\[z=\pm(2n+a)\pi\mathrm{i}+\ln\left(-\frac{\Gamma\left(a\right)}{\Gamma\left(b-a%
\right)}\left(\pm 2n\pi\mathrm{i}\right)^{b-2a}\right)+O\left(n^{-1}\ln n%
\right),\]