10 Bessel Functions Kelvin Functions 10.70 Zeros 10.72 Mathematical Applications

§10.71 Integrals

Referenced by:: §10.1, §10.61(i)
Permalink:: http://dlmf.nist.gov/10.71

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

§10.71(i) Indefinite Integrals

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

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

Symbols:: : 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

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

Symbols:: : 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

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

Symbols:: : differential of , $\int$ : integral, : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function
A&S Ref:: 9.9.23
Permalink:: http://dlmf.nist.gov/10.71.E3
Encodings:: TeX, pMML, png

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

Symbols:: : differential of , $\int$ : integral, : real variable, $\nu$ : complex parameter, $f_{\nu}$ a Kelvin function and $g_{\nu}$ a Kelvin function
A&S Ref:: 9.9.24
Permalink:: http://dlmf.nist.gov/10.71.E4
Encodings:: TeX, pMML, png

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

Symbols:: : 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

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

Symbols:: : 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

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

Symbols:: : 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

¶ Examples

10.71.8

$\int x{\mathop{M_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right)dx=x(\mathop{% \mathrm{ber}_{{\nu}}\/}\nolimits x{\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{% \prime}}}x-{\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{\prime}}}x\mathop{% \mathrm{bei}_{{\nu}}\/}\nolimits x),$

$\int x{\mathop{N_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right)dx=x(\mathop{% \mathrm{ker}_{{\nu}}\/}\nolimits x{\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits^{{% \prime}}}x-{\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits^{{\prime}}}x\mathop{% \mathrm{kei}_{{\nu}}\/}\nolimits x),$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : differential of , $\int$ : integral, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.71.E8
Encodings:: TeX, TeX, pMML, pMML, png, png

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

§10.71(ii) Definite Integrals

Keywords:: Kelvin functions, integrals
Permalink:: http://dlmf.nist.gov/10.71.ii

See Kerr (1978) and Glasser (1979).

§10.71(iii) Compendia

Keywords:: Kelvin functions, integrals
Permalink:: http://dlmf.nist.gov/10.71.iii

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).

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§10.71 Integrals

Contents

§10.71(i) Indefinite Integrals

¶ Examples

§10.71(ii) Definite Integrals

§10.71(iii) Compendia