U=sin(x)cosh(y)+2cosh(x)sin(y)+x²-y²+4xy.
We need to prove:
∂²U/∂x²+∂²U/∂y²=0, if U is a harmonic function (Laplace).
∂U/∂x=cos(x)cosh(y)+2sinh(x)sin(y)+2x+4y,
∂U/∂y=sin(x)sinh(y)+2cosh(x)cos(y)-2y+4x,
∂²U/∂x²=-sin(x)cosh(y)+2cosh(x)sin(y)+2,
∂²U/∂y²=sin(x)cosh(y)-2cosh(x)sin(y)-2.
If we add the previous two equations, we get zero, proving that U is a harmonic function.