Cauchy's theorem in short says for a holomorphic function $f$ which is holomorphic on and inside a path $\gamma$ the path integral is $0$
I have calculated a path integral around a path where there are three singularities inside- I have Cauchy's Residue Theorem to find the integral- but it has given me $0$- the residues all cancel out. This is perfectly possible right?- the Converse of Cauchy is not necessarily true is it? The integral being $0$ doesn't imply that $f$ is holomorphic on and inside $\gamma$. If it is true- I have made a mistake in arithmetic?