Definition of Harmonic Conjugates This definition of Harmonic Conjugates is taken from Complex Variables and Applications by James Ward Brown, Ruel V. Churchill


That is due to the above text $u,v:D(\subset\mathbb R^2)\to\mathbb R,D$ being a domain, are said to be the harmonic conjugates of each other if (1) $u,v$ have continuous partials of 1st and 2nd order in $D,$ (2) $u,v$ satisfy the Laplace equations in $D,$ (3) $u,v$ satisfy the C-R equations in $D.$

But I think condition (2) is an overstatement. Here's my logic. Please comment on my thoughts:
Due to the sufficient condition of differentiability, (1) and (3) implies the analycity of the function $u+iv$ in $D.$ As a result $u,v$ must satisfy the Laplace equations in $D.$
 A: You are correct. In fact, if
$$
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\tag{1}
$$
and
$$
\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}\tag{2}
$$
then taking the partial of $(1)$ with respect to $x$, we get
$$
\frac{\partial^2u}{\partial x^2}=\frac{\partial^2v}{\partial x\partial y}\tag{3}
$$
and taking the partial of $(2)$ with respect to $y$, we get
$$
\frac{\partial^2v}{\partial y\partial x}=-\frac{\partial^2u}{\partial y^2}\tag{4}
$$
Subtracting $(4)$ from $(3)$ and using the equality of mixed partials yields
$$
\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0\tag{5}
$$
Similarly for $v$, by taking the partial of $(1)$ with respect to $y$ and adding the partial of $(2)$ with respect to $x$ yields.
$$
\frac{\partial^2v}{\partial x^2}+\frac{\partial^2v}{\partial y^2}=0\tag{6}
$$
A: The relationship of harmonic conjugacy is not symmetric though. If $f(z)=u+iv$ is analytic then $v$ is called harmonic conjugate of $u$. However, you can see why $-u$ is harmonic conjugate of $v$ as the analyticity of $f(z)=u+iv$ implies that $g(z)=v-iu$ is analytic.
