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?


$$\int\limits_{\partial S} A(z,w) \otimes A(w,v) dw \propto A(z,v)$$ 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.