# §10.5 Wronskians and Cross-Products

 10.5.1 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{J_{\nu}\/}\nolimits\!\left(z% \right),\mathop{J_{-\nu}\/}\nolimits\!\left(z\right)\right\}=\mathop{J_{\nu+1}% \/}\nolimits\!\left(z\right)\mathop{J_{-\nu}\/}\nolimits\!\left(z\right)+% \mathop{J_{\nu}\/}\nolimits\!\left(z\right)\mathop{J_{-\nu-1}\/}\nolimits\!% \left(z\right)=-2\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)/(\pi z),$
 10.5.2 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{J_{\nu}\/}\nolimits\!\left(z% \right),\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)\right\}=\mathop{J_{\nu+1}% \/}\nolimits\!\left(z\right)\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)-% \mathop{J_{\nu}\/}\nolimits\!\left(z\right)\mathop{Y_{\nu+1}\/}\nolimits\!% \left(z\right)=2/(\pi z),$
 10.5.3 $\mathop{\mathscr{W}\/}\nolimits\{\mathop{J_{\nu}\/}\nolimits\!\left(z\right),% \mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)\}=\mathop{J_{\nu+1}\/}% \nolimits\!\left(z\right)\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)-% \mathop{J_{\nu}\/}\nolimits\!\left(z\right)\mathop{{H^{(1)}_{\nu+1}}\/}% \nolimits\!\left(z\right)=2i/(\pi z),$
 10.5.4 $\mathop{\mathscr{W}\/}\nolimits\{\mathop{J_{\nu}\/}\nolimits\!\left(z\right),% \mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)\}=\mathop{J_{\nu+1}\/}% \nolimits\!\left(z\right)\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)-% \mathop{J_{\nu}\/}\nolimits\!\left(z\right)\mathop{{H^{(2)}_{\nu+1}}\/}% \nolimits\!\left(z\right)=-2i/(\pi z),$
 10.5.5 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!% \left(z\right),\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)\right\}=% \mathop{{H^{(1)}_{\nu+1}}\/}\nolimits\!\left(z\right)\mathop{{H^{(2)}_{\nu}}\/% }\nolimits\!\left(z\right)-\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)% \mathop{{H^{(2)}_{\nu+1}}\/}\nolimits\!\left(z\right)=-4i/(\pi z).$