Question

Is product of 2 harmonic fuction is also harmonic fuction?

Answer 1

The easy way to do this is to recognize the expression.

Let f(z)=f(x+iy)=u(x,y)+iv(x,y)f(z)=f(x+iy)=u(x,y)+iv(x,y) and define
g(z)=z3+12z2=(x+iy)3+12(x+iy)2=x3−3xy2+12(x2−y2)+i(3xy2−y3+xy).
g(z)=z3+12z2=(x+iy)3+12(x+iy)2=x3−3xy2+12(x2−y2)+i(3xy2−y3+xy).
In other words, Img(f(z))=3uv2−v3+uvIm⁡g(f(z))=3uv2−v3+uv is the imaginary part of an analytic function, hence harmonic.

The difficult way is to compute the Laplacian and use Cauchy-Riemann's equations to simplify the mess.