About the Project

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

§10.66 Expansions in Series of Bessel Functions

ⓘ

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 $\mathscr{C}=J$ , $\nu=n$ , , , 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 ), $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, : 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
See also:: Annotations for §10.66 and Ch.10

10.66.2

$\operatorname{ber}_{n}\left(x\sqrt{2}\right)=\sum_{k=-\infty}^{\infty}(-1)^{n+% k}J_{n+2k}\left(x\right)I_{2k}\left(x\right),$

$\operatorname{bei}_{n}\left(x\sqrt{2}\right)=\sum_{k=-\infty}^{\infty}(-1)^{n+% k}J_{n+2k+1}\left(x\right)I_{2k+1}\left(x\right).$

ⓘ

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, : 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
See also:: Annotations for §10.66 and Ch.10

10.65 Power Series 10.67 Asymptotic Expansions for Large Argument

© 2010–2026 NIST / Disclaimer / Feedback; Version 1.2.7; Release date 2026-06-15 Math-Image version (See MathML) .

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov