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When is a function satisfying the Cauchy-Riemann equations holomorphic?
If real the functions $u(x,y)$ and $v(x,y)$ satisfy the Cauchy-Riemann equations and have continuous partial derivatives in an open set $U$, then the function $f(z)=u(x,y)+iv(x,y)$, where $z=x+iy$, is analytic in $U$. Are there less restrictive conditions on $u$ and $v$ to ensure the analyticity of $f$? Thanks.
There actually are none. One of the more beautiful theorems of Complex Analysis is that a function is holomorphic if and only if it is continuous and satisfies the Cauchy-Riemann equations.