# §10.71 Integrals

## §10.71(i) Indefinite 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 $\displaystyle\int x^{1+\nu}f_{\nu}dx$ $\displaystyle=-\frac{x^{1+\nu}}{\sqrt{2}}(f_{\nu+1}-g_{\nu+1})=-x^{1+\nu}\left% (\frac{\nu}{x}g_{\nu}-g_{\nu}^{\prime}\right),$ 10.71.2 $\displaystyle\int x^{1-\nu}f_{\nu}dx$ $\displaystyle=\frac{x^{1-\nu}}{\sqrt{2}}(f_{\nu-1}-g_{\nu-1})=x^{1-\nu}\left(% \frac{\nu}{x}g_{\nu}+g_{\nu}^{\prime}\right).$
 10.71.3 $\displaystyle\int x(f_{\nu}\widehat{g}_{\nu}-g_{\nu}\widehat{f}_{\nu})dx$ $\displaystyle=\frac{x}{2\sqrt{2}}\left(\widehat{f}_{\nu}(f_{\nu+1}+g_{\nu+1})-% \widehat{g}_{\nu}(f_{\nu+1}-g_{\nu+1})-f_{\nu}(\widehat{f}_{\nu+1}+\widehat{g}% _{\nu+1})+g_{\nu}(\widehat{f}_{\nu+1}-\widehat{g}_{\nu+1})\right)$ $\displaystyle=\tfrac{1}{2}x(f_{\nu}^{\prime}\widehat{f}_{\nu}-f_{\nu}\widehat{% f}_{\nu}^{\prime}+g_{\nu}^{\prime}\widehat{g}_{\nu}-g_{\nu}\widehat{g}^{\prime% }_{\nu}),$ 10.71.4 $\displaystyle\int x(f_{\nu}\widehat{g}_{\nu}+g_{\nu}\widehat{f}_{\nu})dx$ $\displaystyle=\tfrac{1}{4}x^{2}(2f_{\nu}\widehat{g}_{\nu}-f_{\nu-1}\widehat{g}% _{\nu+1}-f_{\nu+1}\widehat{g}_{\nu-1}+2g_{\nu}\widehat{f}_{\nu}-g_{\nu-1}% \widehat{f}_{\nu+1}-g_{\nu+1}\widehat{f}_{\nu-1}).$
 10.71.5 $\int x(f_{\nu}^{2}+g_{\nu}^{2})dx=x(f_{\nu}g_{\nu}^{\prime}-f_{\nu}^{\prime}g_% {\nu})=-\frac{x}{\sqrt{2}}(f_{\nu}f_{\nu+1}+g_{\nu}g_{\nu+1}-f_{\nu}g_{\nu+1}+% f_{\nu+1}g_{\nu}),$
 10.71.6 $\displaystyle\int xf_{\nu}g_{\nu}dx$ $\displaystyle=\tfrac{1}{4}x^{2}\left(2f_{\nu}g_{\nu}-f_{\nu-1}g_{\nu+1}-f_{\nu% +1}g_{\nu-1}\right),$ 10.71.7 $\displaystyle\int x(f_{\nu}^{2}-g_{\nu}^{2})dx$ $\displaystyle=\tfrac{1}{2}x^{2}\left(f_{\nu}^{2}-f_{\nu-1}f_{\nu+1}-g_{\nu}^{2% }+g_{\nu-1}g_{\nu+1}\right).$

### Examples

 10.71.8 $\displaystyle\int x{\mathop{M_{\nu}\/}\nolimits^{2}}\!\left(x\right)dx$ $\displaystyle=x(\mathop{\mathrm{ber}_{\nu}\/}\nolimits x\mathop{\mathrm{bei}_{% \nu}\/}\nolimits'x-\mathop{\mathrm{ber}_{\nu}\/}\nolimits'x\mathop{\mathrm{bei% }_{\nu}\/}\nolimits x),$ $\displaystyle\int x{\mathop{N_{\nu}\/}\nolimits^{2}}\!\left(x\right)dx$ $\displaystyle=x(\mathop{\mathrm{ker}_{\nu}\/}\nolimits x\mathop{\mathrm{kei}_{% \nu}\/}\nolimits'x-\mathop{\mathrm{ker}_{\nu}\/}\nolimits'x\mathop{\mathrm{kei% }_{\nu}\/}\nolimits x),$

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

See Kerr (1978) and Glasser (1979).

## §10.71(iii) Compendia

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).