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

§10.65 Power Series

Keywords:: Kelvin functions, power series
Permalink:: http://dlmf.nist.gov/10.65

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

§10.65(i) $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x$ and $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x$

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

10.65.1

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\mathop{\sin\/}\nolimits z$ : 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
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10.65.2

$\mathop{\mathrm{ber}\/}\nolimits x=1-\frac{(\frac{1}{4}x^{2})^{2}}{(2!)^{2}}+% \frac{(\frac{1}{4}x^{2})^{4}}{(4!)^{2}}-\cdots,$

$\mathop{\mathrm{bei}\/}\nolimits x=\tfrac{1}{4}x^{2}-\frac{(\frac{1}{4}x^{2})^% {3}}{(3!)^{2}}+\frac{(\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\cdots.$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : : factorial and : real variable
A&S Ref:: 9.9.10
Permalink:: http://dlmf.nist.gov/10.65.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

§10.65(ii) $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x$ and $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x$

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

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

10.65.3 $\mathop{\mathrm{ker}_{{n}}\/}\nolimits x=\tfrac{1}{2}(\tfrac{1}{2}x)^{{-n}}% \sum_{{k=0}}^{{n-1}}\frac{(n-k-1)!}{k!}\mathop{\cos\/}\nolimits\!\left(\tfrac{% 3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-\mathop{\ln\/}% \nolimits\!\left(\tfrac{1}{2}x\right)\mathop{\mathrm{ber}_{{n}}\/}\nolimits x+% \tfrac{1}{4}\pi\mathop{\mathrm{bei}_{{n}}\/}\nolimits x+\tfrac{1}{2}(\tfrac{1}% {2}x)^{n}\sum_{{k=0}}^{\infty}\frac{\mathop{\psi\/}\nolimits\!\left(k+1\right)% +\mathop{\psi\/}\nolimits\!\left(n+k+1\right)}{k!(n+k)!}\mathop{\cos\/}% \nolimits\!\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{% k},$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\ln\/}\nolimits z$ : 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

10.65.4 $\mathop{\mathrm{kei}_{{n}}\/}\nolimits x=-\tfrac{1}{2}(\tfrac{1}{2}x)^{{-n}}% \sum_{{k=0}}^{{n-1}}\frac{(n-k-1)!}{k!}\mathop{\sin\/}\nolimits\!\left(\tfrac{% 3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-\mathop{\ln\/}% \nolimits\!\left(\tfrac{1}{2}x\right)\mathop{\mathrm{bei}_{{n}}\/}\nolimits x-% \tfrac{1}{4}\pi\mathop{\mathrm{ber}_{{n}}\/}\nolimits x+\tfrac{1}{2}(\tfrac{1}% {2}x)^{n}\sum_{{k=0}}^{\infty}\frac{\mathop{\psi\/}\nolimits\!\left(k+1\right)% +\mathop{\psi\/}\nolimits\!\left(n+k+1\right)}{k!(n+k)!}\mathop{\sin\/}% \nolimits\!\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{% k}.$

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : 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

10.65.5

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

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

Symbols:: $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer and : real variable
A&S Ref:: 9.9.12
Permalink:: http://dlmf.nist.gov/10.65.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

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

Notes:: See Whitehead (1911).
Keywords:: Kelvin functions, power series
Permalink:: http://dlmf.nist.gov/10.65.iii

10.65.6 ${\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{2}}}x+{\mathop{\mathrm{bei}_{{\nu}% }\/}\nolimits^{{2}}}x=(\tfrac{1}{2}x)^{{2\nu}}\sum_{{k=0}}^{\infty}\frac{1}{% \mathop{\Gamma\/}\nolimits\!\left(\nu+k+1\right)\mathop{\Gamma\/}\nolimits\!% \left(\nu+2k+1\right)}\frac{(\frac{1}{4}x^{2})^{{2k}}}{k!},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : : factorial, : 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

10.65.7 $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x{\mathop{\mathrm{bei}_{{\nu}}\/}% \nolimits^{{\prime}}}x-{\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{\prime}}}x% \mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x=(\tfrac{1}{2}x)^{{2\nu+1}}\sum_{{k=% 0}}^{\infty}\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\nu+k+1\right)\mathop{% \Gamma\/}\nolimits\!\left(\nu+2k+2\right)}\frac{(\frac{1}{4}x^{2})^{{2k}}}{k!},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : : factorial, : 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

10.65.8 $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x{\mathop{\mathrm{ber}_{{\nu}}\/}% \nolimits^{{\prime}}}x+\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x{\mathop{% \mathrm{bei}_{{\nu}}\/}\nolimits^{{\prime}}}x=\tfrac{1}{2}(\tfrac{1}{2}x)^{{2% \nu-1}}\sum_{{k=0}}^{\infty}\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\nu+k+1% \right)\mathop{\Gamma\/}\nolimits\!\left(\nu+2k\right)}\frac{(\frac{1}{4}x^{2}% )^{{2k}}}{k!},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : : factorial, : 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

10.65.9 $\left({\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits^{{\prime}}}x\right)^{2}+\left(% {\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits^{{\prime}}}x\right)^{2}=(\tfrac{1}{2% }x)^{{2\nu-2}}\sum_{{k=0}}^{\infty}\frac{2k^{2}+2\nu k+\frac{1}{4}\nu^{2}}{% \mathop{\Gamma\/}\nolimits\!\left(\nu+k+1\right)\mathop{\Gamma\/}\nolimits\!% \left(\nu+2k+1\right)}\frac{(\frac{1}{4}x^{2})^{{2k}}}{k!}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, : : factorial, : nonnegative integer, : real variable and $\nu$ : complex parameter
A&S Ref:: 9.9.31
Permalink:: http://dlmf.nist.gov/10.65.E9
Encodings:: TeX, pMML, png

§10.65(iv) Compendia

Keywords:: Kelvin functions, power series
Permalink:: http://dlmf.nist.gov/10.65.iv

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

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§10.65 Power Series

Contents

§10.65(i) and

§10.65(ii) and

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

§10.65(iv) Compendia

§10.65(i) $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x$ and $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x$

§10.65(ii) $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x$ and $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x$