In the text Special Functions and their Applications by N.N. Lebedev (pp.172-173), a derivation is presented for a certain integral representation of the Legendre function of the first kind $P_{\nu }(\cosh (\alpha ))$ for $-1<\Re(\nu )<0.$ The starting point is the contour integral $$\frac{1}{\pi } \int_C \frac{e^{\left(\nu +\frac{1}{2}\right) t}}{\sqrt{2 \cosh (\alpha )-2 \cosh (t)}} \, dt$$ where the contour C is the rectangle bounded by $\Re(t)=-R,\Re(t)=R,\Im(t)=\pi$ and the real t axis. Since the integrand contains branch points at $t=\alpha$ and $t=-\alpha$, we must indent the contour at these points with a small semicircle of radius $\rho$. We then take the limit as $R\to \infty$ and $\rho \to 0$.
The problem I am having is defining a single valued branch of the square root function appearing in the denominator. Where do we place the branch cut(s)? and how will this unambiguously determine the argument of the square root along the contour?. If the integrand contained for example, the multivalued function $\left(t^2-1\right)^{1/2}$, then we could write $t-1=\rho _1 e^{i \phi _1}$ and $t+1=\rho _2 e^{i \phi _2}$ and by taking $0\leq \phi _1<2 \pi$ and $0\leq \phi _2<2 \pi$ we have a branch cut on the segment (-1,1) of the real axis. We could then write $\left(t^2-1\right)^{1/2 }=\sqrt {\rho _1 \rho _2} e^{\frac{1}{2} i \left(\phi _1+\phi _2\right)}$ where $\sqrt{\rho _1 \rho _2}>0.$