# §10.4 Connection Formulas

Other solutions of (10.2.1) include $\mathop{J_{-\nu}\/}\nolimits\!\left(z\right)$, $\mathop{Y_{-\nu}\/}\nolimits\!\left(z\right)$, $\mathop{{H^{(1)}_{-\nu}}\/}\nolimits\!\left(z\right)$, and $\mathop{{H^{(2)}_{-\nu}}\/}\nolimits\!\left(z\right)$.

 10.4.1 $\displaystyle\mathop{J_{-n}\/}\nolimits\!\left(z\right)$ $\displaystyle=(-1)^{n}\mathop{J_{n}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{Y_{-n}\/}\nolimits\!\left(z\right)$ $\displaystyle=(-1)^{n}\mathop{Y_{n}\/}\nolimits\!\left(z\right),$
 10.4.2 $\displaystyle\mathop{{H^{(1)}_{-n}}\/}\nolimits\!\left(z\right)$ $\displaystyle=(-1)^{n}\mathop{{H^{(1)}_{n}}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{{H^{(2)}_{-n}}\/}\nolimits\!\left(z\right)$ $\displaystyle=(-1)^{n}\mathop{{H^{(2)}_{n}}\/}\nolimits\!\left(z\right).$
 10.4.3 $\displaystyle\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)$ $\displaystyle=\mathop{J_{\nu}\/}\nolimits\!\left(z\right)+i\mathop{Y_{\nu}\/}% \nolimits\!\left(z\right),$ $\displaystyle\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)$ $\displaystyle=\mathop{J_{\nu}\/}\nolimits\!\left(z\right)-i\mathop{Y_{\nu}\/}% \nolimits\!\left(z\right),$
 10.4.4 $\displaystyle\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{1}{2}\left(\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z% \right)+\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)\right),$ $\displaystyle\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{1}{2i}\left(\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z% \right)-\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)\right).$
 10.4.5 $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)=\mathop{\csc\/}\nolimits(\nu\pi)% \left(\mathop{Y_{-\nu}\/}\nolimits\!\left(z\right)-\mathop{Y_{\nu}\/}\nolimits% \!\left(z\right)\mathop{\cos\/}\nolimits(\nu\pi)\right).$
 10.4.6 $\displaystyle\mathop{{H^{(1)}_{-\nu}}\/}\nolimits\!\left(z\right)$ $\displaystyle=e^{\nu\pi i}\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{{H^{(2)}_{-\nu}}\/}\nolimits\!\left(z\right)$ $\displaystyle=e^{-\nu\pi i}\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right).$
 10.4.7 $\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)=i\mathop{\csc\/}\nolimits(% \nu\pi)\left(e^{-\nu\pi i}\mathop{J_{\nu}\/}\nolimits\!\left(z\right)-\mathop{% J_{-\nu}\/}\nolimits\!\left(z\right)\right)=\mathop{\csc\/}\nolimits(\nu\pi)% \left(\mathop{Y_{-\nu}\/}\nolimits\!\left(z\right)-e^{-\nu\pi i}\mathop{Y_{\nu% }\/}\nolimits\!\left(z\right)\right),$
 10.4.8 $\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)=i\mathop{\csc\/}\nolimits(% \nu\pi)\left(\mathop{J_{-\nu}\/}\nolimits\!\left(z\right)-e^{\nu\pi i}\mathop{% J_{\nu}\/}\nolimits\!\left(z\right)\right)=\mathop{\csc\/}\nolimits(\nu\pi)% \left(\mathop{Y_{-\nu}\/}\nolimits\!\left(z\right)-e^{\nu\pi i}\mathop{Y_{\nu}% \/}\nolimits\!\left(z\right)\right).$

In (10.4.5), (10.4.7), and (10.4.8) limiting values are taken when $\nu=n$; compare (10.2.3) and (10.2.4).