Recall that a function 
of two variables defined on an open domain 
is harmonic iff 
is 
(i.e., all four second order derivatives 
exist and are continuous in ), and satisfies Laplace equation
As mentioned in problem 6 of the Fall 2008 Calculus III final exam, a function 
is a harmonic conjugate of iff is defined on , 
and 
exist, and the Cauchy-Riemann equations hold:
and 
.
It follows immediately from the Cauchy-Riemann equations that if is a harmonic conjugate of a harmonic function , then is also , with 
, 
, 
and 
. It is also immediate that satisfies Laplace equation because 
, since continuity guarantees that the mixed partial derivatives commute. Thus is also harmonic.
In fact, modulo continuity of the second order derivatives, the harmonic functions are precisely the functions that (locally) admit harmonic conjugates.
To see this, assume first that is in and that it admits a harmonic conjugate . Then 
and 
so 
and was harmonic to begin with.
Conversely, assume that is harmonic in . Suppose first that is (connected and) simply connected. I claim that then admits a harmonic conjugate in . To see this, letting 
, notice that the existence of is equivalent to the claim that 
is a gradient vector field, since 
iff is a harmonic conjugate of 
But, since is simply connected, then is a gradient iff it is conservative, i.e., 
for any simple piecewise smooth loop 
in . Fix such a , and let 
denote its interior. Then, by Green’s theorem,
where the 
sign was to be chosen depending on the orientation of . It follows that is indeed conservative and therefore a gradient, so admits a harmonic conjugate.
Finally, if is not simply connected, we cannot guarantee that such a exists in all of , but the argument above shows that it does in any open (connected) simply connected subset of , for example, any open ball contained in . That we cannot extend this to all of follows from considering, for example, 
in 
. This is a harmonic function but it does not admit a harmonic conjugate in , since there is no continuous 
in . This example can be easily adapted (via a translation) to any non-simply connected .
I close by remarking that, as mentioned in my previous post on average values of harmonic functions, one can use Green’s theorem to prove that harmonic functions satisfy the average (or mean) value property, and this property characterizes harmonicity as well, implies that is actually 
(i.e., admits partial derivatives of all orders, and they are all continuous) and has the additional advantage that it only requires that is continuous, rather than . Similarly, one can show that the Cauchy-Riemann equations on suffice to guarantee that and are harmonic (and in particular, 
). However, one needs to require that the equations hold everywhere on . A pointwise requirement would not suffice. But I won’t address this issue here (I mention it in the notes in complex analysis that I hope to post some day).
Posted by andrescaicedo 
Teaching blog 