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