f(x,y)=ln(4x^2+4y^2) show that the function is a harmonic series.
Do you mean to show that the function is harmonic?
To do so, show that $\displaystyle f\!\left(x,y\right)$ satisfies Laplace's Equation $\displaystyle \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}=0$