Editing Cauchy's integral formula (section)

==Generalizations==

===Smooth functions===
A version of Cauchy's integral formula is the Cauchy–[[Dimitrie Pompeiu|Pompeiu]] formula,<ref>{{harvnb|Pompeiu|1905}}</ref> and holds for [[smooth function]]s as well, as it is based on [[Stokes' theorem]].  Let {{math|''D''}} be a disc in {{math|'''C'''}} and suppose that {{math|''f''}} is a complex-valued {{math|[[continuously differentiable function|''C''{{isup|1}}]]}} function on the [[closure (topology)|closure]] of {{math|''D''}}.  Then<ref>{{cite web | url = https://people.math.carleton.ca/~ckfong/S32.pdf | title = §2. Complex 2-Forms: Cauchy-Pompeiu's Formula}}</ref><ref>{{harvnb|Hörmander|1966|loc=Theorem 1.2.1}}</ref><ref>{{cite web | url = https://www.jirka.org/scv/scv.pdf#thm.4.1.1 | title = Theorem 4.1.1 (Cauchy–Pompeiu).}}</ref>
<math display="block">f(\zeta) = \frac{1}{2\pi i}\int_{\partial D} \frac{f(z) \,dz}{z-\zeta} - \frac{1}{\pi}\iint_D \frac{\partial f}{\partial \bar{z}}(z) \frac{dx\wedge dy}{z-\zeta}.</math>

One may use this representation formula to solve the inhomogeneous [[Cauchy–Riemann equations]] in {{math|''D''}}.  Indeed, if {{math|''φ''}} is a function in {{math|''D''}}, then a particular solution {{math|''f''}} of the equation is a holomorphic function outside the support of {{math|''μ''}}.  Moreover, if in an open set {{math|''D''}},
<math display="block">d\mu = \frac{1}{2\pi i}\varphi \, dz\wedge d\bar{z}</math>
for some {{math|''φ'' ∈ ''C''{{isup|''k''}}(''D'')}} (where {{math|''k''&nbsp;≥&nbsp;1}}), then {{math|''f''(''ζ'', {{overline|''ζ''}})}} is also in {{math|''C''{{isup|''k''}}(''D'')}} and satisfies the equation
<math display="block">\frac{\partial f}{\partial\bar{z}} = \varphi(z,\bar{z}).</math>

The first conclusion is, succinctly, that the [[convolution]] {{math|''μ'' ∗ ''k''(''z'')}} of a compactly supported measure with the '''Cauchy kernel'''
<math display="block">k(z) = \operatorname{p.v.}\frac{1}{z}</math>
is a holomorphic function off the support of {{math|''μ''}}.  Here {{math|p.v.}} denotes the [[Cauchy principal value|principal value]]. The second conclusion asserts that the Cauchy kernel is a [[fundamental solution]] of the Cauchy–Riemann equations. Note that for smooth complex-valued functions {{math|''f''}} of compact support on {{math|'''C'''}} the generalized Cauchy integral formula simplifies to
<math display="block">f(\zeta) =  \frac{1}{2\pi i}\iint \frac{\partial f}{\partial \bar{z}}\frac{dz\wedge d\bar{z}}{z-\zeta},</math>
and is a restatement of the fact that, considered as a [[distribution (mathematics)|distribution]], {{math|(π''z'')<sup>−1</sup>}} is a [[fundamental solution]] of the [[Cauchy–Riemann operator]] {{math|{{sfrac|∂|∂''z̄''}}}}.<ref>{{harvnb|Hörmander|1983|pp=63, 81}}</ref>

