Morera's Theorem and independence of path Determine whether 
$$
G(z)=\int_{1}^{z}\frac{1}{t} dt
$$
is independent of the path joining $1$ and $z$ , and discuss the relationship of your answer to Morera's Theorem .....  I believe that as long as the paths goes through a region that excludes the origin ,the integral is independent of the path.... but I am not sure if this is correct or what type of argument is required here , in particular the relation of the answer to Morera's theorem is not clear to me .

Have you thought about traversing the unit circle for $z=1$?
$$
\int_C\frac 1zdz=\int_0^{2\pi}\frac{1}{e^{it}}(ie^{it})dt=\int_0^{2\pi}idt=2\pi i
$$
