Let $f(z)$ be a complex valued function defined on an open subset $U$ of the complex plane. We say $f(z)$ is analytic if it can be expressed by a convergent power series on a neighborhood of every point of $U$. Suppose $U$ is simply connected. Then it is easy to prove that the contour integral of $f(z)$ on a piecewise continuously differentiable closed curve in $U$ is zero. It is easy because you can reduce it in the trivial case when $f(z) = z^n$. Let's call this the weak version of Cauchy's theorem.
I think we can get most of the important results of complex function theory of one variable by using this theorem. The reason is as follows. In most applications of complex analysis, it is easy to prove that the function is analytic in our sense. For example, it is easy to show that the Riemann zeta function is analytic in our sense.
Am I mistaken?