If $ \dfrac{1}{2\pi i}\int_{C}f(z)dz$ exists for any circle of radius $r>0$ centered at origin, the point $0$ is a singularity for $f$, and always the integral has the same value, say $k = \dfrac{1}{2\pi i}\int_{C}f(z)dz,$ and $f$ is analitic outside $C$ with $|f(z)|\leq \dfrac{m}{|z|}$ ($m$ constant), is true that we can shift path of integration from $C$ to the some line $Re(z)= \gamma$ using Cauchy's Theorem?
My doubt arises because of the lemma 7.1 in the PAZY book (Semigroups of linear operators and applications to partial differential equations), which shows that the integral exists for any circle centered on the origin and then says that it is possible to use Cauchy's Theorem to change the path of integration from a circle to a line. Is that really true? I know it can be exchanged for another closed contour, more the way it did I’ve never seen.