Cauchy–Riemann criterion

Statement

Let $U\subseteq\mathbb C$ be open and write $f=u+iv:U\to\mathbb C$ . If $u$ and $v$ are $C^1$ and satisfy the Cauchy–Riemann equations throughout $U$ , then $f$ is holomorphic. Conversely, a holomorphic function satisfies those equations; indeed, holomorphic functions are smooth.

Pointwise form

At a single point, real differentiability of $f$ together with the Cauchy–Riemann equations is equivalent to existence of the complex derivative at that point. Continuity of the partial derivatives near the point is a convenient sufficient condition for real differentiability.

Indeed, if $Df(a)$ is real-linear, the complex difference quotient can have a limit only when $Df(a)(h)=ch$ for one complex number $c$ . This complex-linearity condition is exactly $u_x=v_y$ and $u_y=-v_x$ ; under the $C^1$ hypothesis the first-order remainder is $o(|h|)$ , so the quotient tends to $c=u_x+iv_x$ .

Regularity warning

Merely having partial derivatives that satisfy the equations pointwise is not, by itself, a safe holomorphicity criterion: existence of partial derivatives need not imply real differentiability. Weaker hypotheses are available in distributional and Sobolev formulations, but they are separate regularity theorems rather than the elementary $C^1$ criterion.

References

Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979. Relevant: Chapter 2, §2.