Solution: Let \[ U = \frac{\partial u}{\partial y} - \frac{\partial v}{\partial x} = u_y - v_x \text{ and } V = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = u_x + v_y. \] Consider the function $f = U(x,y) + \iota V(x,y)$. Then we have \begin{align*} U_x & = u_{yx} - v_{xx}, \kern 0.4cm U_y = u_{yy} - v_{xy} \\ V_x & = u_{xx} + v_{yx}, \kern 0.4cm V_y = u_{xy} + v_{yy}. \end{align*} Since $u$ and $v$ are harmonic, they both have continuous derivatives up to the second order. Therefore, the first derivatives of $U$ and $V$ are continuous. Also, $u_{xx} + u_{yy} = 0 $ and $v_{xx} + v_{yy} = 0$ implies that $U_x = V_y$ and $U_y = -V_x$. Therefore, $f$ satisfies the Cauchy-Riemann equation. Thus, $f$ is analytic in $\mathcal{R} $.