[proofplan]
We reduce the problem to the Identity Theorem for holomorphic functions of one variable. The composition $h(z) = R(f(z), g(z))$ is holomorphic on the connected open set $\Omega$ (as a composition of holomorphic maps), vanishes on $S$, and $S$ has a limit point in $\Omega$. The Identity Theorem then forces $h \equiv 0$ on $\Omega$.
[/proofplan]
[step:Establish that the composition $h(z) = R(f(z), g(z))$ is holomorphic on $\Omega$]
Define the function
\begin{align*}
h: \Omega &\to \mathbb{C} \\
z &\mapsto R(f(z), g(z)).
\end{align*}
We verify that $h$ is holomorphic on $\Omega$. Consider the map
\begin{align*}
\Phi: \Omega &\to \mathbb{C}^2 \\
z &\mapsto (f(z), g(z)).
\end{align*}
Since $f$ and $g$ are holomorphic on $\Omega$, the map $\Phi$ is holomorphic as a map from $\Omega \subset \mathbb{C}$ to $\mathbb{C}^2$ (a map into $\mathbb{C}^n$ is holomorphic if and only if each component function is holomorphic). The function $R: \mathbb{C}^2 \to \mathbb{C}$ is holomorphic by hypothesis. The composition of holomorphic maps is holomorphic: $h = R \circ \Phi$ is holomorphic on $\Omega$.
[guided]
Why is the composition holomorphic? We are composing two maps:
1. $\Phi: \Omega \to \mathbb{C}^2$, which is holomorphic because its components $f$ and $g$ are holomorphic functions of one complex variable.
2. $R: \mathbb{C}^2 \to \mathbb{C}$, which is holomorphic in two complex variables by hypothesis.
The chain rule for holomorphic functions of several variables guarantees that the composition $R \circ \Phi$ is holomorphic. Concretely, the derivative of $h$ at $z_0 \in \Omega$ is
\begin{align*}
h'(z_0) = \frac{\partial R}{\partial w_1}(f(z_0), g(z_0)) \cdot f'(z_0) + \frac{\partial R}{\partial w_2}(f(z_0), g(z_0)) \cdot g'(z_0),
\end{align*}
which exists at every $z_0 \in \Omega$ since all the constituent functions and partial derivatives are holomorphic. So $h$ is holomorphic on $\Omega$.
[/guided]
[/step]
[step:Apply the Identity Theorem to conclude $h \equiv 0$]
The function $h: \Omega \to \mathbb{C}$ is holomorphic on the connected open set $\Omega$. By hypothesis, $h(z) = R(f(z), g(z)) = 0$ for all $z \in S$, and $S$ has a limit point $z^* \in \Omega$. In particular, the zero set $h^{-1}(\{0\}) \supset S$ has a limit point in $\Omega$.
By the [Identity Theorem](/theorems/???), a holomorphic function on a connected open set that vanishes on a set with a limit point in that set must vanish identically. The hypotheses are satisfied: $\Omega$ is connected and open, $h$ is holomorphic on $\Omega$, and $h$ vanishes on $S$ which has a limit point $z^* \in \Omega$. Therefore $h(z) = 0$ for all $z \in \Omega$, i.e.,
\begin{align*}
R(f(z), g(z)) = 0 \quad \text{for all } z \in \Omega.
\end{align*}
[guided]
The Identity Theorem (also called the Uniqueness Theorem for holomorphic functions) states: if $h$ is holomorphic on a connected open set $\Omega$ and $h$ vanishes on a subset $S \subset \Omega$ that has a limit point in $\Omega$, then $h \equiv 0$ on $\Omega$.
Why does having a limit point matter? The proof of the Identity Theorem proceeds by showing the zero set of a non-constant holomorphic function consists of isolated points. If the zeros have a limit point in $\Omega$, the function cannot be non-constant, so it must be identically zero.
We verify the hypotheses:
- $\Omega$ is connected and open: given in the theorem statement.
- $h$ is holomorphic on $\Omega$: established in the previous step.
- $h$ vanishes on $S$: we are told $R(f(z), g(z)) = 0$ for all $z \in S$.
- $S$ has a limit point in $\Omega$: given in the theorem statement.
All conditions are met, so the Identity Theorem yields $h \equiv 0$ on $\Omega$. This means $R(f(z), g(z)) = 0$ for every $z \in \Omega$, which is the desired conclusion.
The power of this result is that a functional equation $R(f, g) = 0$ that holds on an arbitrarily small set with a limit point (such as a convergent sequence, or a small arc, or even just a sequence tending to a boundary point from inside $\Omega$) automatically persists to the entire connected domain. This is a direct manifestation of the rigidity of holomorphic functions.
[/guided]
[/step]
