General solution of laplace equation

Question

I want to know that: What is the general solution of the (2 and 3 dimentional)Laplace equation $f_{xx}+ f_{yy}=0$ and $f_{xx}+ f_{yy} +f_{zz}=0$? With many thanks for your help.

Answer 1

For $f_{xx}+f_{yy}=0$ , the general solution is $f(x,y)=A(x+iy)+B(x-iy)$ .

For $f_{xx}+f_{yy}+f_{zz}=0$ , the general solution is related to the factorized form of $x^2+y^2+z^2$ in $\mathbb{C}$ .