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