Okay. So, my confusion around harmonic functions as they are treated in Conway's stems from trying to properly interpret the following problem:
Use the Cauchy integral formula to derive the mean value property of harmonic functions, that $$u(z_0) = \int_{0}^{2 \pi} u(z_0+\rho e^{i \theta}) d{\theta}$$ where $u(z)$ is harmonic in a domain $U$ and the closed disk $\{z : |z-z_0 | \le \rho \}$ is contained in $U$.
Here is where my confusion arises and why I think the above statement is slightly ambiguous:
The above comes from Conway's book on complex analysis. So, Conway offers a definition of a real-valued harmonic function and then states some facts about them. In particular, if $u$ is a harmonic function on a simply connected domain $G$, then there is another harmonic function $v$ on $G$ such that $f = u + iv$ is analytic on $G$. But in proposition 1.3 we are suddenly dealing with complex-valued harmonic functions, which, as far as I can tell, have not been defined. A consequence of proposition 1.3 is that these complex-valued harmonic functions are analytic. And in the proof they apparently have the same property as real-valued harmonic functions on simply connected domains--namely, that there is another complex-valued harmonic function on the same simply connected domain $v$ such that $f = u + iv$ is analytic.
Am I reading this correctly or is there an error in the book?
If I am understanding all this correctly, if we assume that the the above problem means for $u$ to be complex-valued, then proposition 1.3 states that $u$ is analytic and solving the problem is a very straightforward application of Cauchy's integral formula. But if $u$ is meant to be real-valued, then it's not as straightforward but still doable (just use the fact that $u$ has a harmonic conjugate $v$ such that $f = u + iv$ is analytic, and then apply Cauchy's integral formula to $f$).
So, what (if anything) am I misunderstanding?