[proofplan] Since $f$ is holomorphic, it is infinitely [differentiable](/page/Derivative) by the [Holomorphicity of Power Series](/theorems/335) theorem (applied to the local [power series](/page/Power%20Series) representation from Cauchy's [integral](/page/Integral) formula). The [Cauchy--Riemann equations](/theorems/333) give $u_x = v_y$ and $u_y = -v_x$. Differentiating these once more and using the [Symmetry of Second Derivatives](/theorems/332) ($v_{xy} = v_{yx}$) shows $u_{xx} + u_{yy} = 0$. The same argument with the roles of $u$ and $v$ swapped gives $v_{xx} + v_{yy} = 0$. [/proofplan] [step:Establish that $u$ and $v$ have continuous second partial derivatives] Since $f$ is holomorphic on $U$, it admits a local [power series](/page/Power%20Series) representation at each point (by Cauchy's integral formula). By the [Holomorphicity of Power Series](/theorems/335) theorem, $f$ is infinitely [differentiable](/page/Derivative) on $U$. In particular, the real and imaginary parts $u$ and $v$ have [continuous](/page/Continuity) partial derivatives of all orders on $U$. [/step] [step:Differentiate the Cauchy--Riemann equations to show $u$ is harmonic] The [Cauchy--Riemann equations](/theorems/333) give $u_x = v_y$ and $u_y = -v_x$. Differentiating the first with respect to $x$ and the second with respect to $y$: \begin{align*} u_{xx} = v_{yx}, \qquad u_{yy} = -v_{xy}. \end{align*} Since $v$ has [continuous](/page/Continuity) second partial [derivatives](/page/Derivative), the [Symmetry of Second Derivatives](/theorems/332) gives $v_{xy} = v_{yx}$. Adding: \begin{align*} u_{xx} + u_{yy} = v_{yx} - v_{xy} = 0. \end{align*} Therefore $u$ is harmonic. [/step] [step:Show $v$ is harmonic by the same argument with roles swapped] Differentiate $u_x = v_y$ with respect to $y$ and $u_y = -v_x$ with respect to $x$: \begin{align*} u_{xy} = v_{yy}, \qquad u_{yx} = -v_{xx}. \end{align*} By the [Symmetry of Second Derivatives](/theorems/332), $u_{xy} = u_{yx}$, so $v_{yy} = -v_{xx}$, giving $v_{xx} + v_{yy} = 0$. Therefore $v$ is harmonic. [/step]