§10.65 Power Series

10.65.1		${\mathrm{ber}}_{\nu }x$	$={(\frac{1}{2}x)}^{\nu }{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{\cos \left(\frac{3}{4}\nu \pi +\frac{1}{2}k\pi \right)}{k!\mathrm{\Gamma }\left(\nu +k+1\right)}}{(\frac{1}{4}{x}^2)}^k,$
		${\mathrm{bei}}_{\nu }x$	$={(\frac{1}{2}x)}^{\nu }{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{\sin \left(\frac{3}{4}\nu \pi +\frac{1}{2}k\pi \right)}{k!\mathrm{\Gamma }\left(\nu +k+1\right)}}{(\frac{1}{4}{x}^2)}^k.$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.9 Referenced by: §10.65(ii) Permalink: http://dlmf.nist.gov/10.65.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.65(i), §10.65 and Ch.10

10.65.2		$\mathrm{ber}x$	$=1-{\displaystyle \frac{{(\frac{1}{4}{x}^2)}^2}{{(2!)}^2}}+{\displaystyle \frac{{(\frac{1}{4}{x}^2)}^4}{{(4!)}^2}}-\mathrm{\cdots },$
		$\mathrm{bei}x$	$=\frac{1}{4}{x}^2-{\displaystyle \frac{{(\frac{1}{4}{x}^2)}^3}{{(3!)}^2}}+{\displaystyle \frac{{(\frac{1}{4}{x}^2)}^5}{{(5!)}^2}}-\mathrm{\cdots }.$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: 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

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

10.65.3		${\ker }_{n}x$	$=\begin{array}{l}\frac{1}{2}{(\frac{1}{2}x)}^{-n}{\displaystyle \sum _{k=0}^{n-1}}{\displaystyle \frac{(n-k-1)!}{k!}}\cos \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k-\ln \left(\frac{1}{2}x\right){\mathrm{ber}}_{n}x\\ +\frac{1}{4}\pi {\mathrm{bei}}_{n}x+\frac{1}{2}{(\frac{1}{2}x)}^n{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{\psi \left(k+1\right)+\psi \left(n+k+1\right)}{k!(n+k)!}}\cos \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k,\end{array}$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $x$: real variable A&S Ref: 9.9.11 Permalink: http://dlmf.nist.gov/10.65.E3 Encodings: TeX, pMML, png See also: Annotations for §10.65(ii), §10.65 and Ch.10
10.65.4		${\mathrm{kei}}_{n}x$	$=\begin{array}{l}\begin{array}{l}-\frac{1}{2}{(\frac{1}{2}x)}^{-n}{\displaystyle \sum _{k=0}^{n-1}}{\displaystyle \frac{(n-k-1)!}{k!}}\sin \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k\\ -\ln \left(\frac{1}{2}x\right){\mathrm{bei}}_{n}x-\frac{1}{4}\pi {\mathrm{ber}}_{n}x\end{array}\\ +\frac{1}{2}{(\frac{1}{2}x)}^n{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{\psi \left(k+1\right)+\psi \left(n+k+1\right)}{k!(n+k)!}}\sin \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k.\end{array}$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $n$: integer, $k$: nonnegative integer and $x$: real variable A&S Ref: 9.9.11 Permalink: http://dlmf.nist.gov/10.65.E4 Encodings: TeX, pMML, png See also: Annotations for §10.65(ii), §10.65 and Ch.10

10.65.5		$\ker x$	$=-\ln \left(\frac{1}{2}x\right)\mathrm{ber}x+\frac{1}{4}\pi \mathrm{bei}x+{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{\displaystyle \frac{\psi \left(2k+1\right)}{{((2k)!)}^2}}{(\frac{1}{4}{x}^2)}^{2k},$
		$\mathrm{kei}x$	$=-\ln \left(\frac{1}{2}x\right)\mathrm{bei}x-\frac{1}{4}\pi \mathrm{ber}x+{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{\displaystyle \frac{\psi \left(2k+2\right)}{{((2k+1)!)}^2}}{(\frac{1}{4}{x}^2)}^{2k+1}.$
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $x$: real variable A&S Ref: 9.9.12 Permalink: http://dlmf.nist.gov/10.65.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.65(ii), §10.65 and Ch.10

10.65.6		$${\mathrm{ber}}_{\nu }^{2}x+{\mathrm{bei}}_{\nu }^{2}x={(\frac{1}{2}x)}^{2\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{1}{\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\nu +2k+1\right)}\frac{{(\frac{1}{4}{x}^2)}^{2k}}{k!},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.28 Permalink: http://dlmf.nist.gov/10.65.E6 Encodings: TeX, pMML, png See also: Annotations for §10.65(iii), §10.65 and Ch.10

10.65.7		$${\mathrm{ber}}_{\nu }x{\mathrm{bei}}_{\nu }^{\prime }x-{\mathrm{ber}}_{\nu }^{\prime }x{\mathrm{bei}}_{\nu }x={(\frac{1}{2}x)}^{2\nu +1}\sum _{k=0}^{\mathrm{\infty }}\frac{1}{\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\nu +2k+2\right)}\frac{{(\frac{1}{4}{x}^2)}^{2k}}{k!},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.29 Permalink: http://dlmf.nist.gov/10.65.E7 Encodings: TeX, pMML, png See also: Annotations for §10.65(iii), §10.65 and Ch.10

10.65.8		$${\mathrm{ber}}_{\nu }x{\mathrm{ber}}_{\nu }^{\prime }x+{\mathrm{bei}}_{\nu }x{\mathrm{bei}}_{\nu }^{\prime }x=\frac{1}{2}{(\frac{1}{2}x)}^{2\nu -1}\sum _{k=0}^{\mathrm{\infty }}\frac{1}{\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\nu +2k\right)}\frac{{(\frac{1}{4}{x}^2)}^{2k}}{k!},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.30 Permalink: http://dlmf.nist.gov/10.65.E8 Encodings: TeX, pMML, png See also: Annotations for §10.65(iii), §10.65 and Ch.10

10.65.9		$${\left({\mathrm{ber}}_{\nu }^{\prime }x\right)}^2+{\left({\mathrm{bei}}_{\nu }^{\prime }x\right)}^2={(\frac{1}{2}x)}^{2\nu -2}\sum _{k=0}^{\mathrm{\infty }}\frac{2k^2+2\nu k+\frac{1}{4}{\nu }^2}{\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\nu +2k+1\right)}\frac{{(\frac{1}{4}{x}^2)}^{2k}}{k!}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter A&S Ref: 9.9.31 Permalink: http://dlmf.nist.gov/10.65.E9 Encodings: TeX, pMML, png See also: Annotations for §10.65(iii), §10.65 and Ch.10

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