Question: If harmonic functions $u$ and $v$ satisfy Cauchy-Riemann equations, then $u+iv$ is an analytic function.
Am a bit confused here as we already have a theorem which says that if a function satisfies Cauchy Riemann equations for each point in the domain and each of its four partial derivatives $u_x, v_x, u_y, v_y$ are continuous then it is analytic in that domain.
So can we say that if $u$ and $v$ are harmonic then each of its four partial derivatives $u_x, v_x, u_y, v_y$ are continuous thereby implying that the function is analytic in $D.$
Best Answer
There are harmonic functions that do not satisfy the Cauchy-reimann equations. (x+y)+i(x-y) is an example.