# §10.34 Analytic Continuation

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

 10.34.1 $I_{\nu}\left(ze^{m\pi i}\right)=e^{m\nu\pi i}I_{\nu}\left(z\right),$
 10.34.2 $K_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}K_{\nu}\left(z\right)-\pi i\sin% \left(m\nu\pi\right)\csc\left(\nu\pi\right)I_{\nu}\left(z\right).$
 10.34.3 $\displaystyle I_{\nu}\left(ze^{m\pi i}\right)$ $\displaystyle=(i/\pi)\left(\pm e^{m\nu\pi i}K_{\nu}\left(ze^{\pm\pi i}\right)% \mp e^{(m\mp 1)\nu\pi i}K_{\nu}\left(z\right)\right),$ 10.34.4 $\displaystyle K_{\nu}\left(ze^{m\pi i}\right)$ $\displaystyle=\csc\left(\nu\pi\right)\left(\pm\sin\left(m\nu\pi\right)K_{\nu}% \left(ze^{\pm\pi i}\right)\mp\sin\left((m\mp 1)\nu\pi\right)K_{\nu}\left(z% \right)\right).$

If $\nu=n(\in\mathbb{Z})$, then limiting values are taken in (10.34.2) and (10.34.4):

 10.34.5 $K_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}K_{n}\left(z\right)+(-1)^{n(m-1)-1}m\pi iI% _{n}\left(z\right),$
 10.34.6 $K_{n}\left(ze^{m\pi i}\right)=\pm(-1)^{n(m-1)}mK_{n}\left(ze^{\pm\pi i}\right)% \mp(-1)^{nm}(m\mp 1)K_{n}\left(z\right).$

For real $\nu$,

 10.34.7 $\displaystyle I_{\nu}\left(\overline{z}\right)$ $\displaystyle=\overline{I_{\nu}\left(z\right)},$ $\displaystyle K_{\nu}\left(\overline{z}\right)$ $\displaystyle=\overline{K_{\nu}\left(z\right)}.$

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