Editing Cauchy's integral formula

{{short description|Provides integral formulas for all derivatives of a holomorphic function}}
{{Distinguish|Cauchy's integral theorem|Cauchy formula for repeated integration}}
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In mathematics, '''Cauchy's integral formula''', named after [[Augustin-Louis Cauchy]], is a central statement in [[complex analysis]]. It expresses the fact that a [[holomorphic function]] defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under [[uniform convergence|uniform limits]] – a result that does not hold in [[real analysis]].

==Theorem==
Let {{math|''U''}} be an [[open subset]] of the [[complex plane]] {{math|'''C'''}}, and suppose the closed disk {{math|''D''}} defined as
<math display="block">D = \bigl\{z:|z - z_0| \leq r\bigr\}</math>
is completely contained in {{math|''U''}}.  Let {{math|''f'' : ''U'' → '''C'''}} be a [[holomorphic function]], and let {{math|''γ''}} be the [[circle]], oriented [[Curve orientation|counterclockwise]], forming the [[boundary (topology)|boundary]] of {{math|''D''}}. Then for every {{math|''a''}} in the [[interior (topology)|interior]] of {{math|''D''}},
<math display="block">f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,dz.\,</math>

The proof of this statement uses the [[Cauchy integral theorem]] and like that theorem, it only requires {{math|''f''}} to be [[complex differentiable]].  Since <math>1/(z-a)</math> can be expanded as a [[power series]] in the variable <math>a</math>
<math display="block">\frac{1}{z-a} = \frac{1+\frac{a}{z}+\left(\frac{a}{z}\right)^2+\cdots}{z}</math>
it follows that [[holomorphic functions are analytic]], i.e. they can be expanded as convergent power series.
In particular {{math|''f''}} is actually infinitely differentiable, with
<math display="block">f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\,dz.</math>

This formula is sometimes referred to as '''Cauchy's differentiation formula'''.

The theorem stated above can be generalized. The circle {{math|''γ''}} can be replaced by any closed [[rectifiable curve]] in {{math|''U''}} which has [[winding number]] one about {{math|''a''}}. Moreover, as for the Cauchy integral theorem, it is sufficient to require that {{math|''f''}} be holomorphic in the open region enclosed by the path and continuous on its [[closure (topology)|closure]].

Note that not every continuous function on the boundary can be used to produce a function inside the boundary that fits the given boundary function. For instance, if we put the function  {{math|''f''(''z'') {{=}} {{sfrac|1|''z''}}}}, defined for {{math|1={{abs|''z''}} = 1}}, into the Cauchy integral formula, we get zero for all points inside the circle. In fact, giving just the real part on the boundary of a holomorphic function is enough to determine the function [[up to]] an imaginary constant — there is only one imaginary part on the boundary that corresponds to the given real part, up to addition of a constant. We can use a combination of a [[Möbius transformation]] and the [[Stieltjes_transformation|Stieltjes inversion formula]] to construct the holomorphic function from the real part on the boundary. For example, the function {{math|1=''f''(''z'') = ''i'' − ''iz''}} has real part {{math|1=Re ''f''(''z'') = Im ''z''}}. On the unit circle this can be written {{math|{{sfrac|{{sfrac|''i''|''z''}} − ''iz''|2}}}}. Using the Möbius transformation and the Stieltjes formula we construct the function inside the circle. The {{math|{{sfrac|''i''|''z''}}}} term makes no contribution, and we find the function {{math|−''iz''}}. This has the correct real part on the boundary, and also gives us the corresponding imaginary part, but off by a constant, namely {{math|''i''}}.

== Proof sketch ==

By using the [[Cauchy integral theorem]], one can show that the integral over {{math|''C''}} (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around {{math|''a''}}. Since {{math|''f''(''z'')}} is continuous, we can choose a circle small enough on which {{math|''f''(''z'')}} is arbitrarily close to {{math|''f''(''a'')}}. On the other hand, the integral
<math display="block">\oint_C  \frac{1}{z-a} \,dz = 2 \pi i,</math>
over any circle {{math|''C''}} centered at {{math|''a''}}.  This can be calculated directly via a parametrization ([[integration by substitution]]) {{math|''z''(''t'') {{=}} ''a'' + ''εe<sup>it</sup>''}} where {{math|0 ≤ ''t'' ≤ 2π}} and {{math|''ε''}} is the radius of the circle.

