Question

Show that U= sin x cosh y +2 cosh x sin y +x² – y² +4xy is a harmonic function

Answer 1

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.