u(x,y)=x2+y2∂u∂x=2x,∂2u∂x2=2∂u∂y=2y,∂2u∂y2=2Since ∂2u∂x2+∂2u∂y2=4 ,u is not harmonic.u is not regular and has singularitiesat every point except at x=y=0.\displaystyle u(x, y) = x^2 + y^2\\ \frac{\partial u}{\partial x} = 2x, \frac{\partial^2 u}{\partial x^2} = 2\\ \frac{\partial u}{\partial y} = 2y, \frac{\partial^2 u}{\partial y^2} = 2\\ \textsf{Since}\, \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 4\,,u\, \textsf{is not harmonic.}\\ u\, \textsf{is not regular and has singularities}\\ \textsf{at every point except at}\,x = y = 0.u(x,y)=x2+y2∂x∂u=2x,∂x2∂2u=2∂y∂u=2y,∂y2∂2u=2Since∂x2∂2u+∂y2∂2u=4,uis not harmonic.uis not regular and has singularitiesat every point except atx=y=0.
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