§5.5 The Reflection Principle

5.5 The Reflection Principle

We have previously considered analytic continuations over two regions with an intersection. Under certain conditions, analytic continuations can be derived across a curve, given by the following theorem.

Theorem 5.5.1 (Painlevé).

Let $𝑈_{1}$ and $𝑈_{2}$ be two disjoint simply connected open regions in $ℂ$ such that $𝜕 𝑈_{1} \cap 𝜕 𝑈_{2}$ is a simple curve $𝛾$ without its endpoints. Let $𝑓_{1} : 𝑈_{1} \to ℂ$ and $𝑓_{2} : 𝑈_{2} \to ℂ$ be two holomorphic functions that are continuous on $𝑈_{1} \cup 𝛾$ and $𝑈_{2} \cup 𝛾$ , respectively, such that $𝑓_{1} \equiv 𝑓_{2}$ on $𝛾$ . Then there exists a unique holomorphic function

𝑓 = {\begin{cases} 𝑓_{1} & on 𝑈_{1} \\ 𝑓_{2} & on 𝑈_{2} \\ 𝑓_{1} \equiv 𝑓_{2} & on 𝛾 \end{cases} .

𝑈_{1} \cup 𝑈_{2} \cup 𝛾

Figure 5.2: The two regions

𝑈_{1}

and

𝑈_{2}

sharing a boundary curve

𝛾

, the disk

𝐷 (𝑧, 𝑅_{𝑧})

for

𝑧 \in 𝛾

, and the curve¹

Γ

. Solid large arrows and arrowheads denote orientation of the

{\tilde{Γ}}_{1}

and

{\tilde{Γ}}_{2}

regions.

Proof.

We aim to prove that the constructed function $𝑓$ is holomorphic on $𝑈_{1} \cup 𝑈_{2} \cup 𝛾$ . In particular, we only need to prove that $𝑓$ is holomorphic on (a neighborhood of) $𝛾$ , after which the Identity Theorem (Theorem 3.3.3 $Theorem 3.3.3 Theorem 3.3.3$ ) applies.

Let $𝑧 \in 𝛾$ be fixed, and choose $𝑅 = 𝑅_{𝑧} > 0$ such that $𝐷 (𝑧, 𝑅) \subseteq 𝑈_{1} \cup 𝑈_{2} \cup 𝛾$ . Let $Γ$ be any simple closed curve in $𝐷 (𝑧, 𝑅)$ . If $Γ$ is fully contained in $𝑈_{1} \cup 𝛾 (\cap 𝐷 (𝑧, 𝑅))$ , then by Cauchy–Goursat (Theorem 3.1.9 $Theorem 3.1.9 Theorem 3.1.9$ ),

\oint_{Γ} 𝑓 (𝑧) d 𝑧 = \oint_{Γ} 𝑓_{1} (𝑧) d 𝑧 = 0 .

Similarly, if $Γ$ is fully contained in $𝑈_{2} \cup 𝛾$ , then

\oint_{Γ} 𝑓 (𝑧) d 𝑧 = \oint_{Γ} 𝑓_{2} (𝑧) d 𝑧 = 0 .

If $Γ$ intersects $𝛾$ , then we can decompose $Γ = Γ_{1} \cup Γ_{2}$ , where $Γ_{1}$ is the part of $Γ$ that lies in $𝑈_{1} \cup 𝛾$ and $Γ_{2}$ is the part of $Γ$ that lies in $𝑈_{2} \cup 𝛾$ . The set $\tilde{Γ} = 𝛾 \cap int (Γ)$ closes $Γ_{1}$ and $Γ_{2}$ in the sense that ${\tilde{Γ}}_{1} = Γ_{1} \cup \tilde{Γ}$ and ${\tilde{Γ}}_{2} = Γ_{2} \cup \tilde{Γ}$ are both simple closed curves, or unions of simple closed curves (where $\tilde{Γ}$ in each of the two curves have opposite orientations, see Figure 5.2 $Figure 5.2 Figure 5.2$ ). By Cauchy–Goursat (Theorem 3.1.9 $Theorem 3.1.9 Theorem 3.1.9$ ), we have

