# §10.11 Analytic Continuation

When $m\in\mathbb{Z}$,

 10.11.1 $\mathop{J_{\nu}\/}\nolimits\!\left(ze^{m\pi i}\right)=e^{m\nu\pi i}\mathop{J_{% \nu}\/}\nolimits\!\left(z\right),$
 10.11.2 $\mathop{Y_{\nu}\/}\nolimits\!\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}\mathop{Y_% {\nu}\/}\nolimits\!\left(z\right)+2i\mathop{\sin\/}\nolimits\!\left(m\nu\pi% \right)\mathop{\cot\/}\nolimits\!\left(\nu\pi\right)\mathop{J_{\nu}\/}% \nolimits\!\left(z\right).$
 10.11.3 $\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{{H^{(1)}_{\nu}}\/}% \nolimits\!\left(ze^{m\pi i}\right)=-\mathop{\sin\/}\nolimits\!\left((m-1)\nu% \pi\right)\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)-e^{-\nu\pi i}% \mathop{\sin\/}\nolimits\!\left(m\nu\pi\right)\mathop{{H^{(2)}_{\nu}}\/}% \nolimits\!\left(z\right),$
 10.11.4 $\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{{H^{(2)}_{\nu}}\/}% \nolimits\!\left(ze^{m\pi i}\right)=e^{\nu\pi i}\mathop{\sin\/}\nolimits\!% \left(m\nu\pi\right)\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)+% \mathop{\sin\/}\nolimits\!\left((m+1)\nu\pi\right)\mathop{{H^{(2)}_{\nu}}\/}% \nolimits\!\left(z\right).$
 10.11.5 $\displaystyle\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(ze^{\pi i}\right)$ $\displaystyle=-e^{-\nu\pi i}\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(ze^{-\pi i}\right)$ $\displaystyle=-e^{\nu\pi i}\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right).$

If $\nu=n$ $(\in\mathbb{Z})$, then limiting values are taken in (10.11.2)–(10.11.4):

 10.11.6 $\mathop{Y_{n}\/}\nolimits\!\left(ze^{m\pi i}\right)=(-1)^{mn}(\mathop{Y_{n}\/}% \nolimits\!\left(z\right)+2im\mathop{J_{n}\/}\nolimits\!\left(z\right)),$
 10.11.7 $\mathop{{H^{(1)}_{n}}\/}\nolimits\!\left(ze^{m\pi i}\right)=(-1)^{mn-1}((m-1)% \mathop{{H^{(1)}_{n}}\/}\nolimits\!\left(z\right)+m\mathop{{H^{(2)}_{n}}\/}% \nolimits\!\left(z\right)),$
 10.11.8 $\mathop{{H^{(2)}_{n}}\/}\nolimits\!\left(ze^{m\pi i}\right)=(-1)^{mn}(m\mathop% {{H^{(1)}_{n}}\/}\nolimits\!\left(z\right)+(m+1)\mathop{{H^{(2)}_{n}}\/}% \nolimits\!\left(z\right)).$

For real $\nu$,

 10.11.9 $\displaystyle\mathop{J_{\nu}\/}\nolimits\!\left(\overline{z}\right)$ $\displaystyle=\overline{\mathop{J_{\nu}\/}\nolimits\!\left(z\right)},$ $\displaystyle\hskip 10.0pt\mathop{Y_{\nu}\/}\nolimits\!\left(\overline{z}\right)$ $\displaystyle=\overline{\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)},$ $\displaystyle\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(\overline{z}\right)$ $\displaystyle=\overline{\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)},$ $\displaystyle\hskip 10.0pt\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(\overline% {z}\right)$ $\displaystyle=\overline{\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)}.$

For complex $\nu$ replace $\nu$ by $\overline{\nu}$ on the right-hand sides.