\[\frac{2\pi\mathrm{i}{\mathrm{e}}^{-z}}{\Gamma\left(b\right)\Gamma\left(a-b+1%
\right)}M\left(b-a,b,z\right)={\mathrm{e}}^{b\pi\mathrm{i}}U\left(a,b,{\mathrm%
{e}}^{\pi\mathrm{i}}z\right)-{\mathrm{e}}^{-b\pi\mathrm{i}}U\left(a,b,{\mathrm%
{e}}^{-\pi\mathrm{i}}z\right),\]