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.