§10.71 Integrals

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		${\displaystyle \int x^{1+\nu }{f}_{\nu }dx}$	$=-{\displaystyle \frac{x^{1+\nu }}{\sqrt{2}}}(f_{\nu +1}-g_{\nu +1})=-x^{1+\nu }\left({\displaystyle \frac{\nu }{x}}{g}_{\nu }-g_{\nu }^{\prime }\right),$
ⓘ Symbols: $dx$: 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		${\displaystyle \int x^{1-\nu }{f}_{\nu }dx}$	$={\displaystyle \frac{x^{1-\nu }}{\sqrt{2}}}(f_{\nu -1}-g_{\nu -1})=x^{1-\nu }\left({\displaystyle \frac{\nu }{x}}{g}_{\nu }+g_{\nu }^{\prime }\right).$
ⓘ Symbols: $dx$: 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		${\displaystyle \int x(f_{\nu }{\widehat{g}}_{\nu }-g_{\nu }{\widehat{f}}_{\nu })dx}$	$={\displaystyle \frac{x}{2\sqrt{2}}}\left({\widehat{f}}_{\nu }(f_{\nu +1}+g_{\nu +1})-{\widehat{g}}_{\nu }(f_{\nu +1}-g_{\nu +1})-f_{\nu }({\widehat{f}}_{\nu +1}+{\widehat{g}}_{\nu +1})+g_{\nu }({\widehat{f}}_{\nu +1}-{\widehat{g}}_{\nu +1})\right)$
			$=\frac{1}{2}x(f_{\nu }^{\prime }{\widehat{f}}_{\nu }-f_{\nu }{\widehat{f}}_{\nu }^{\prime }+g_{\nu }^{\prime }{\widehat{g}}_{\nu }-g_{\nu }{\widehat{g}}_{\nu }^{\prime }),$
ⓘ Symbols: $dx$: 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		${\displaystyle \int x(f_{\nu }{\widehat{g}}_{\nu }+g_{\nu }{\widehat{f}}_{\nu })dx}$	$=\frac{1}{4}{x}^2(2f_{\nu }{\widehat{g}}_{\nu }-f_{\nu -1}{\widehat{g}}_{\nu +1}-f_{\nu +1}{\widehat{g}}_{\nu -1}+2g_{\nu }{\widehat{f}}_{\nu }-g_{\nu -1}{\widehat{f}}_{\nu +1}-g_{\nu +1}{\widehat{f}}_{\nu -1}).$
ⓘ Symbols: $dx$: 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		$$\int x(f_{\nu }^2+g_{\nu }^2)dx=x(f_{\nu }{g}_{\nu }^{\prime }-f_{\nu }^{\prime }{g}_{\nu })=-\frac{x}{\sqrt{2}}(f_{\nu }{f}_{\nu +1}+g_{\nu }{g}_{\nu +1}-f_{\nu }{g}_{\nu +1}+f_{\nu +1}{g}_{\nu }),$$
ⓘ Symbols: $dx$: 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		${\displaystyle \int x{f}_{\nu }{g}_{\nu }dx}$	$=\frac{1}{4}{x}^2\left(2f_{\nu }{g}_{\nu }-f_{\nu -1}{g}_{\nu +1}-f_{\nu +1}{g}_{\nu -1}\right),$
ⓘ Symbols: $dx$: 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		${\displaystyle \int x(f_{\nu }^2-g_{\nu }^2)dx}$	$=\frac{1}{2}{x}^2\left(f_{\nu }^2-f_{\nu -1}{f}_{\nu +1}-g_{\nu }^2+g_{\nu -1}{g}_{\nu +1}\right).$
ⓘ Symbols: $dx$: 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

See Kerr (1978) and Glasser (1979).

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).