The generalized Cauchy integral formula can be deduced for any bounded open region {{math|''X''}} with {{math|''C''{{isup|1}}}} boundary {{math|∂''X''}} from this result and the formula for the [[distributional derivative]] of the [[indicator function|characteristic function]] {{math|''χ''<sub>''X''</sub>}} of {{math|''X''}}:
<math display="block"> \frac {\partial \chi_X}{\partial \bar z}= \frac{i}{2} \oint_{\partial X} \,dz,</math>
where the distribution on the right hand side denotes [[contour integration]] along {{math|∂''X''}}.<ref>{{harvnb|Hörmander|1983|pp=62–63}}</ref>
{{Math proof|For <math>\varphi \in \mathcal{D}(X)</math> calculate:
:<math>
\begin{aligned}
\left\langle\frac{\partial}{\partial \bar{z}}\left(\chi_X\right), \varphi\right\rangle & =-\int_X \frac{\partial \varphi}{\partial \bar{z}} \mathrm{~d}(x, y) \\
& =-\frac{1}{2} \int_X\left(\partial_x \varphi+\mathrm{i} \partial_y \varphi\right) \mathrm{d}(x, y) .
\end{aligned}
</math>
then traverse <math>\partial X</math> in the anti-clockwise direction. Fix a point <math>p \in \partial X</math> and let <math>s</math> denote arc length on <math>\partial X</math> measured from <math>p</math> anti-clockwise. Then, if <math>\ell</math> is the length of <math>\partial X,[0, \ell] \ni s \mapsto(x(s), y(s))</math> is a parametrization of <math>\partial X</math>. The derivative <math>\tau=\left(x'(s), y'(s)\right)</math> is a unit tangent to <math>\partial X</math> and <math>\nu:=\left(-y'(s), x'(s)\right)</math> is the unit outward normal on <math>\partial X</math>. We are lined up for use of the [[divergence theorem]]: put <math>V=(\varphi, \mathrm{i} \varphi) \in \mathcal{D}(X)^2</math> so that <math>\operatorname{div} V=\partial_x \varphi+\mathrm{i} \partial_y \varphi</math> and we get
:<math>
\begin{aligned}
-\frac{1}{2} \int_X\left(\partial_x \varphi+\mathrm{i} \partial_y \varphi\right) \mathrm{d}(x, y) & =-\frac{1}{2} \int_{\partial X} V \cdot \nu \mathrm{d} S \\
& =-\frac{1}{2} \int_0^{\ell}\left(\varphi \nu_1+\mathrm{i} \varphi \nu_2\right) \mathrm{d} s \\
& =-\frac{1}{2} \int_0^{\ell} \varphi(x(s), y(s))\left(y'(s)-\mathrm{i} x'(s)\right) \mathrm{d} s \\
& =\frac{1}{2} \int_0^{\ell} \mathrm{i} \varphi(x(s), y(s))\left(x'(s)+\mathrm{i} y'(s)\right) \mathrm{d} s \\
& =\frac{\mathrm{i}}{2} \int_{\partial X} \varphi \mathrm{d} z
\end{aligned}
</math>
Hence we proved <math> \frac {\partial \chi_X}{\partial \bar z}= \frac{i}{2} \oint_{\partial X} \,dz</math>.
}}
Now we can deduce the generalized Cauchy integral formula:
{{Math proof|Since <math>u=\frac{\chi_X}{\pi\left(z-z_0\right)} \in \mathrm{L}_{\text{loc}}^1(X)</math> and since <math>z_0 \in X</math> this distribution is locally in <math>X</math> of the form "distribution times {{math|C<sup>&infin;</sup>}} function", so we may apply the [[Product rule|Leibniz rule]] to calculate its derivatives:
:<math>\frac{\partial u}{\partial \bar{z}} =\frac{\partial}{\partial \bar{z}}\left(\frac{1}{\pi\left(z-z_0\right)}\right) \chi_X+\frac{1}{\pi\left(z-z_0\right)} \frac{\partial}{\partial \bar{z}}\left(\chi_X\right)</math>
Using that {{math|(π''z'')<sup>−1</sup>}} is a [[fundamental solution]] of the [[Cauchy–Riemann operator]] {{math|{{sfrac|∂|∂''z̄''}}}}, we get <math>\frac{\partial}{\partial \bar{z}}\left(\frac{1}{\pi\left(z-z_0\right)}\right)=\delta_{z_0}</math>:
:<math>\frac{\partial u}{\partial \bar{z}}=\delta_{z_0}+\frac{1}{\pi\left(z-z_0\right)} \frac{\partial}{\partial \bar{z}}\left(\chi_X\right)
</math>
Applying <math>\frac{\partial u}{\partial \bar{z}}</math> to <math>\phi \in \mathcal{D}(X)</math>:
:<math>\begin{aligned}
\left\langle\frac{\partial}{\partial \bar{z}}\left(\frac{\chi_X}{\pi\left(z-z_0\right)}\right), \phi\right\rangle & =\phi\left(z_0\right)+\left\langle\frac{1}{\pi\left(z-z_0\right)} \frac{\partial}{\partial \bar{z}}\left(\chi_X\right), \phi\right\rangle \\
& =\phi\left(z_0\right)+\left\langle\frac{\partial}{\partial \bar{z}}\left(\chi_X\right), \frac{\phi}{\pi\left(z-z_0\right)}\right\rangle \\
& =\phi\left(z_0\right)+\frac{\mathrm{i}}{2} \int_{\partial X} \frac{\phi(z)}{\pi\left(z-z_0\right)} \mathrm{d} z
\end{aligned}</math>
where <math> \frac {\partial \chi_X}{\partial \bar z}= \frac{i}{2} \oint_{\partial X} \,dz</math> is used in the last line.

Rearranging, we get
:<math>\phi(z_0)={\frac {1}{2\pi i}}\int _{\partial X}{\frac {\phi(z)\,dz}{z-z_0 }}-{\frac {1}{\pi }}\iint _X{\frac {\partial \phi}{\partial {\bar {z}}}}(z){\frac {dx\wedge dy}{z-z_0 }}.</math>
as desired.
}}

===Several variables===
In [[Function of several complex variables|several complex variables]], the Cauchy integral formula can be generalized to [[polydisc]]s.<ref>{{harvnb|Hörmander|1966|loc=Theorem 2.2.1}}</ref>  Let {{math|''D''}} be the polydisc given as the [[Cartesian product]] of {{math|''n''}} open discs {{math|''D''<sub>1</sub>, ..., ''D''<sub>''n''</sub>}}:
<math display="block">D = \prod_{i=1}^n D_i.</math>

Suppose that {{math|''f''}} is a holomorphic function in {{math|''D''}} continuous on the closure of {{math|''D''}}.  Then
<math display="block">f(\zeta) = \frac{1}{\left(2\pi i\right)^n}\int\cdots\iint_{\partial D_1\times\cdots\times\partial D_n} \frac{f(z_1,\ldots,z_n)}{(z_1-\zeta_1)\cdots(z_n-\zeta_n)} \, dz_1\cdots dz_n</math>
where {{math|1=''ζ'' = (''ζ''<sub>1</sub>,...,''ζ''<sub>''n''</sub>) ∈ ''D''}}.

===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.