Cauchy's Integral Formula and Ampere's Law - Any Connection?

[brandon_1892]

I know how, by Amp$\grave{e}$re's Law for the loop integral of a magnetic field, $$\oint_C \textbf B \cdot d \textbf s = \mu_0 I$$ and this is zero when there is no current enclosed in the loop; there is a nonzero result when there is a current.

With Cauchy's Integral Theorem, if a function is analytic on and in the loop integrated over, $$\oint_C f(z) dz = 0$$; otherwise there may be a nonzero result.

Since essentially the magnetic field approaches infinity right in the current, I thought maybe Ampere's Law is an instance of Cauchy's Integral Formula/Theorem; when there is no current enclosed, the magnetic field has no such center, so all of it would be analytic.

Is this an actual connection, then?

 

[elfmotat]

If you wanted to get hand-wavy, I suppose you could view current as the residue of the magnetic field.