\[\frac{U\left(a,b,z\right)}{\Gamma\left(c\right)\Gamma\left(c-b+1\right)}=\frac%
{z^{1-c}}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt}t^{-c}{{}_{2}{\mathbf{F}}_%
{1}}\left(a,c;a+c-b+1;1-\frac{1}{t}\right)\,\mathrm{d}t,\]