In Cauchy's Theorem there are two homotopic paths in $\mathbb{C}$ with the same endpoints (the homotopy leaving both endpoints fixed). There is an analytic function $\mathbb{C} \to \mathbb{C}$ and the integral over both paths is the same.
I can imagine a generalization, where there are two homotopic maps from the square $[0,1]^2$ to $\mathbb{C}^2$ (the homotopy leaving the boundary of the square fixed), defining two "patches". And there is an analytic function $\mathbb{C}^2 \to \mathbb{C}$. The integral of the function over both "patches" is the same.
If there is such a generalization, then is it really a form of Stokes Theorem ?
The particular integral I'm interested in is the complex inversion formula for the multivariate Laplace Transform. This formula is a limit of an integral over "patches" centered at $0$ that grow arbitrarily large.
Thank you in advance...