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Suppose u(x,y) and v(x,y) are harmonic on G and c is an element of R. Prove u(x,y) + cv(x,y) is also harmonic.
I tried using the Laplace Equation of Uxx+Uyy=0
I have:
du/dx=Ux
d^2u/dx^2=Uxx
du/dy=Uy
d^2u/dy^2=Uyy
dv/dx=cVx
d^2v/dx^2=cVxx
dv/dy=cVy
d^2v/dy^2=cVyy
I'm not really sure how to prove these are harmonic...am I missing a relationship?
I tried using the Laplace Equation of Uxx+Uyy=0
I have:
du/dx=Ux
d^2u/dx^2=Uxx
du/dy=Uy
d^2u/dy^2=Uyy
dv/dx=cVx
d^2v/dx^2=cVxx
dv/dy=cVy
d^2v/dy^2=cVyy
I'm not really sure how to prove these are harmonic...am I missing a relationship?