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Suppose $f:\mathbb{C}\to\mathbb{C}$ is an analytic function and $f:=u+iv$. Then is it always true that $u_{xx}+u_{yy}=v_{xx}+v_{yy}=0$ ?

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    @lab $f_{xy}=f_{yx}$ when they are both continous. See here :http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Clairaut.27s_theorem2012-08-20

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Analytic function satisfies Cauchy Riemann equation.

i.e.

$u_{x}=v_{y}$ & $u_{y}=-v_{x}$

so, $u_{xx}=v_{xy}$ & $u_{yy}=-v_{yx}$

so, $u_{xx}+u_{yy}=0\ when\ v_{xy}=v_{yx}$