2f_{\nu}g_{\nu}-f_{\nu-1}g_{\nu+1}-f_{\nu% +1}g_{\nu-1}\right),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : 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		$\displaystyle\int x(f_{\nu}^{2}-g_{\nu}^{2})\,\mathrm{d}x$	$\displaystyle=\tfrac{1}{2}x^{2}\left(f_{\nu}^{2}-f_{\nu-1}f_{\nu+1}-g_{\nu}^{2% }+g_{\nu-1}g_{\nu+1}\right).$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : 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		$\displaystyle\int x{M_{\nu}}^{2}\left(x\right)\,\mathrm{d}x$	$\displaystyle=x(\operatorname{ber}_{\nu}x\operatorname{bei}_{\nu}'x-% \operatorname{ber}_{\nu}'x\operatorname{bei}_{\nu}x),$
10.71.8		$\displaystyle\int x{N_{\nu}}^{2}\left(x\right)\,\mathrm{d}x$	$\displaystyle=x(\operatorname{ker}_{\nu}x\operatorname{kei}_{\nu}'x-% \operatorname{ker}_{\nu}'x\operatorname{kei}_{\nu}x),$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $x$ : real variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.71.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.71(i), §10.71(i), §10.71 and Ch.10

where $M_{\nu}\left(x\right)$ and $N_{\nu}\left(x\right)$ are the modulus functions introduced in §10.68(i).

§10.71(ii) Definite Integrals

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Keywords:: Kelvin functions, definite, integrals
Permalink:: http://dlmf.nist.gov/10.71.ii
See also:: Annotations for §10.71 and Ch.10

See Kerr (1978) and Glasser (1979).

§10.71(iii) Compendia

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Keywords:: Kelvin functions, Laplace transforms, compendia, definite, integrals
Permalink:: http://dlmf.nist.gov/10.71.iii
See also:: Annotations for §10.71 and Ch.10

For infinite double integrals involving Kelvin functions see Prudnikov et al. (1986b, pp. 630–631).

For direct and inverse Laplace transforms of Kelvin functions see Prudnikov et al. (1992a, §3.19) and Prudnikov et al. (1992b, §3.19).

10.70 Zeros 10.72 Mathematical Applications

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