6:21 PM
Theorem in Conway's book on complex analysis: Let $u$ and $v$ be real-valued functions defined on a region $G$ and suppose that $u$ and $v$ have continuous partial derivatives. Then $f : G \to \Bbb{C}$ defined by $f(z) = u(z) +iv(z)$ is analytic if and only if $u$ and $v$ satisfy the Cauchy-Riemann equations.
For the backwards direction, he writes $u(x+s,y+t) - u(x,y) = [u(x+s,y+t)-u(x,y+t)] + [u(x,y+t)-u(x,y)]$
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