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

§10.5 Wronskians and 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).
Keywords:: Bessel functions, Hankel functions
Permalink:: http://dlmf.nist.gov/10.5

10.5.1 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{J_{{\nu}}\/}\nolimits\!\left(z% \right),\mathop{J_{{-\nu}}\/}\nolimits\!\left(z\right)\right\}=\mathop{J_{{\nu% +1}}\/}\nolimits\!\left(z\right)\mathop{J_{{-\nu}}\/}\nolimits\!\left(z\right)% +\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{J_{{-\nu-1}}\/}\nolimits% \!\left(z\right)=-2\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)/(\pi z),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.15
Permalink:: http://dlmf.nist.gov/10.5.E1
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10.5.2 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{J_{{\nu}}\/}\nolimits\!\left(z% \right),\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)\right\}=\mathop{J_{{\nu+% 1}}\/}\nolimits\!\left(z\right)\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)-% \mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{Y_{{\nu+1}}\/}\nolimits\!% \left(z\right)=2/(\pi z),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.16
Permalink:: http://dlmf.nist.gov/10.5.E2
Encodings:: TeX, pMML, png

10.5.3 $\mathop{\mathscr{W}\/}\nolimits\{\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)% ,\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)\}=\mathop{J_{{\nu+1}}% \/}\nolimits\!\left(z\right)\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z% \right)-\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{{H^{{(1)}}_{{\nu+% 1}}}\/}\nolimits\!\left(z\right)=2i/(\pi z),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.5.E3
Encodings:: TeX, pMML, png

10.5.4 $\mathop{\mathscr{W}\/}\nolimits\{\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)% ,\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)\}=\mathop{J_{{\nu+1}}% \/}\nolimits\!\left(z\right)\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z% \right)-\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{{H^{{(2)}}_{{\nu+% 1}}}\/}\nolimits\!\left(z\right)=-2i/(\pi z),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.5.E4
Encodings:: TeX, pMML, png

10.5.5 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits% \!\left(z\right),\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)\right% \}=\mathop{{H^{{(1)}}_{{\nu+1}}}\/}\nolimits\!\left(z\right)\mathop{{H^{{(2)}}% _{{\nu}}}\/}\nolimits\!\left(z\right)-\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits% \!\left(z\right)\mathop{{H^{{(2)}}_{{\nu+1}}}\/}\nolimits\!\left(z\right)=-4i/% (\pi z).$

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.17
Permalink:: http://dlmf.nist.gov/10.5.E5
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.4 Connection Formulas 10.6 Recurrence Relations and Derivatives