The Cauchy-Riemann Equations are useful for checking differentiability in the complex plane, and computing them.

Suppose

$$\begin{aligned} f(z) = u(x,y) + i \cdot v(x,y) \end{aligned}$$

is differentiable at $z_0$. Then

1. the partial derivatives $u_x, u_y, v_x, v_y$ exist at $z_0$,
2. $u_x = v_y$, and
3. $u_y = -v_x$.

Conversely, suppose

$$\begin{aligned} f = u + i \cdot v \end{aligned}$$

is defined on a domain $D \in \mathbb{C}$. Then $f$ is analytic in $D$ iff $u(x,y)$ and $v(x,y)$ have continuous first partial derivatives on $D$ that satisfies the Cauchy-Riemann equations. The key point is continuity is necessary for the guarantee to work.

More info at Analysis of a Complex Kind.

# posted by rot13(Unafba Pune) @ 10:48 PM