# §14.23 Values on the Cut

When $-1,

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

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

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

Conversely,

 14.23.4 $\displaystyle\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=e^{\pm\mu\pi i/2}\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\pm i% 0\right),$ 14.23.5 $\displaystyle\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\tfrac{1}{2}\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)% \left(e^{-\mu\pi i/2}\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x+i% 0\right)+e^{\mu\pi i/2}\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x% -i0\right)\right),$

or equivalently,

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

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 $\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ to complex $x$.

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

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