§10.66 Expansions in Series of Bessel Functions

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Keywords:: Kelvin functions, expansions in series of Bessel functions
Notes:: For (10.66.1) apply (10.23.1) with $\mathscr{C}=J$ and $\lambda=e^{3\pi i/4}$ ; also (10.44.1) with $\mathscr{Z}=I$ and $\lambda=e^{\pi i/4}$ . For (10.66.2) apply (10.23.2) with $\mathscr{C}=J$ , $\nu=n$ , $u=-x$ , $v=ix$ , and equate real and imaginary parts.
Permalink:: http://dlmf.nist.gov/10.66
See also:: Annotations for Ch.10

10.66.1		$\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu}x=\sum_{k=0}^{\infty}\frac{% e^{(3\nu+k)\pi i/4}x^{k}J_{\nu+k}\left(x\right)}{2^{k/2}k!}=\sum_{k=0}^{\infty% }\frac{e^{(3\nu+3k)\pi i/4}x^{k}I_{\nu+k}\left(x\right)}{2^{k/2}k!}.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $k$ : nonnegative integer, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 9.9.32 Referenced by: §10.66 Permalink: http://dlmf.nist.gov/10.66.E1 Encodings: TeX, pMML, png See also: Annotations for §10.66 and Ch.10

10.66.2	$\displaystyle\operatorname{ber}_{n}\left(x\sqrt{2}\right)$	$\displaystyle=\sum_{k=-\infty}^{\infty}(-1)^{n+k}J_{n+2k}\left(x\right)I_{2k}% \left(x\right),$
10.66.2	$\displaystyle\operatorname{bei}_{n}\left(x\sqrt{2}\right)$	$\displaystyle=\sum_{k=-\infty}^{\infty}(-1)^{n+k}J_{n+2k+1}\left(x\right)I_{2k% +1}\left(x\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $n$ : integer, $k$ : nonnegative integer and $x$ : real variable A&S Ref: 9.9.33 Referenced by: §10.66 Permalink: http://dlmf.nist.gov/10.66.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.66 and Ch.10