# Kelvin functions

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##### 2: 10.64 Integral Representations
###### Schläfli-Type Integrals
10.64.1 $\operatorname{ber}_{n}\left(x\sqrt{2}\right)=\frac{(-1)^{n}}{\pi}\int_{0}^{\pi% }\cos\left(x\sin t-nt\right)\cosh\left(x\sin t\right)\,\mathrm{d}t,$
10.64.2 $\operatorname{bei}_{n}\left(x\sqrt{2}\right)=\frac{(-1)^{n}}{\pi}\int_{0}^{\pi% }\sin\left(x\sin t-nt\right)\sinh\left(x\sin t\right)\,\mathrm{d}t.$
##### 4: 10.71 Integrals
###### §10.71(iii) Compendia
For infinite double integrals involving Kelvin functions see Prudnikov et al. (1986b, pp. 630–631). …
##### 7: 10.69 Uniform Asymptotic Expansions for Large Order
###### §10.69 Uniform Asymptotic Expansions for Large Order
10.69.2 $\operatorname{ber}_{\nu}\left(\nu x\right)+i\operatorname{bei}_{\nu}\left(\nu x% \right)\sim\frac{e^{\nu\xi}}{(2\pi\nu\xi)^{\ifrac{1}{2}}}\left(\frac{xe^{3\pi i% /4}}{1+\xi}\right)^{\nu}\sum_{k=0}^{\infty}\frac{U_{k}(\xi^{-1})}{\nu^{k}},$
All fractional powers take their principal values. …
##### 8: 10.65 Power Series
###### §10.65(iii) Cross-Products and Sums of Squares
10.65.6 ${\operatorname{ber}_{\nu}}^{2}x+{\operatorname{bei}_{\nu}}^{2}x=(\tfrac{1}{2}x% )^{2\nu}\sum_{k=0}^{\infty}\frac{1}{\Gamma\left(\nu+k+1\right)\Gamma\left(\nu+% 2k+1\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!},$
###### §10.65(iv) Compendia
For further power series summable in terms of Kelvin functions and their derivatives see Hansen (1975).
##### 9: 10.63 Recurrence Relations and Derivatives
###### §10.63(i) $\operatorname{ber}_{\nu}x$, $\operatorname{bei}_{\nu}x$, $\operatorname{ker}_{\nu}x$, $\operatorname{kei}_{\nu}x$
Let $f_{\nu}(x)$, $g_{\nu}(x)$ denote any one of the ordered pairs: …
$\sqrt{2}\operatorname{kei}'x=-\operatorname{ker}_{1}x+\operatorname{kei}_{1}x.$
##### 10: 10.67 Asymptotic Expansions for Large Argument
###### §10.67(ii) Cross-Products and Sums of Squares in the Case $\nu=0$
10.67.9 ${\operatorname{ber}}^{2}x+{\operatorname{bei}}^{2}x\sim\frac{e^{x\sqrt{2}}}{2% \pi x}\left(1+\frac{1}{4\sqrt{2}}\frac{1}{x}+\frac{1}{64}\frac{1}{x^{2}}-\frac% {33}{256\sqrt{2}}\frac{1}{x^{3}}-\frac{1797}{8192}\frac{1}{x^{4}}+\dotsb\right),$
10.67.10 $\operatorname{ber}x\operatorname{bei}'x-\operatorname{ber}'x\operatorname{bei}% x\sim\frac{e^{x\sqrt{2}}}{2\pi x}\left(\frac{1}{\sqrt{2}}+\frac{1}{8}\frac{1}{% x}+\frac{9}{64\sqrt{2}}\frac{1}{x^{2}}+\frac{39}{512}\frac{1}{x^{3}}+\frac{75}% {8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),$