"analytic on $\mathbb{R^2}$" vs. "analytic on $\mathbb{C}$" My course notes (Mathematics BSc, third-year module in complex analysis, unpublished) have,

Definition: A real-valued function $u(x,y)$ on a region $D\subseteq\mathbb{R}^2$ which has continuous second order partial derivatives and satisfies Laplace's equation on $D$ is said to be harmonic on $D$.


Theorem: If $f=u+iv$ is analytic on a region $D$, then $u$ and $v$ are harmonic on $D$.


Examples: In each of the following cases decide whether there is a function $f$ analytic on $\mathbb{C}$ such that Re$(f)=u$.

There seems to have been a slip from "$f$ analytic on $\mathbb{R^2}$" to "$f$ analytic on $\mathbb{C}$". Is the difference unimportant?
 A: You are correct that analyticity of a function $f : \mathbb C \mapsto \mathbb C$ is not the same as analyticity of the corresponding function $F : \mathbb R^2 \mapsto \mathbb C$, where $f$ and $F$ are related by $f(x+iy)=F(x,y)$ (and similarly when the domain of $f$ is an open $D \subset \mathbb C$ with corresponding open domain $D' = \{(x,y) \in \mathbb R^2 \mid x+iy \in D\}$ for $F$). Here I'm using the common definition of analyticity: $f$ is analytic (i.e. complex-analytic) if it is equal to its power series in the single complex variable $z=x+iy$; whereas $F$ is analytic (i.e. real-analytic) if it is equal to its power series in the pair of real variables $x,y$.
Nonetheless, this kind of abuse of language is quite common in complex analysis. In fact, in a discussion of complex analysis, real-analyticity is generally ignored. The very theorem that you cited can in some sense be taken as a justification for this abuse of language: if $f$ is analytic on $D$ then each of the two coordinate functions of $f$, thought of as 2-real-variable functions (i.e. as the two coordinate functions of $F$), are harmonic on $D'$. There's also a converse to this theorem: if the two coordinate functions $u,v$ of $F$ are harmonic, and if those two functions are related in a certain manner (e.g. the integral relation written down here), then $f$ is analytic.
A: A complex function $f = u + i v$ is analytic if and only if it is differentiable. This is quite different from what happens in $\mathbb{R}^2$, where a function is analytic if it has derivatives of all orders and is representable as a power series.
