Say we have an integral $\tfrac{1}{2\pi i}\int_{\gamma}\tfrac{1}{z-a}dz$
I know this can be parametrised as
$$\tfrac{1}{2\pi i}\int^1_0\tfrac{\gamma'(t)}{\gamma(t)-a}dt$$.
My question :
So long as $a\notin \{\gamma\}$ is it true that
$$\tfrac{1}{2\pi i}\int^1_0\tfrac{\gamma'(t)}{\gamma(t)-a}dt=\tfrac{1}{2\pi i}[F(1)-F(0)]$$(as the integrand is holomorphic)
and also what doesthe expression for F(1) look like ?