∳_{Γ} 𝑓 (𝑧) d 𝑧 = (\int_{Γ_{1}} + \int_{Γ_{2}} + \int_{\tilde{Γ}} - \int_{\tilde{Γ}}) 𝑓 (𝑧) d 𝑧 = (∳_{{\tilde{Γ}}_{1}} + ∳_{{\tilde{Γ}}_{2}}) 𝑓 (𝑧) d 𝑧 = 0 .

Hence, by Morera’s Theorem (Theorem 3.2.6 $Theorem 3.2.6 Theorem 3.2.6$ ), $𝑓$ is holomorphic on $⋃_{𝑧 \in 𝛾} 𝐷 (𝑧, 𝑅_{𝑧})$ , and the assertion follows. ⁠ $□$

A consequent result was discovered by Schwarz, known as the reflection principle, is a unique result derived from the above theorem for when the shared boundary curve lies in the real axis under certain conditions.

Theorem 5.5.2 (Schwarz Reflection Principle).

Let $𝑈 \subseteq ℂ$ be a connected region on one side of the real axis such that there exists a non-degenerate curve $𝛾 \subseteq 𝜕 𝑈$ such that $𝛾 \subseteq ℝ$ . Let $𝑓 : 𝑈 \to ℂ$ be holomorphic with continuity up to $𝑈 \cup 𝛾$ such that $𝑓$ is real-valued on $𝛾$ , and let $\tilde{𝑈} = {\overline{𝑧} : 𝑧 \in 𝑈}$ be the reflection of $𝑈$ across the real axis. Then there exists a unique holomorphic function

\tilde{𝑓} (𝑧) = {\begin{cases} 𝑓 (𝑧) & if 𝑧 \in 𝑈 \\ \overline{𝑓 (\overline{𝑧})} & if 𝑧 \in \tilde{𝑈} \\ 𝑓 (𝑧) \equiv \overline{𝑓 (\overline{𝑧})} & if 𝑧 \in 𝛾 \end{cases} .

𝑈 \cup \tilde{𝑈} \cup 𝛾

Proof.

If $𝑧 \in ℝ$ , then $\overline{𝑧} = 𝑧$ , and since $𝑓$ is real on $𝛾$ , it follows that $𝑓 (𝑧) = \overline{𝑓 (\overline{𝑧})}$ for $𝑧 \in 𝛾$ . Thus, we are left to prove that $𝑧 \mapsto \overline{𝑓 (\overline{𝑧})}$ is holomorphic on $\tilde{𝑈}$ . Let $𝑧_{0} \in \tilde{𝑈}$ . It follows that

lim_{\begin{matrix} 𝑧 \to 𝑧_{0} \\ 𝑧 \in \tilde{𝑈} \end{matrix}} \frac{\overline{𝑓 (\overline{𝑧})} - \overline{𝑓 (\overline{𝑧_{0}})}}{𝑧 - 𝑧_{0}} = lim_{\begin{matrix} 𝑧 \to 𝑧_{0} \\ 𝑧 \in \tilde{𝑈} \end{matrix}} \overline{(\frac{𝑓 (\overline{𝑧}) - 𝑓 (\overline{𝑧_{0}})}{\overline{𝑧} - \overline{𝑧_{0}}})} = \overline{𝑓^{'} (\overline{𝑧_{0}})} .

