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:
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 is 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).