Showing that (x+iy)/r is an eigenfunction of the angular momentum operator

Edgarngg

1. The problem statement, all variables and given/known data
I know that,if (operator)(function)=(value)(samefunction)
that function is said to be eigenfunction of the operator.
in this case i need to show this function to be eigenfunction of the Lz angular momentum:


2. Relevant equations
function:
ψ=(x+iy)/r
operator:
Lz= (h bar)/i (x [itex]\partial[/itex]/[itex]\partial[/itex]y - y [itex]\partial[/itex]/[itex]\partial[/itex]x)


3. The attempt at a solution
My question is how do i treat "r", do i have to change to polar coordinates? or is it possible to do it like this.
i know that i have to apply the operator over the function, and that is (h bar/i) (x(partial derivative for y)- y(partial der for x)) and then see if i get the same function multiplied by an eigenvalue.
the problem i have is that i dont know how to treat that function, since i see an "r" there. So when d/dx "r" and "y" would be constant, and that doesnt make sense to me.
Thank you very much

Dick

r=sqrt(x^2+y^2), isn't it?