Mellin transform definition

Mellin transform definition

is thre a possible meaning or definition to the Mellin transform
$$ \int_{0}^{\infty} \frac{dt}{t-1}t^{s-1}= F(s) $$
i know that $$ \int_{0}^{\infty} \frac{dt}{t+1}t^{s-1}= F(s)=
\frac{\pi}{sin(\pi s)} $$
howver can one overcome the pole at $ t=1 $ ?