= Solution
Suppose first that $u$ is <harmonic function>[harmonic]. Since
$$
\frac{\partial}{\partial y}u_x
=\frac{\partial}{\partial x}u_y
\quad\text{and}\quad
\frac{\partial}{\partial x}u_x
=-\frac{\partial}{\partial y}u_y,
$$
the <Cauchy-Riemann equations> show that
$$
g:=u_x-iu_y
$$
is <holomorphic function>[holomorphic] on $D$. An open disc is a <simply connected domain>, so $g$ has a <primitive of a holomorphic function on a simply connected domain>[holomorphic primitive] $f$. If $f=p+iq$, then
$$
p_x=\operatorname{Re}f'=u_x,
\qquad
p_y=-\operatorname{Im}f'=u_y.
$$
Thus $p-u$ has zero <gradient> and is constant on the connected disc. Subtracting that real constant from $f$ gives
$$
\boxed{u=\operatorname{Re}f}.
$$
Conversely, if $u=\operatorname{Re}f$ for a holomorphic $f=u+iv$, differentiating the Cauchy-Riemann equations gives
$$
\Delta u=u_{xx}+u_{yy}=v_{yx}-v_{xy}=0.
$$
\b[Hence $u$ is harmonic. This proves the <harmonic function as the real part of a holomorphic function> criterion on $D$.]
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