Editing Cauchy–Riemann equations (section)

==Generalizations==
=== Goursat's theorem and its generalizations ===
{{See also|Cauchy–Goursat theorem}}

Suppose that {{math|1=''f'' = ''u'' + i''v''}} is a complex-valued function which is [[Fréchet derivative|differentiable]] as a function <math>f : \mathbb{R}^2 \rarr \mathbb{R}^2</math>. Then [[Édouard Goursat|Goursat]]'s theorem asserts that ''f'' is analytic in an open complex domain Ω if and only if it satisfies the Cauchy–Riemann equation in the domain.{{sfn|Rudin|1966|loc=Theorem 11.2}} In particular, continuous differentiability of ''f'' need not be assumed.{{r|Dieudonné1969_para910Ex1}}

The hypotheses of Goursat's theorem can be weakened significantly. If {{math|1=''f'' = ''u'' + i''v''}} is continuous in an open set Ω and the [[partial derivative]]s of ''f'' with respect to ''x'' and ''y'' exist in Ω, and satisfy the Cauchy–Riemann equations throughout Ω, then ''f'' is holomorphic (and thus analytic). This result is the [[Looman–Menchoff theorem]].

The hypothesis that ''f'' obey the Cauchy–Riemann equations throughout the domain Ω is essential. It is possible to construct a continuous function satisfying the Cauchy–Riemann equations at a point, but which is not analytic at the point (e.g., {{math|1=''f''(''z'') = ''z''<sup>5</sup>/{{!}}z{{!}}<sup>4</sup>)}}. Similarly, some additional assumption is needed besides the Cauchy–Riemann equations (such as continuity), as the following example illustrates{{sfn|Looman|1923|p=107}}

<math display="block">f(z) = \begin{cases}
 \exp\left(-z^{-4}\right) & \text{if }z \not= 0\\
                        0 & \text{if }z = 0
\end{cases}</math>

which satisfies the Cauchy–Riemann equations everywhere, but fails to be continuous at ''z''&nbsp;=&nbsp;0.

Nevertheless, if a function satisfies the Cauchy–Riemann equations in an open set in a [[weak derivative|weak sense]], then the function is analytic. More precisely:{{sfn|Gray|Morris|1978|loc=Theorem 9}}
: If {{math|''f''(''z'')}} is locally integrable in an open domain <math>\Omega \isin \mathbb{C},</math> and satisfies the Cauchy–Riemann equations weakly, then {{mvar|f}} agrees [[almost everywhere]] with an analytic function in {{math|Ω}}.

This is in fact a special case of a more general result on the regularity of solutions of [[hypoelliptic operator|hypoelliptic]] partial differential equations.

===Several variables===
There are Cauchy–Riemann equations, appropriately generalized, in the theory of [[Function of several complex variables|several complex variables]]. They form a significant [[overdetermined system]] of PDEs. This is done using a straightforward generalization of the [[Wirtinger derivative]], where the function in question is required to have the (partial) Wirtinger derivative with respect to each complex variable vanish.

===Complex differential forms===
As often formulated, the ''d-bar operator''
<math display="block">\bar{\partial}</math>
annihilates holomorphic functions. This generalizes most directly the formulation
<math display="block">{\partial f \over \partial \bar z} = 0,</math>
where
<math display="block">{\partial f \over \partial \bar z} = {1 \over 2}\left({\partial f \over \partial x} + i{\partial f \over \partial y}\right).</math>

=== Bäcklund transform ===
Viewed as [[conjugate harmonic functions]], the Cauchy–Riemann equations are a simple example of a [[Bäcklund transform]]. More complicated, generally non-linear Bäcklund transforms, such as in the [[sine-Gordon equation]], are of great interest in the theory of [[soliton]]s and [[integrable system]]s.

=== Definition in Clifford algebra ===
In the [[Clifford algebra]] <math>C\ell(2)</math>, the complex number <math>z = x+iy </math> is represented as <math>z \equiv x + J y</math> where <math>J \equiv \sigma_1 \sigma_2</math>, (<math>\sigma_1^2=\sigma_2^2=1, \sigma_1 \sigma_2 + \sigma_2 \sigma_1 = 0</math>, so <math>J^2=-1</math>). The [[Dirac operator]] in this Clifford algebra is defined as <math>\nabla \equiv \sigma_1 \partial_x + \sigma_2\partial_y</math>. The function <math>f=u + J v</math> is considered analytic if and only if <math>\nabla f = 0</math>, which can be calculated in the following way:

<math display="block">
\begin{align}
0 & =\nabla f= ( \sigma_1 \partial_x + \sigma_2 \partial_y )(u + \sigma_1 \sigma_2 v) \\[4pt]
& =\sigma_1 \partial_x u + \underbrace{\sigma_1 \sigma_1 \sigma_2}_{=\sigma_2} \partial_x v + \sigma_2 \partial_y u + \underbrace{\sigma_2 \sigma_1 \sigma_2}_{=-\sigma_1} \partial_y v =0
\end{align}
</math>

Grouping by <math>\sigma_1</math> and <math>\sigma_2</math>:

<math display="block">\nabla f = \sigma_1 ( \partial_x u - \partial_y v) + \sigma_2 ( \partial_x v + \partial_y u) = 0 \Leftrightarrow \begin{cases}
\partial_x u - \partial_y v = 0\\[4pt]
\partial_x v + \partial_y u = 0
\end{cases}</math>

Hence, in traditional notation:

<math display="block">\begin{cases}
\dfrac{ \partial u }{ \partial x } = \dfrac{ \partial v }{ \partial y }\\[12pt]
\dfrac{ \partial u }{ \partial y } = -\dfrac{ \partial v }{ \partial x }
\end{cases}</math>

=== Conformal mappings in higher dimensions ===
Let Ω be an open set in the Euclidean space <math>\mathbb{R}^n</math>. The equation for an orientation-preserving mapping <math>f:\Omega\to\mathbb{R}^n</math> to be a [[conformal mapping]] (that is, angle-preserving) is that
<math display="block">Df^\mathsf{T} Df = (\det(Df))^{2/n}I</math>

where ''Df'' is the Jacobian matrix, with transpose <math>Df^\mathsf{T}</math>, and ''I'' denotes the identity matrix.{{r|IwaniecMartin2001_32}} For {{math|1=''n'' = 2}}, this system is equivalent to the standard Cauchy–Riemann equations of complex variables, and the solutions are holomorphic functions. In dimension {{math|''n'' > 2}}, this is still sometimes called the Cauchy–Riemann system, and [[Liouville's theorem (conformal mappings)|Liouville's theorem]] implies, under suitable smoothness assumptions, that any such mapping is a [[Möbius transformation]].

=== Lie pseudogroups ===
{{See also|Pseudogroup}}

One might seek to generalize the Cauchy-Riemann equations instead by asking more generally when are the solutions of a system of PDEs closed under composition. The theory of Lie Pseudogroups addresses these kinds of questions.