How to show the real part of a holomorphic function is harmonic So I know that for a real valued function $u$, it's harmonic if it's continuously differentiable and it satisfies $$u_{xx}+u_{yy}=0$$
How do I show that for a generic holomorphic function $f$, that the real part $Re(f)(z)$ is harmonic?

Let $f(z) = f(x + iy) = u(x, y) + iv(x, y)$. Then, if $f$ is holomorphic, it satisfies the Cauchy-Riemann equations:
$$\begin{cases}
u_x = v_y\\
u_y = -v_x
\end{cases}$$
Hence it follows that
$$\begin{cases}
u_{xx} = v_{yx}\\
u_{yy} = -v_{xy}
\end{cases}$$
Now, it is well known that holomorphic functions are infinitely differentiable (this is usually proved after introducing complex integrals), and thus $u, v \in \mathcal C^2$. From Schwarz's theorem we have that $v_{yx} = v_{xy}$ and so we can conclude
$$u_{xx} + u_{yy} = v_{yx} - v_{xy} = 0.$$
The same can be said about $v(x, y)$.
