Is writing $f(z)=u(x,y)+i v(x,y)$ a big abuse of notation in complex analysis?

Question

I am a bit uncomfortable with the habit of writing $$f(z)=u(x,y)+i v(x,y)$$ where $f:U\subseteq \mathbb C \to \mathbb C$ is a complex function and $z=x+iy$. This is especially done when Cauchy-Riemann equations and harmonic functions are introduced. The issue is that I think authors switch from subsets of $\mathbb C$ to $\mathbb R^2$ ad vice-versa improperly.

For example, in Theorem VIII, $2.2$ of his book "Complex Analysis", Conway writes that the connected and open set $G\subseteq \mathbb C$ is simply connected if and only if every harmonic function $u: G\to \mathbb C$ has a harmonic conjugate, i.e. there is another harmonic function $v: G\to \mathbb C$ such that $f=u+iv$ is holomorphic in $G$.

This approach does not convince me, because for a function to be harmonic one should be able to perform derivatives with respect to $x$ and $y$, so should consider a function of two real variables.

This also makes me question about Wirtinger derivatives $$\frac{\partial}{\partial z}=\frac{\partial}{\partial x}-i\frac{\partial}{\partial y} \\ \frac{\partial}{\partial \bar z}=\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}$$ because this would be appropriate for a function of the three variables $x,y,z$.

Here we discussed the meaning of $\partial/\partial \bar z$ and it looks to me that it is only a trick that works.

How can one be more rigorous? I thought we could introduce a set $\tilde G=\{(x,y) \in \mathbb R^2: z=x+iy \in G\}$ and define $u,v$ there. Is there another approach, perhaps more frequent in literature?

Answer 1

There are a lot of "standard embeddings" in mathematics which we use to make "identifications", such as:

$\mathbb N$ embedded into $\mathbb Z$ embedded into $\mathbb Q$ embedded into $\mathbb R$ embedded into $\mathbb C$.

Your objections regarding the last embedding of this sequence would probably apply as well to the earlier ones. The objections in those cases would be even more serious if one stuck to the most formal definitions. For instance, an element of $\mathbb R$ is an equivalence class of Cauchy sequences of rational numbers... (which is a lot more complicated than regarding an element of $\mathbb C$ as an ordered pair of elements of $\mathbb R$...)

... but to force oneself to always think about an element of $\mathbb R$ like that? That way lies madness.

So, for these most important of all embeddings, what the mathematical community does to stay sane is something like this.

Formally, $\mathbb N$ is defined in set theory using the Von Neumann definition of natural number.

Next, $\mathbb Z$ is defined in terms of $\mathbb N$ [as equivalence classes of ordered pairs of natural numbers, using the equivalence relation $(a,b) \sim (c,d) \iff a+d=c+b$]. We then use the standard embedding of $\mathbb N$ into $\mathbb Z$ [using $n \mapsto (n,0)$]. Finally we as a community agree to conveniently forget about that embedding and to treat $\mathbb N$ as a subset of $\mathbb Z$.

Next, $\mathbb Q$ is defined in terms of $\mathbb Z$ [details elided], and then we use a standard embedding of $\mathbb Z$ into $\mathbb Q$ [more details elided], and then we as a community agree to conveniently forget about the embedding and to treat $\mathbb Z$ (and hence also $\mathbb N$) as a subset of $\mathbb Q$.

And the same for the next two embeddings of $\mathbb Q$ into $\mathbb R$, and of $\mathbb R$ into $\mathbb C$.

In the end, the outcome is that we, as a community, agree to treat $\mathbb N$ as a subset of $\mathbb Z$ which is a subset of $\mathbb Q$ which is a subset of $\mathbb R$ which is a subset of $\mathbb C$. Every now and then it is good to remind ourselves of the formalities of the definitions and of the embeddings... and then to once again re-forget those formalities.

Answer 2

From the formal point of view there is indeed an irresolvable conflict.

Let $U \subset \mathbb C$.

1. Writing a function $\phi : U \to \mathbb K$, where $\mathbb K = \mathbb R, \mathbb C$, in the form $\phi(x,y)$ only makes sense if we understand $\mathbb C$ as $\mathbb R^2$.

2. Writing $f : U \to \mathbb C$ in the form $f = u + iv$ with $u, v : U \to \mathbb R$ only makes sense if we regard $\mathbb R$ as a genuine subset of $\mathbb C$.
   This is an absolutely legit approach; see Lee Mosher's answer.

But $\mathbb R \subset\mathbb C$ and $\mathbb C = \mathbb R^2$ cannot be true at the same time.

Therefore it is an abuse of notation to write 2. in the form $f(z)=u(x,y)+i v(x,y)$.

Let us write $\mathbb C_1$ for the $\mathbb R \subset\mathbb C$ interpretation and $\mathbb C_2$ for the $\mathbb C = \mathbb R^2$ interpretation. These are isomorphic fields, but technically they are distinct models of the complex numbers. The above conflict is due to the fact that $\mathbb C_1 \ne \mathbb C_2$; the abuse of notation is to write $\mathbb C_1 = \mathbb C_2$. However, we have the obvious field isomorphism $$\rho : \mathbb C_2 \to \mathbb C_1, \rho(x,y) = x + iy .$$

So what can be done if we really want to be pedantic?

• We could agree to work with two different interpretations of $\mathbb C$ when we consider functions: On the domain we use the $\mathbb C_2$ interpretation, on the codomain the $\mathbb C_1$ interpretation.
  However, in that case the definition $f'(z_0) = \lim_{z\to z_0} \dfrac{f(z) - f(z_0)}{z - z_0}$ would not make sense because numerator and denominator do not live in the same model of $\mathbb C$.

• In the $\mathbb C_1$ interpretation we must use $\rho$ and get $f(z) = f(\rho(x,y)) = u(x,y) + iv(x,y)$. In that case the expressions $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ are defined as $\dfrac{\partial (f \circ \rho)}{\partial x}$ and $\dfrac{\partial (f \circ \rho)}{\partial y}$.

• In the $\mathbb C_2$ interpretation we also must use $\rho$ and get $f(z) = f(x,y) = (u(x,y), v(x,y))$ or $f(z) = f(x,y) = \rho^{-1}(u(x,y) + iv(x,y)) = \rho^{-1}(u(z) + iv(z)) $ $= (\rho^{-1} \circ u)(z) + (0,1) (\rho^{-1} \circ v)(z) $.

I do not think that this kind of formalism makes life easier. I prefer the abuse $\mathbb C_1 = \mathbb C_2$. It does not cause any problems and everybody understands what this "streamlined" notation means.