# §14.23 Values on the Cut

When $-1,

 14.23.1 $P^{\mu}_{\nu}\left(x\pm i0\right)=e^{\mp\mu\pi i/2}\mathsf{P}^{\mu}_{\nu}\left% (x\right),$
 14.23.2 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)=\frac{e^{\pm\mu\pi i/2}}{\Gamma% \left(\nu+\mu+1\right)}\left(\mathsf{Q}^{\mu}_{\nu}\left(x\right)\mp\tfrac{1}{% 2}\pi i\mathsf{P}^{\mu}_{\nu}\left(x\right)\right).$

In terms of the hypergeometric function $\mathbf{F}$14.3(i))

 14.23.3 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)=\frac{e^{\mp\nu\pi i/2}\pi^{3/2% }\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\frac{x\mathbf{F}\left(\frac{1}% {2}\mu-\frac{1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};% x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)% \Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}\mp i\frac{% \mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+% \frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}% \mu+1\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}\right).$

Conversely,

 14.23.4 $\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)$ $\displaystyle=e^{\pm\mu\pi i/2}P^{\mu}_{\nu}\left(x\pm i0\right),$ 14.23.5 $\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ $\displaystyle=\tfrac{1}{2}\Gamma\left(\nu+\mu+1\right)\left(e^{-\mu\pi i/2}% \boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)+e^{\mu\pi i/2}\boldsymbol{Q}^{\mu}% _{\nu}\left(x-i0\right)\right),$

or equivalently,

 14.23.6 $\mathsf{Q}^{\mu}_{\nu}\left(x\right)=e^{\mp\mu\pi i/2}\Gamma\left(\nu+\mu+1% \right)\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)\pm\tfrac{1}{2}\pi ie^{% \pm\mu\pi i/2}P^{\mu}_{\nu}\left(x\pm i0\right).$

If cuts are introduced along the intervals $(-\infty,-1]$ and $[1,\infty)$, then (14.23.4) and (14.23.6) could be used to extend the definitions of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ to complex $x$.

The conical function defined by (14.20.2) can be represented similarly by

 14.23.7 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\tfrac{1}{2}e^{% 3\mu\pi i/2}Q^{-\mu}_{-\frac{1}{2}+i\tau}\left(x-i0\right)+\tfrac{1}{2}e^{-3% \mu\pi i/2}Q^{-\mu}_{-\frac{1}{2}-i\tau}\left(x+i0\right).$