Clearly, the Green's Theorem proof does not hold as that relies on the specific conditions of complex differentiability. However, Goursat's proof involving the triangle case and building up from there invokes the Cauchy-Riemann equations nowhere, and just requires a basic definition of differentiability that satisfies:
$$f(x+h)-f(x)-f'(x)h=\psi_x(x+h)h$$
Where $\psi_x(x+h)h$ can be made as small as desired.
It also requires some basic topology and geometric properties of triangles, so I believe any $\Bbb R^2$-type space with the Euclidean topology would satisfy Goursat's lemma. Moreover, the use of this lemma to prove the Cauchy Integral Theorem again requires only some basic notion of contour (or line) integrability, and antiderivatives, which the space $\Bbb R^2$ also has. The homotopic invariance theorem relies on properties of compact sets, which exist in the Euclidean topology, and not really any much else.
This leads to believe that if you have a space which is:
- Complete
- Has a notion of differentiability satisfying the above
- Has a notion of antiderivative, Riemann sum and line integral
- Is two-dimensional with the Euclidean topology
- Has compact sets and uniform continuity of continuous functions on them
Then you have homotopic invariance of line integrals of differentiable functions. You can't quite have the Cauchy Integral Formula, as $2\pi i$ is not defined without letting $i$ be complex, however similar expressions to do with winding numbers could arise in other spaces.
For example, by my reasoning, we have the above theorems valid in:
- $\Bbb R^2$
- The split-complex numbers
- The dual numbers
Yet these theorems are always touted as important results of complex analysis, implying that I'm missing something here. What about the proofs of any of these theorems is complex-specific?