§10.66 Expansions in Series of Bessel Functions

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

10.66.1 $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x+i\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x=\sum _{{k=0}}^{\infty}\frac{e^{{(3\nu+k)\pi i/4}}x^{k}\mathop{J_{{\nu+k}}\/}\nolimits\!\left(x\right)}{2^{{k/2}}k!}=\sum _{{k=0}}^{\infty}\frac{e^{{(3\nu+3k)\pi i/4}}x^{k}\mathop{I_{{\nu+k}}\/}\nolimits\!\left(x\right)}{2^{{k/2}}k!}.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $k$ : 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
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10.66.2

$\mathop{\mathrm{ber}_{{n}}\/}\nolimits\!\left(x\sqrt{2}\right)=\sum _{{k=-\infty}}^{\infty}(-1)^{{n+k}}\mathop{J_{{n+2k}}\/}\nolimits\!\left(x\right)\mathop{I_{{2k}}\/}\nolimits\!\left(x\right),$

$\mathop{\mathrm{bei}_{{n}}\/}\nolimits\!\left(x\sqrt{2}\right)=\sum _{{k=-\infty}}^{\infty}(-1)^{{n+k}}\mathop{J_{{n+2k+1}}\/}\nolimits\!\left(x\right)\mathop{I_{{2k+1}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Kelvin function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $n$ : 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