User:Muphrid15/Cauchy integral draft

In non-complex algebras
The Cauchy integral formula is generalizable to real vector spaces of two or more dimensions. The insight into this property comes from geometric algebra, where objects beyond scalars and vectors (such as planar bivectors and volumetric trivectors) are considered, and a proper generalization of Stokes theorem.

Geometric calculus defines a derivative operator $$\nabla = \hat e_i \partial_i$$ under its geometric product--that is, for a $$k$$-vector field $$\psi(\vec r)$$, the derivative $$\nabla \psi$$ generally contains terms of grade $$k+1$$ and $$k-1$$. For example, a vector field ($$k=1$$) generally has in its derivative a scalar part, the divergence ($$k=0$$), and a bivector part, the curl ($$k=2$$). This particular derivative operator has a Green's function:


 * $$G(\vec r, \vec r') = \frac{1}{S_n} \frac{\vec r - \vec r'}{|\vec r - \vec r'|^n}$$

where $$S_n$$ is the surface area of a unit ball in the space (that is, $$S_2=2\pi$$, the circumference of a circle with radius 1, and $$S_3 = 4\pi$$, the surface area of a sphere with radius 1). By definition of a Green's function, $$\nabla_{\vec r} G(\vec r, \vec r') = \delta(\vec r- \vec r')$$. It is this useful property that can be used, in conjunction with the generalized Stokes theorem:


 * $$\oint_{\partial V} d\vec S \; f(\vec r) = \int_V d\vec V \; \nabla f(\vec r)$$

where, for an $$n$$-dimensional vector space, $$d\vec S$$ is an $$n-1$$-vector and $$d\vec V$$ is an $$n-$$vector. The function $$f(\vec r)$$ can, in principle, be composed of any combination of multivectors. The proof of Cauchy's integral theorem for higher dimensional spaces relies on the using the generalized Stokes theorem on the quantity $$G(\vec r,\vec r') f(\vec r')$$ and use of the product rule:


 * $$\oint_{\partial V'} G(\vec r, \vec r')\; d\vec S' \; f(\vec r') = \int_V \left([\nabla' G(\vec r, \vec r')] f(\vec r') + G(\vec r, \vec r') \nabla' f(\vec r')\right) \; d\vec V$$

when $$\nabla \vec f = 0$$, $$f(\vec r)$$ is called a monogenic function, the generalization of holomorphic functions to higher-dimensional spaces--indeed, it can be shown that the Cauchy-Riemann condition is just the two-dimensional expression of the monogenic condition. When that condition is met, the second term in the right-hand integral vanishes, leaving only


 * $$\oint_{\partial V'} G(\vec r, \vec r')\; d\vec S' \; f(\vec r') = \int_V [\nabla' G(\vec r, \vec r')] f(\vec r') = \int_V \delta(\vec r - \vec r') f(\vec r') \; d\vec V = i_n f(\vec r)$$

where $$i_n$$ is that algebra's unit $$n$$-vector, the pseudoscalar. The result is


 * $$f(\vec r) = \frac{1}{i_n} \oint_{\partial V} G(\vec r, \vec r')\; d\vec S \; f(\vec r') = \frac{1}{i_n} \oint_{\partial V} \frac{\vec r - \vec r'}{S_n |\vec r - \vec r'|^n} \; d\vec S \; f(\vec r')$$

Thus, as in the two-dimensional (complex analysis) case, the value of an analytic (monogenic) function at a point can be found by an integral over the surface surrounding the point, and this is valid not only for scalar functions but vector and general multivector functions as well.