10 Bessel Functions Kelvin Functions 10.65 Power Series 10.67 Asymptotic Expansions for Large Argument

§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 $\mathop{\mathscr{C}\/}\nolimits=\mathop{J\/}\nolimits$ , $\nu=n$ , , , 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, : base of exponential function, : : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : nonnegative integer, : 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

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, : integer, : nonnegative integer and : 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

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