# §10.5 Wronskians and Cross-Products

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