Letting {{math|''ε'' → 0}} gives the desired estimate
<math display="block">\begin{align}
\left | \frac{1}{2 \pi i} \oint_C \frac{f(z)}{z-a} \,dz  - f(a) \right |
&= \left | \frac{1}{2 \pi i} \oint_C \frac{f(z)-f(a)}{z-a} \,dz \right | \\[1ex]
&= \left | \frac{1}{2\pi i}\int_0^{2\pi}\left(\frac{f\bigl(z(t)\bigr)-f(a)}{\varepsilon e^{it}}\cdot\varepsilon e^{it} i\right )\,dt\right | \\[1ex]
&\leq \frac{1}{2 \pi} \int_0^{2\pi} \frac{ \left|f\bigl(z(t)\bigr) - f(a)\right| } {\varepsilon} \,\varepsilon\,dt \\[1ex]
&\leq \max_{|z-a|=\varepsilon} \left|f(z) - f(a)\right|
~~ \xrightarrow[\varepsilon\to 0]{} ~~ 0.
\end{align}</math>

== Example ==
[[File:ComplexResiduesExample.png|thumb|300px|Surface of the real part of the function {{math|1=''g''(''z'') = {{sfrac|''z''<sup>2</sup>|''z''<sup>2</sup> + 2''z'' + 2}}}} and its singularities, with the contours described in the text.]]
Let
<math display="block">g(z) = \frac{z^2}{z^2+2z+2},</math>
and let {{math|''C''}} be the contour described by {{math|1={{abs|''z''}} = 2}} (the circle of radius 2).

To find the integral of {{math|''g''(''z'')}} around the contour {{math|''C''}}, we need to know the singularities of {{math|''g''(''z'')}}. Observe that we can rewrite {{math|''g''}} as follows:
<math display="block">g(z) = \frac{z^2}{(z-z_1)(z-z_2)}</math>
where {{math|1=''z''<sub>1</sub> = − 1 + ''i''}} and {{math|1=''z''<sub>2</sub> = − 1 − ''i''}}.

Thus, {{math|''g''}} has poles at {{math|''z''<sub>1</sub>}} and {{math|''z''<sub>2</sub>}}. The [[absolute value|moduli]] of these points are less than 2 and thus lie inside the contour. This integral can be split into two smaller integrals by [[Cauchy–Goursat theorem]]; that is, we can express the integral around the contour as the sum of the integral around {{math|''z''<sub>1</sub>}} and {{math|''z''<sub>2</sub>}} where the contour is a small circle around each pole<!-- diagram works best -->. Call these contours {{math|''C''<sub>1</sub>}} around {{math|''z''<sub>1</sub>}} and {{math|''C''<sub>2</sub>}} around {{math|''z''<sub>2</sub>}}.

Now, each of these smaller integrals can be evaluated by the Cauchy integral formula, but they first must be rewritten to apply the theorem. For the integral around {{math|''C''<sub>1</sub>}}, define {{math|''f''<sub>1</sub>}} as {{math|1=''f''<sub>1</sub>(''z'') = (''z'' − ''z''<sub>1</sub>)''g''(''z'')}}. This is [[holomorphic function|analytic]] (since the contour does not contain the other singularity). We can simplify {{math|''f''<sub>1</sub>}} to be:
<math display="block">f_1(z) = \frac{z^2}{z-z_2}</math>
and now
<math display="block">g(z) = \frac{f_1(z)}{z-z_1}.</math>

Since the Cauchy integral formula says that:
<math display="block">\oint_C \frac{f_1(z)}{z-a}\, dz=2\pi i\cdot f_1(a),</math>
we can evaluate the integral as follows:
<math display="block">
  \oint_{C_1} g(z)\,dz
 =\oint_{C_1} \frac{f_1(z)}{z-z_1}\,dz
 =2\pi i\frac{z_1^2}{z_1-z_2}.
</math>

Doing likewise for the other contour:
<math display="block">f_2(z) = \frac{z^2}{z-z_1},</math>
we evaluate
<math display="block">
  \oint_{C_2} g(z)\,dz
 =\oint_{C_2} \frac{f_2(z)}{z-z_2}\,dz
 =2\pi i\frac{z_2^2}{z_2-z_1}.
</math>

The integral around the original contour {{math|''C''}} then is the sum of these two integrals:
<math display="block">\begin{align}
     \oint_C g(z)\,dz
&{}= \oint_{C_1} g(z)\,dz
   + \oint_{C_2} g(z)\,dz \\[.5em]
&{}= 2\pi i\left(\frac{z_1^2}{z_1-z_2}+\frac{z_2^2}{z_2-z_1}\right) \\[.5em]
&{}= 2\pi i(-2) \\[.3em]
&{}=-4\pi i.
\end{align}</math>

An elementary trick using [[partial fraction decomposition]]:
<math display="block">
  \oint_C g(z)\,dz
 =\oint_C \left(1-\frac{1}{z-z_1}-\frac{1}{z-z_2}\right) \, dz
 =0-2\pi i-2\pi i
 =-4\pi i
</math>

==Consequences==
The integral formula has broad applications.  First, it implies that a function which is holomorphic in an open set is in fact [[infinitely differentiable]] there.  Furthermore, it is an [[analytic function]], meaning that it can be represented as a [[power series]].  The proof of this uses the [[dominated convergence theorem]] and the [[geometric series]] applied to

<math display="block">f(\zeta) = \frac{1}{2\pi i}\int_C \frac{f(z)}{z-\zeta}\,dz.</math>

The formula is also used to prove the [[residue theorem]], which is a result for [[meromorphic function]]s, and a related result, the [[argument principle]].  It is known from [[Morera's theorem]] that the uniform limit of holomorphic functions is holomorphic. This can also be deduced from Cauchy's integral formula: indeed the formula also holds in the limit and the integrand, and hence the integral, can be expanded as a power series. In addition the Cauchy formulas for the higher order derivatives show that all these derivatives also converge uniformly.

