Editing Cauchy–Riemann equations (section)

=== 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.