# What is the generalisation of this residue theorem to N-dimensions?

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.