Problem 1545 (difficulty: 5/10)

Let \(\displaystyle D\subset\RR^2\) be an open domain and \(\displaystyle u,v:D\to\RR^2\) twice differentiable for which the map \(\displaystyle x+yi\mapsto u(x,y)+iv(x,y)\) is regular on \(\displaystyle D\). Show that

\(\displaystyle \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0. \)

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