$$\frac{\partial u}{\partial x} = cosx coshy - 2sinx sinh y + 2x + 4y$$ $$\frac{\partial^2 u}{\partial x^2} = -sinx coshy - 2cosx sinh y + 2$$ $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = -sinx* coshy - 2cosxsinhy + 2+ sinx * coshy + 2cosx * sinh y - 2$$ $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$

$\therefore$ function u satisfies Laplace equation.

$f'(z) = ux - iuy \\ = -2sinx sinh y + 2x + 4y + cosx coshy - i(sinx sinhy + 2cosx cosh y - 2y + 4x)$

$= 2z + cosz - i(2cosz + 4z)$ [x = z, y = 0, Milne Thompson]

$\therefore f(z) = z^2 + sinz - i(2sinz + 2z^2)$