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I am really new to this area and a selflearner and sadly enough I cannot understand the following: Suppose $f:\Bbb{R^2}\to \Bbb{C}$, $f(x,y)=u(x,y)+iv(x,y)$ is a complex valued function of $2$ real variables and $L$ the linear operator $L=\nabla\cdot p\nabla +q$ , then what is the $Lf$ ?

For the case where $f$ takes real values, I can find that $Lf=\nabla p\cdot \nabla f + p\Delta f +qf$ , but in the case where $f$ is complex valued I do not have a clue...Any help is appreciated.

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    $\begingroup$ $Lf$ will be given by $Lu + iLv$ at all $(x,y)$ in the domain. This is because differentiation and pointwise multiplication are linear even for complex valued functions. $\endgroup$ – rolandcyp Apr 1 at 18:28
  • $\begingroup$ @rolandcyp, great that is nice and simple, thanks! $\endgroup$ – dmtri Apr 2 at 2:48

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