# §10.65 Power Series

## §10.65(i) $\operatorname{ber}_{\nu}x$ and $\operatorname{bei}_{\nu}x$

 10.65.1 $\displaystyle\operatorname{ber}_{\nu}x$ $\displaystyle=(\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\cos\left(\frac{3}% {4}\nu\pi+\frac{1}{2}k\pi\right)}{k!\Gamma\left(\nu+k+1\right)}(\tfrac{1}{4}x^% {2})^{k},$ $\displaystyle\operatorname{bei}_{\nu}x$ $\displaystyle=(\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\sin\left(\frac{3}% {4}\nu\pi+\frac{1}{2}k\pi\right)}{k!\Gamma\left(\nu+k+1\right)}(\tfrac{1}{4}x^% {2})^{k}.$
 10.65.2 $\displaystyle\operatorname{ber}x$ $\displaystyle=1-\frac{(\frac{1}{4}x^{2})^{2}}{(2!)^{2}}+\frac{(\frac{1}{4}x^{2% })^{4}}{(4!)^{2}}-\dotsb,$ $\displaystyle\operatorname{bei}x$ $\displaystyle=\tfrac{1}{4}x^{2}-\frac{(\frac{1}{4}x^{2})^{3}}{(3!)^{2}}+\frac{% (\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\dotsi.$ ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$: Kelvin function, $!$: factorial (as in $n!$) and $x$: real variable A&S Ref: 9.9.10 Permalink: http://dlmf.nist.gov/10.65.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.65(i), §10.65 and Ch.10

## §10.65(ii) $\operatorname{ker}_{\nu}x$ and $\operatorname{kei}_{\nu}x$

When $\nu$ is not an integer combine (10.65.1) with (10.61.6). Also, with $\psi\left(x\right)=\Gamma'\left(x\right)/\Gamma\left(x\right)$,

 10.65.3 $\displaystyle\operatorname{ker}_{n}x$ $\displaystyle=\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{% k!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-% \ln\left(\tfrac{1}{2}x\right)\operatorname{ber}_{n}x+\tfrac{1}{4}\pi% \operatorname{bei}_{n}x+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}% \frac{\psi\left(k+1\right)+\psi\left(n+k+1\right)}{k!(n+k)!}\cos\left(\tfrac{3% }{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k},$ 10.65.4 $\displaystyle\operatorname{kei}_{n}x$ $\displaystyle=-\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}% {k!}\sin\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-% \ln\left(\tfrac{1}{2}x\right)\operatorname{bei}_{n}x-\tfrac{1}{4}\pi% \operatorname{ber}_{n}x+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}% \frac{\psi\left(k+1\right)+\psi\left(n+k+1\right)}{k!(n+k)!}\sin\left(\tfrac{3% }{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}.$
 10.65.5 $\displaystyle\operatorname{ker}x$ $\displaystyle=-\ln\left(\tfrac{1}{2}x\right)\operatorname{ber}x+\tfrac{1}{4}% \pi\operatorname{bei}x+\sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(2k+1\right)}% {((2k)!)^{2}}(\tfrac{1}{4}x^{2})^{2k},$ $\displaystyle\operatorname{kei}x$ $\displaystyle=-\ln\left(\tfrac{1}{2}x\right)\operatorname{bei}x-\tfrac{1}{4}% \pi\operatorname{ber}x+\sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(2k+2\right)}% {((2k+1)!)^{2}}(\tfrac{1}{4}x^{2})^{2k+1}.$

## §10.65(iii) Cross-Products and Sums of Squares

 10.65.6 ${\operatorname{ber}_{\nu}}^{2}x+{\operatorname{bei}_{\nu}}^{2}x=(\tfrac{1}{2}x% )^{2\nu}\sum_{k=0}^{\infty}\frac{1}{\Gamma\left(\nu+k+1\right)\Gamma\left(\nu+% 2k+1\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!},$
 10.65.7 $\operatorname{ber}_{\nu}x\operatorname{bei}_{\nu}'x-\operatorname{ber}_{\nu}'x% \operatorname{bei}_{\nu}x=(\tfrac{1}{2}x)^{2\nu+1}\sum_{k=0}^{\infty}\frac{1}{% \Gamma\left(\nu+k+1\right)\Gamma\left(\nu+2k+2\right)}\frac{(\frac{1}{4}x^{2})% ^{2k}}{k!},$
 10.65.8 $\operatorname{ber}_{\nu}x\operatorname{ber}_{\nu}'x+\operatorname{bei}_{\nu}x% \operatorname{bei}_{\nu}'x=\tfrac{1}{2}(\tfrac{1}{2}x)^{2\nu-1}\sum_{k=0}^{% \infty}\frac{1}{\Gamma\left(\nu+k+1\right)\Gamma\left(\nu+2k\right)}\frac{(% \frac{1}{4}x^{2})^{2k}}{k!},$
 10.65.9 $\left(\operatorname{ber}_{\nu}'x\right)^{2}+\left(\operatorname{bei}_{\nu}'x% \right)^{2}=(\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}^{\infty}\frac{2k^{2}+2\nu k+% \frac{1}{4}\nu^{2}}{\Gamma\left(\nu+k+1\right)\Gamma\left(\nu+2k+1\right)}% \frac{(\frac{1}{4}x^{2})^{2k}}{k!}.$

## §10.65(iv) Compendia

For further power series summable in terms of Kelvin functions and their derivatives see Hansen (1975).