It appears to me that there is a tight connection between the Cauchy integral theorem and Poincare's Lemma(in 2 dimensions). While the former requires the function to be holomorphic, i.e. to be differentiable in the real sense and fulfill the Cauchy-Riemann equations, the latter stipulates that the function have a 2-dimensional curl of 0(in vector-analysis formulation, naturally). Additionally, for both to apply, the space has to be contractible, only that this is merely implicitly stated in the Cauchy integral theorem since an everywhere holomorphic function certainly is defined at every point.
Now, i am only a student of physics, but it irks me that this apparently deep connection(especially considering that the CR-equations appear to me to basically be a complex 2-dim-curl equal to 0, or, equivalently, an exterior derivative of the one-form equal to 0) is never commented on in our watered-down mathematics courses.
So here is the question: what is the actual connection between these two marvelous theorems?
Thanks in advance!