Since this limit exists, it follows that $\overline{𝑓 (\overline{𝑧})}$ is holomorphic on $\tilde{𝑈}$ . Assume that $𝑧_{0} \in 𝛾$ . Since

lim_{𝑧 \to 𝑧_{0}} \overline{𝑓 (\overline{𝑧})} = \overline{𝑓 (lim_{𝑧 \to 𝑧_{0}} \overline{𝑧})} = \overline{𝑓 (𝑧_{0})} = 𝑓 (𝑧_{0}),

it follows that

\overline{𝑓 (\overline{𝑧})}

is continuous on

\tilde{𝑈} \cup 𝛾

. Therefore, by the Painlevé Theorem,

\tilde{𝑓}

is holomorphic on

𝑈 \cup \tilde{𝑈} \cup 𝛾

. ⁠

□

This conjugate-symmetry can be generalized by transforming $𝛾$ :

Theorem 5.5.3 (Symmetry Principle).

Let

𝐿 \subset ℂ

be an (infinite) straight line, and let

𝑈 \subset ℂ

be an open region lying entirely on one side of

𝐿

. Suppose

𝛾 \subseteq 𝐿

is a non-degenerate open curve contained in

𝜕 𝑈

. If

𝑓

is holomorphic on

𝑈

, continuous on

𝑈 \cup 𝛾

, and satisfies

𝑓 (𝛾) \subseteq Γ

, where

Γ \subset ℂ

is a straight line, then there exists a unique holomorphic function

\tilde{𝑓} : 𝑈 \cup \tilde{𝑈} \cup 𝛾 \to ℂ

such that

\tilde{𝑓} \equiv 𝑓

𝑈

, where

\tilde{𝑈}

is the reflection of

𝑈

across

𝐿

. Moreover, for any pair

𝑧_{1}, 𝑧_{2} \in 𝑈 \cup \tilde{𝑈} \cup 𝛾

symmetric with respect to

𝐿

, the values

\tilde{𝑓} (𝑧_{1})

and

\tilde{𝑓} (𝑧_{2})

are symmetric with respect to

Γ

Proof.

There exist

𝑎, 𝑐 \in ℂ^{*}

and

𝑏, 𝑑 \in ℂ

such that

𝜙 (𝑧) = 𝑎 𝑧 + 𝑏

maps

𝐿

ℝ

and

𝜓 (𝑧) = 𝑐 𝑧 + 𝑑

maps

Γ

ℝ

. Let

𝑈^{'} = 𝜙 (𝑈)

, which lies entirely on one side of the real axis, and let

𝛾^{'} = 𝜙 (𝛾)

, a curve on the real axis. The function

𝜑 = 𝜓 \circ 𝑓 \circ 𝜙^{- 1}

is holomorphic on

𝑈^{'}

and continuous on

𝑈^{'} \cup 𝛾^{'}

. By the Schwarz Reflection Principle (Theorem 5.5.2

Theorem 5.5.2 Theorem 5.5.2

), there exists a unique holomorphic function

\tilde{𝜑} : 𝑈^{'} \cup \tilde{𝑈^{'}} \cup 𝛾^{'} \to ℂ

such that

\tilde{𝜑} \equiv 𝜑

𝑈^{'}

, where

\tilde{𝑈^{'}}

is the reflection of

𝑈^{'}

across the real axis. Then

\tilde{𝑓} = 𝜓^{- 1} \circ \tilde{𝜑} \circ 𝜙

is a holomorphic function on

𝑈 \cup \tilde{𝑈} \cup 𝛾

such that

\tilde{𝑓} \equiv 𝑓

𝑈

. Since linear transformations preserves symmetry, for any pair

𝑧_{1}, 𝑧_{2} \in 𝑈 \cup \tilde{𝑈} \cup 𝛾

symmetric with respect to

𝐿

, we have

𝜙 (𝑧_{1}) = \overline{𝜙 (𝑧_{2})}

, and thus

\tilde{𝜑} \circ 𝜙 (𝑧_{1})

and

\tilde{𝜑} \circ 𝜙 (𝑧_{2})

are symmetric with respect to

ℝ

. Hence,

\tilde{𝑓} (𝑧_{1})

and

\tilde{𝑓} (𝑧_{2})

are symmetric with respect to

𝜓^{- 1} (ℝ) = Γ

. ⁠

□