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Consider the contour integral equation:

$$\int_C \frac{1}{(z-w)} \frac{1}{(w-v)} dw = \frac{\pi}{z-v}$$

Where $C$ is a contour that completely surrounds the complex point $z$ or $v$.

Is there a generalisation of this to N dimensions. For example the most obvious thing I can think of is to replace the complex numbers with quaternions and replace the contour with a 3-manifold. Would something like that work?

Is there a generalisatio of this equation into N dimensions using vector calculus?

e.g.

$$\int\limits_{\partial S} A(z,w) \otimes A(w,v) dw \propto A(z,v)$$ Where $\partial S$ is some $N-1$ dimensional curved surface and $z$, $w$ and $v$ are in $N$ dimensional space? And the integral is zero unless $\partial S$ enclosed either the point $z$ or $w$. $A$ are vector fields and $\otimes$ is some tensor product of some kind.

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