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DLMF: §10.65 Power Series ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions

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10 Bessel Functions Kelvin Functions 10.64 Integral Representations 10.66 Expansions in Series of Bessel Functions

§10.65 Power Series

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Keywords:: Kelvin functions, power series
Permalink:: http://dlmf.nist.gov/10.65
See also:: Annotations for Ch.10

Contents

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

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Notes:: Combine (10.2.2) and (10.61.1). See also Whitehead (1911).
Permalink:: http://dlmf.nist.gov/10.65.i
See also:: Annotations for §10.65 and Ch.10

10.65.1		$\displaystyle\operatorname{ber}_{\nu}x$	$\displaystyle=(\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\cos\left(\frac{3}% {4}\nu\pi+\frac{1}{2}k\pi\right)}{k!\Gamma\left(\nu+k+1\right)}(\tfrac{1}{4}x^% {2})^{k},$
10.65.1		$\displaystyle\operatorname{bei}_{\nu}x$	$\displaystyle=(\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\sin\left(\frac{3}% {4}\nu\pi+\frac{1}{2}k\pi\right)}{k!\Gamma\left(\nu+k+1\right)}(\tfrac{1}{4}x^% {2})^{k}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : 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		$\displaystyle\operatorname{ber}x$	$\displaystyle=1-\frac{(\frac{1}{4}x^{2})^{2}}{(2!)^{2}}+\frac{(\frac{1}{4}x^{2% })^{4}}{(4!)^{2}}-\dotsb,$
10.65.2		$\displaystyle\operatorname{bei}x$	$\displaystyle=\tfrac{1}{4}x^{2}-\frac{(\frac{1}{4}x^{2})^{3}}{(3!)^{2}}+\frac{% (\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\dotsi.$
ⓘ 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 $\operatorname{kei}_{\nu}x$

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Notes:: Combine (10.31.1), (10.61.1), and (10.61.2). See also Whitehead (1911) and Young and Kirk (1964, p. x).
Permalink:: http://dlmf.nist.gov/10.65.ii
See also:: Annotations for §10.65 and Ch.10

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

10.65.3		$\displaystyle\operatorname{ker}_{n}x$	$\displaystyle=\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{% k!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-% \ln\left(\tfrac{1}{2}x\right)\operatorname{ber}_{n}x+\tfrac{1}{4}\pi% \operatorname{bei}_{n}x+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}% \frac{\psi\left(k+1\right)+\psi\left(n+k+1\right)}{k!(n+k)!}\cos\left(\tfrac{3% }{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k},$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : 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		$\displaystyle\operatorname{kei}_{n}x$	$\displaystyle=-\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}% {k!}\sin\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-% \ln\left(\tfrac{1}{2}x\right)\operatorname{bei}_{n}x-\tfrac{1}{4}\pi% \operatorname{ber}_{n}x+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}% \frac{\psi\left(k+1\right)+\psi\left(n+k+1\right)}{k!(n+k)!}\sin\left(\tfrac{3% }{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}.$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : 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		$\displaystyle\operatorname{ker}x$	$\displaystyle=-\ln\left(\tfrac{1}{2}x\right)\operatorname{ber}x+\tfrac{1}{4}% \pi\operatorname{bei}x+\sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(2k+1\right)}% {((2k)!)^{2}}(\tfrac{1}{4}x^{2})^{2k},$
10.65.5		$\displaystyle\operatorname{kei}x$	$\displaystyle=-\ln\left(\tfrac{1}{2}x\right)\operatorname{bei}x-\tfrac{1}{4}% \pi\operatorname{ber}x+\sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(2k+2\right)}% {((2k+1)!)^{2}}(\tfrac{1}{4}x^{2})^{2k+1}.$
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : 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(iii) Cross-Products and Sums of Squares

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Keywords:: Kelvin functions, cross-products and sums of squares, power series
Notes:: See Whitehead (1911).
Permalink:: http://dlmf.nist.gov/10.65.iii
See also:: Annotations for §10.65 and Ch.10

10.65.6		${\operatorname{ber}_{\nu}}^{2}x+{\operatorname{bei}_{\nu}}^{2}x=(\tfrac{1}{2}x% )^{2\nu}\sum_{k=0}^{\infty}\frac{1}{\Gamma\left(\nu+k+1\right)\Gamma\left(\nu+% 2k+1\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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		$\operatorname{ber}_{\nu}x\operatorname{bei}_{\nu}'x-\operatorname{ber}_{\nu}'x% \operatorname{bei}_{\nu}x=(\tfrac{1}{2}x)^{2\nu+1}\sum_{k=0}^{\infty}\frac{1}{% \Gamma\left(\nu+k+1\right)\Gamma\left(\nu+2k+2\right)}\frac{(\frac{1}{4}x^{2})% ^{2k}}{k!},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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		$\operatorname{ber}_{\nu}x\operatorname{ber}_{\nu}'x+\operatorname{bei}_{\nu}x% \operatorname{bei}_{\nu}'x=\tfrac{1}{2}(\tfrac{1}{2}x)^{2\nu-1}\sum_{k=0}^{% \infty}\frac{1}{\Gamma\left(\nu+k+1\right)\Gamma\left(\nu+2k\right)}\frac{(% \frac{1}{4}x^{2})^{2k}}{k!},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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		$\left(\operatorname{ber}_{\nu}'x\right)^{2}+\left(\operatorname{bei}_{\nu}'x% \right)^{2}=(\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}^{\infty}\frac{2k^{2}+2\nu k+% \frac{1}{4}\nu^{2}}{\Gamma\left(\nu+k+1\right)\Gamma\left(\nu+2k+1\right)}% \frac{(\frac{1}{4}x^{2})^{2k}}{k!}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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

§10.65(iv) Compendia

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Keywords:: Kelvin functions, compendia, power series
Permalink:: http://dlmf.nist.gov/10.65.iv
See also:: Annotations for §10.65 and Ch.10

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

10.64 Integral Representations 10.66 Expansions in Series of Bessel Functions

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