Editing Cauchy's integral formula (section)

===In real 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 [[bivector]]s and volumetric [[trivector]]s) are considered, and a proper generalization of [[Stokes' theorem]].

Geometric calculus defines a derivative operator {{math|1=∇ = '''ê'''<sub>''i''</sub> ∂<sub>''i''</sub>}} under its geometric product — that is, for a {{math|''k''}}-vector field {{math|''ψ''('''r''')}}, the derivative {{math|∇''ψ''}} generally contains terms of grade {{math|''k'' + 1}} and {{math|''k'' − 1}}.  For example, a vector field ({{math|1=''k'' = 1}}) generally has in its derivative a scalar part, the [[divergence]] ({{math|1=''k'' = 0}}), and a bivector part, the [[curl (mathematics)|curl]] ({{math|1=''k'' = 2}}).  This particular derivative operator has a [[Green's function]]:
<math display="block">G\left(\mathbf r, \mathbf r'\right) = \frac{1}{S_n} \frac{\mathbf r - \mathbf r'}{\left|\mathbf r - \mathbf r'\right|^n}</math>
where {{math|''S<sub>n</sub>''}} is the surface area of a unit {{math|''n''}}-[[ball (mathematics)|ball]] in the space (that is, {{math|1=''S''<sub>2</sub> = 2π}}, the circumference of a circle with radius 1, and {{math|1=''S''<sub>3</sub> = 4π}}, the surface area of a sphere with radius 1).  By definition of a Green's function,
<math display="block">\nabla G\left(\mathbf r, \mathbf r'\right) = \delta\left(\mathbf r- \mathbf r'\right).</math>

It is this useful property that can be used, in conjunction with the generalized Stokes theorem:
<math display="block">\oint_{\partial V} d\mathbf S \; f(\mathbf r) = \int_V d\mathbf V \; \nabla f(\mathbf r)</math>
where, for an {{math|''n''}}-dimensional vector space, {{math|''d'''''S'''}} is an {{math|(''n'' − 1)}}-vector and {{math|''d'''''V'''}} is an {{math|''n''}}-vector.  The function {{math|''f''('''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 {{math|''G''('''r''', '''r'''′) ''f''('''r'''′)}} and use of the product rule:
<math display="block">\oint_{\partial V'} G\left(\mathbf r, \mathbf r'\right)\;  d\mathbf S' \; f\left(\mathbf r'\right)
= \int_V \left(\left[\nabla' G\left(\mathbf r, \mathbf r'\right)\right] f\left(\mathbf r'\right) + G\left(\mathbf r, \mathbf r'\right) \nabla' f\left(\mathbf r'\right)\right) \; d\mathbf V</math>

When {{math|1=∇''f'' = 0}}, {{math|''f''('''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
<math display="block">\oint_{\partial V'} G\left(\mathbf r, \mathbf r'\right)\;  d\mathbf S' \; f\left(\mathbf r'\right)
= \int_V \left[\nabla' G\left(\mathbf r, \mathbf r'\right)\right] f\left(\mathbf r'\right)
= -\int_V \delta\left(\mathbf r - \mathbf r'\right) f\left(\mathbf r'\right) \; d\mathbf V
=- i_n f(\mathbf r)</math>
where {{math|''i<sub>n</sub>''}} is that algebra's unit {{math|''n''}}-vector, the [[pseudoscalar]].  The result is
<math display="block">f(\mathbf r)
=- \frac{1}{i_n} \oint_{\partial V} G\left(\mathbf r, \mathbf r'\right)\;  d\mathbf S \; f\left(\mathbf r'\right)
= -\frac{1}{i_n} \oint_{\partial V} \frac{\mathbf r - \mathbf r'}{S_n \left|\mathbf r - \mathbf r'\right|^n} \; d\mathbf S \; f\left(\mathbf r'\right)</math>

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.