The analog of the Cauchy integral formula in real analysis is the [[Poisson integral formula]] for [[harmonic function]]s; many of the results for holomorphic functions carry over to this setting. No such results, however, are valid for more general classes of differentiable or real analytic functions.  For instance, the existence of the first derivative of a real function need not imply the existence of higher order derivatives, nor in particular the analyticity of the function.  Likewise, the uniform limit of a sequence of (real) differentiable functions may fail to be differentiable, or may be differentiable but with a derivative  which is not the limit of the derivatives of the members of the sequence.

Another consequence is that if {{math|1=''f''(''z'') = Σ ''a''<sub>''n''</sub> ''z''<sup>''n''</sup> }} is holomorphic in {{math|{{abs|''z''}} < ''R''}} and {{math|0 < ''r'' < ''R''}} then the coefficients {{math|''a''<sub>''n''</sub>}} satisfy '''[[Cauchy's estimate]]'''<ref>{{harvnb|Titchmarsh|1939|p=84}}</ref>
<math display="block">|a_n|\le r^{-n} \sup_{|z|=r}|f(z)|.</math>

From Cauchy's estimate, one can easily deduce that every bounded entire function must be constant (which is [[Liouville's theorem (complex analysis)|Liouville's theorem]]).

The formula can also be used to derive '''Gauss's Mean-Value Theorem''', which states<ref>{{WolframAlpha |title=Gauss's Mean-Value Theorem |id=GausssMean-ValueTheorem}}</ref>
<math display="block">f(z) = \frac{1}{2\pi} \int_{0}^{2\pi} f(z + r e^{i\theta}) \, d\theta.</math>

In other words, the average value of {{math|''f''}} over the circle centered at {{math|''z''}} with radius {{math|''r''}} is {{math|''f''(''z'')}}. This can be calculated directly via a parametrization of the circle.

==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.

==See also==
{{div col|colwidth=20em}}
*[[Cauchy–Riemann equations]]
*[[Methods of contour integration]]
*[[Nachbin's theorem]]
*[[Morera's theorem]]
*[[Mittag-Leffler's theorem]]
*[[Green's function]] generalizes this idea to the non-linear setup
*[[Schwarz integral formula]]
*[[Parseval–Gutzmer formula]]
*[[Bochner–Martinelli formula]]
*[[Helffer–Sjöstrand formula]]
{{div col end}}

==Notes==
{{reflist}}

==References==
{{refbegin}}
* {{cite book|first=Lars|last=Ahlfors|author-link=Lars Ahlfors|title=Complex analysis|publisher=McGraw Hill|edition=3rd|year=1979|isbn=978-0-07-000657-7}}
* {{cite journal|url=http://archive.numdam.org/ARCHIVE/AFST/AFST_1905_2_7_3/AFST_1905_2_7_3_265_0/AFST_1905_2_7_3_265_0.pdf|first=D.|last=Pompeiu|title=Sur la continuité des fonctions de variables complexes|journal=Annales de la Faculté des Sciences de Toulouse |series=Série 2|volume=7| issue=3|date=1905 | pages=265–315}}
*{{cite book|first=E. C.|last=Titchmarsh|author-link=Edward Charles Titchmarsh|title=Theory of functions|url=https://archive.org/details/in.ernet.dli.2015.2588 | publisher=[[Oxford University Press]]|year=1939|edition=2nd}}
* {{cite book|first=Lars|last=Hörmander|author-link=Lars Hörmander|title=An Introduction to Complex Analysis in Several Variables|publisher=Van Nostrand | year=1966}}
* {{cite book|first=Lars|last=Hörmander|author-link=Lars Hörmander|title=The Analysis of Linear Partial Differential Operators I|year=1983|publisher=Springer| isbn=3-540-12104-8}}
* {{cite book|last1=Doran|first1=Chris |last2=Lasenby |first2=Anthony |title=Geometric Algebra for Physicists | publisher=Cambridge University Press|year=2003 |isbn=978-0-521-71595-9 }}
{{refend}}

== External links ==
* {{springer|title=Cauchy integral|id=p/c020890}}
* {{MathWorld | urlname= CauchyIntegralFormula | title= Cauchy Integral Formula }}

{{DEFAULTSORT:Cauchy's Integral Formula}}
[[Category:Augustin-Louis Cauchy]]
[[Category:Theorems in complex analysis]]