**Proof plan.** The structure follows the proof of the [Schwarz–Christoffel Half-Plane Formula](/theorems/683) for bounded polygons. Steps 1 and 2 (argument jumps and constancy of $\arg g$ on $\mathbb{R}$) are identical — they depend only on the local geometry near each prevertex and do not use the angle-sum identity. Step 3 (extension of $g$ to an entire [function](/page/Function)) also carries over: the reflection principle and removability of singularities use the same local arguments. The only modification is Step 4: the growth estimate at infinity. In the bounded case, $f$ is bounded, giving $f'(z) = O(|z|^{-2})$ and hence $|g| = O(1)$ (bounded), so [Liouville's Theorem](/theorems/346) applies directly. For unbounded $P$, $f$ is not bounded, but we show $g$ has at most polynomial growth, and then use the fact that an entire function of polynomial growth is a polynomial, and a polynomial with no zeros is a nonzero constant.
**Steps 1–3: Identical to the bounded case.**
Define the ratio
\begin{align*}
g(z) = \frac{f'(z)}{\displaystyle\prod_{k=1}^{n}(z - x_k)^{\alpha_k - 1}}.
\end{align*}
By the same argument as in the proof of the [Schwarz–Christoffel Half-Plane Formula](/theorems/683):
- $\arg f'(t)$ is constant on each interval $(x_k, x_{k+1})$ and jumps by $(\alpha_k - 1)\pi$ at each prevertex (since the image traces straight edges and turns through the exterior angle at each vertex — this is purely local and does not depend on whether $P$ is bounded).
- The factors $(\zeta - x_k)^{\alpha_k - 1}$ produce matching jumps, so $\arg g(t)$ is constant on all of $\mathbb{R} \setminus \{x_1, \ldots, x_n\}$.
- By the Schwarz reflection principle, $g$ extends to a holomorphic function on $\mathbb{C} \setminus \{x_1, \ldots, x_n\}$.
- Near each prevertex $x_k$, the local behavior $f(z) - w_k \sim A_k(z - x_k)^{\alpha_k}$ gives $g(z) \to \alpha_k A_k / h_k(x_k) \neq 0$, so each singularity is removable.
After removing all singularities, $g$ is entire and has no zeros in $\mathbb{C}$ (since $f'(z) \neq 0$ on $\mathbb{H}$ and the [limits](/page/Limit) at prevertices are nonzero).
**Step 4: Growth estimate at infinity (modified for unbounded domains).**
[claim:Polynomial Growth Of The Auxiliary Function]
The entire function $g$ satisfies $|g(z)| = O(|z|^M)$ for some $M > 0$ as $|z| \to \infty$.
[/claim]
[proof]
Since $P$ is a simply connected unbounded polygonal domain, it is contained in a half-plane (after rotation and translation). More precisely, the [boundary](/page/Boundary) rays of $P$ extend to infinity in at most two directions, and $P$ is contained in a sector $S = \{w \in \mathbb{C} : |\arg(w - w_0) - \theta_0| < \beta\pi/2\}$ for some $w_0 \in \mathbb{C}$, $\theta_0 \in \mathbb{R}$, and $\beta \leq 2$.
By standard distortion estimates for [conformal maps](/page/Conformal%20Maps) (the Koebe quarter theorem applied at scale $\operatorname{Im}(z)$), the derivative satisfies
\begin{align*}
|f'(z)| \leq \frac{4\,\operatorname{dist}(f(z), \partial P)}{\operatorname{Im}(z)}
\end{align*}
for all $z \in \mathbb{H}$. Since $P$ is contained in a sector, $\operatorname{dist}(f(z), \partial P) \leq |f(z) - w_0| + C$ for some constant $C$ depending on the geometry of $P$ near its finite vertices. Furthermore, the conformal map $f : \mathbb{H} \to P \subset S$ has at most polynomial growth: the comparison map from $\mathbb{H}$ onto a sector of opening $\beta\pi$ is $z \mapsto z^\beta$, and by the Schwarz–Pick lemma (comparing hyperbolic metrics), $|f(z)| = O(|z|^N)$ for some $N > 0$ depending on $\beta$ and the geometry of $P$.
Combining: $|f'(z)| = O(|z|^N / \operatorname{Im}(z))$ in $\mathbb{H}$.
The denominator satisfies $\prod_{k=1}^n |z - x_k|^{\alpha_k - 1} \sim |z|^{-\sigma}$ for large $|z|$, where $\sigma = \sum(1 - \alpha_k)$. Therefore
\begin{align*}
|g(z)| = \frac{|f'(z)|}{\prod |z - x_k|^{\alpha_k - 1}} = O\!\left(\frac{|z|^N}{|z|^{-\sigma} \operatorname{Im}(z)}\right) = O\!\left(\frac{|z|^{N + \sigma}}{\operatorname{Im}(z)}\right).
\end{align*}
This estimate holds in $\mathbb{H}$, and by the reflection symmetry $|g(\bar{z})| = |g(z)|$, it holds in the lower half-plane as well. On the real axis (away from prevertices), $g$ is continuous with constant argument, and the same polynomial bound holds by [continuity](/page/Continuity). Therefore $|g(z)| = O(|z|^{N + \sigma + 1})$ throughout $\mathbb{C}$ (the extra power of $|z|$ absorbs the $1/\operatorname{Im}(z)$ factor on the [set](/page/Set) $\{z : \operatorname{Im}(z) \geq 1/|z|\}$, while for $|\operatorname{Im}(z)| < 1/|z|$ the bound follows from the maximum principle on thin rectangles near $\mathbb{R}$).
[/proof]
**Step 5: Conclusion.**
By the extended Liouville theorem (Cauchy's estimate for [derivatives](/page/Derivative)), an entire function satisfying $|g(z)| = O(|z|^M)$ is a polynomial of degree at most $\lfloor M \rfloor$. But $g$ has no zeros in $\mathbb{C}$: we showed $g(z) \neq 0$ for all $z \in \mathbb{H}$, the limits at prevertices are nonzero, and the reflection relation preserves this. The only polynomial with no zeros is a nonzero constant, so $g \equiv c_1 \in \mathbb{C} \setminus \{0\}$.
Therefore $f'(z) = c_1 \prod_{k=1}^n (z - x_k)^{\alpha_k - 1}$, and integrating — path-independently, by [Cauchy's Theorem for Simply Connected Domains](/theorems/344) — gives the stated formula.
[remark:Comparison With The Bounded Case]
The proof above subsumes the bounded case: when $\sigma = 2$, the polygon is bounded, $f$ is bounded, and the polynomial growth estimate simplifies to $|g| = O(1)$. The "polynomial with no zeros is constant" step is then equivalent to the direct application of [Liouville's Theorem](/theorems/346) used in the proof of the [Schwarz–Christoffel Half-Plane Formula](/theorems/683). The advantage of the polynomial-growth approach is that it does not require an independent estimate of the exact decay rate of $f'$ at infinity.
[/remark]
