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dingo_d

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Since complex field is 'extension' to the real field, and in electrodynamics we use things like Stokes theorem, or Gauss theorem, that are being done on real field (differential manifolds and things like that, right?), can we use theorems in complex analysis, and say that some feature can be described because of that?

An example.

In complex analysis there is famous Cauchy-Goursat theorem which states that if we have some analytical function on a convex set, and if we have some closed path in that set the integral:

[tex]\oint_\gamma f(z)dz=0[/tex]

And in electrostatics we have the irrotational electric field

[tex]\vec{\nabla}\times \vec{E}=0[/tex] which if we use Kelvin-Stokes becomes:

[tex]\oint \vec{E}\cdot d\vec{\ell}=0[/tex].

Is this just a generalization of CG theorem and could we use all the theorems in complex analysis, or should we need to be more careful (like with analytic functions etc.)?