How to test whether the function v(x,y)=e^x(xcosy-ysiny) is harmonic

If it is,determine its conjugate harmonic function such that f(z)=u+jv analytic.

Thanks for the help!!

∂v/∂x = e^x (x cos y - y sin y) + e^x (cos y).

∂²v/∂x² = [e^x (x cos y - y sin y) + e^x cos y] + e^x cos y.
..........= e^x (x cos y - y sin y) + 2e^x cos y

∂v/∂y = e^x (-x sin y - (sin y + y cos y))

∂²v/∂y² = e^x (-x cos y - (cos y + (cos y - y sin y)))
..........= e^x (-x cos y - 2 cos y + y sin y).

Since ∂²v/∂x² + ∂²v/∂y² ≠ 0, v is not harmonic.

I hope this helps!

Put the wabbajack in the glory hole.