Content-Length: 77960 | pFad | https://dlmf.nist.gov/./.././bib/../././bib/.././././10.5#info

DLMF: §10.5 Wronskians and Cross-Products ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions

About the Project

10 Bessel Functions Bessel and Hankel Functions 10.4 Connection Formulas 10.6 Recurrence Relations and Derivatives

§10.5 Wronskians and Cross-Products

ⓘ

Keywords:: Bessel functions, Hankel functions, Wronskians, cross-product, cross-products
Notes:: For the Wronskians use (1.13.5) and the limiting forms in §10.7. Then for the cross-products apply (10.6.2).
Permalink:: http://dlmf.nist.gov/10.5
See also:: Annotations for Ch.10

10.5.1		$\mathscr{W}\left\{J_{\nu}\left(z\right),J_{-\nu}\left(z\right)\right\}=J_{\nu+% 1}\left(z\right)J_{-\nu}\left(z\right)+J_{\nu}\left(z\right)J_{-\nu-1}\left(z% \right)=-2\sin\left(\nu\pi\right)/(\pi z),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.15 Permalink: http://dlmf.nist.gov/10.5.E1 Encodings: TeX, pMML, png See also: Annotations for §10.5 and Ch.10

10.5.2		$\mathscr{W}\left\{J_{\nu}\left(z\right),Y_{\nu}\left(z\right)\right\}=J_{\nu+1% }\left(z\right)Y_{\nu}\left(z\right)-J_{\nu}\left(z\right)Y_{\nu+1}\left(z% \right)=2/(\pi z),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.16 Permalink: http://dlmf.nist.gov/10.5.E2 Encodings: TeX, pMML, png See also: Annotations for §10.5 and Ch.10

10.5.3		$\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(1)}_{\nu}}\left(z\right)\right\}=% J_{\nu+1}\left(z\right){H^{(1)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{% (1)}_{\nu+1}}\left(z\right)=2i/(\pi z),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.5.E3 Encodings: TeX, pMML, png See also: Annotations for §10.5 and Ch.10

10.5.4		$\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)\right\}=% J_{\nu+1}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{% (2)}_{\nu+1}}\left(z\right)=-2i/(\pi z),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.5.E4 Encodings: TeX, pMML, png See also: Annotations for §10.5 and Ch.10

10.5.5		$\mathscr{W}\left\{{H^{(1)}_{\nu}}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)% \right\}={H^{(1)}_{\nu+1}}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-{H^{(1)}% _{\nu}}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)=-4i/(\pi z).$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.17 Permalink: http://dlmf.nist.gov/10.5.E5 Encodings: TeX, pMML, png See also: Annotations for §10.5 and Ch.10

10.4 Connection Formulas 10.6 Recurrence Relations and Derivatives

© 2010–2024 NIST / Disclaimer / Feedback; Version 1.2.3; Release date 2024-12-15.

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