# An operator of a complex valued function

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.

• $Lf$ 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. – rolandcyp Apr 1 at 18:28
• @rolandcyp, great that is nice and simple, thanks! – dmtri Apr 2 at 2:48