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

§10.65(i) $\operatorname{ber}_{\nu}x$ and $\operatorname{bei}_{\nu}x$
§10.65(ii) $\operatorname{ker}_{\nu}x$ and $\operatorname{kei}_{\nu}x$
§10.65(iii) Cross-Products and Sums of Squares
§10.65(iv) Compendia

§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

$\operatorname{ber}_{\nu}x=(\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\cos% \left(\frac{3}{4}\nu\pi+\frac{1}{2}k\pi\right)}{k!\Gamma\left(\nu+k+1\right)}(% \tfrac{1}{4}x^{2})^{k},$

$\operatorname{bei}_{\nu}x=(\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\sin% \left(\frac{3}{4}\nu\pi+\frac{1}{2}k\pi\right)}{k!\Gamma\left(\nu+k+1\right)}(% \tfrac{1}{4}x^{2})^{k}.$

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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 ), $\sin\NVar{z}$ : sine function, : nonnegative integer, : 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

$\operatorname{ber}x=1-\frac{(\frac{1}{4}x^{2})^{2}}{(2!)^{2}}+\frac{(\frac{1}{% 4}x^{2})^{4}}{(4!)^{2}}-\dotsb,$

$\operatorname{bei}x=\tfrac{1}{4}x^{2}-\frac{(\frac{1}{4}x^{2})^{3}}{(3!)^{2}}+% \frac{(\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\dotsi.$

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, : factorial (as in ) and : 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 $\operatorname{ker}_{n}x=\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{% (n-k-1)!}{k!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^% {2})^{k}-\ln\left(\tfrac{1}{2}x\right)\operatorname{ber}_{n}x+\tfrac{1}{4}\pi% \operatorname{bei}_{n}x+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}% \frac{\psi\left(k+1\right)+\psi\left(n+k+1\right)}{k!(n+k)!}\cos\left(\tfrac{3% }{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k},$

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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 ), $\ln\NVar{z}$ : principal branch of logarithm function, : integer, : nonnegative integer and : 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 $\operatorname{kei}_{n}x=-\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac% {(n-k-1)!}{k!}\sin\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x% ^{2})^{k}-\ln\left(\tfrac{1}{2}x\right)\operatorname{bei}_{n}x-\tfrac{1}{4}\pi% \operatorname{ber}_{n}x+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}% \frac{\psi\left(k+1\right)+\psi\left(n+k+1\right)}{k!(n+k)!}\sin\left(\tfrac{3% }{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}.$

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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 ), $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, : integer, : nonnegative integer and : 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

$\operatorname{ker}x=-\ln\left(\tfrac{1}{2}x\right)\operatorname{ber}x+\tfrac{1% }{4}\pi\operatorname{bei}x+\sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(2k+1% \right)}{((2k)!)^{2}}(\tfrac{1}{4}x^{2})^{2k},$

$\operatorname{kei}x=-\ln\left(\tfrac{1}{2}x\right)\operatorname{bei}x-\tfrac{1% }{4}\pi\operatorname{ber}x+\sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(2k+2% \right)}{((2k+1)!)^{2}}(\tfrac{1}{4}x^{2})^{2k+1}.$

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

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

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

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

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

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