# §10.11 Analytic Continuation

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

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

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

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

For real $\nu$,

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

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