10 Bessel Functions Modified Bessel Functions 10.27 Connection Formulas 10.29 Recurrence Relations and Derivatives

§10.28 Wronskians and Cross-Products

Notes:: For the Wronskians use (1.13.5) and the limiting forms in §10.30. For the cross-products apply (10.29.2).
Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.28

10.28.1 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{I_{{\nu}}\/}\nolimits\!\left(z% \right),\mathop{I_{{-\nu}}\/}\nolimits\!\left(z\right)\right\}=\mathop{I_{{\nu% }}\/}\nolimits\!\left(z\right)\mathop{I_{{-\nu-1}}\/}\nolimits\!\left(z\right)% -\mathop{I_{{\nu+1}}\/}\nolimits\!\left(z\right)\mathop{I_{{-\nu}}\/}\nolimits% \!\left(z\right)=-2\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)/(\pi z),$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.14
Permalink:: http://dlmf.nist.gov/10.28.E1
Encodings:: TeX, pMML, png

10.28.2 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{K_{{\nu}}\/}\nolimits\!\left(z% \right),\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)\right\}=\mathop{I_{{\nu}% }\/}\nolimits\!\left(z\right)\mathop{K_{{\nu+1}}\/}\nolimits\!\left(z\right)+% \mathop{I_{{\nu+1}}\/}\nolimits\!\left(z\right)\mathop{K_{{\nu}}\/}\nolimits\!% \left(z\right)=1/z.$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.15
Permalink:: http://dlmf.nist.gov/10.28.E2
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. 10.27 Connection Formulas 10.29 Recurrence Relations and Derivatives