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