I'm trying to understand Cauchy’s integral theorem and I've encountered with two statements for that:
If $f(z)$ is analytic in some simply connected region $R$, then $\oint_\gamma f(z)\,dz = 0$ for any closed contour $\gamma$ completely contained in $R$.
And the second statement:
If $f(z)$ is analytic on a simple closed path $\gamma$ and everywhere inside $\gamma$ then $\oint_\gamma f(z)\,dz = 0$.
And I wondered how these statements are equivalent. Certainly simply connected region doesn't contain boundary so we couldn't conclude second statement. I'm new to complex analysis and its concepts are really hard to grasp for me.