\[\frac{1}{\Gamma^{*}\left(z\right)}=1-\frac{1}{2\pi\mathrm{i}}\int_{0}^{\infty}%
\frac{{\mathrm{e}}^{-2\pi t}\Gamma^{*}\left(t{\mathrm{e}}^{\mathrm{i}\pi/2}%
\right)}{t-\mathrm{i}z}\,\mathrm{d}t+\frac{1}{2\pi\mathrm{i}}\int_{0}^{\infty}%
\frac{{\mathrm{e}}^{-2\pi t}\Gamma^{*}\left(t{\mathrm{e}}^{-\mathrm{i}\pi/2}%
\right)}{t+\mathrm{i}z}\,\mathrm{d}t,\]