§13.3 The Monodromy Theorem

13.3 The Monodromy Theorem

Before we address our main issue, we will first formalize the concept of homotopy. Informally, it is a continuous deformation or “smooth interpolation” between two curves (that lies entirely within a provided set).

Definition 13.3.1.

Let $𝑈 \subseteq ℂ$ be an open set. Two curves $𝛾_{1}, 𝛾_{2} : [0, 1] \to 𝑈$ are said to be homotopic if there exists a continuous function $𝐻 : [0, 1] \times [0, 1] \to 𝑈$ such that:

$𝐻 (0, 𝑡) = 𝛾_{1} (𝑡)$ for all $𝑡 \in [0, 1]$ ,
$𝐻 (1, 𝑡) = 𝛾_{2} (𝑡)$ for all $𝑡 \in [0, 1]$ .

The function $𝐻$ is known as a homotopy between $𝛾_{1}$ and $𝛾_{2}$ .

We are primarily concerned with homotopies with fixed endpoints (under the pretense that $𝛾_{1} (0) = 𝛾_{2} (0)$ and $𝛾_{1} (1) = 𝛾_{2} (1)$ ), or that the homotopy $𝐻$ satisfies $𝐻 (𝑠, 0) = 𝛾_{1} (0) = 𝛾_{2} (0)$ and $𝐻 (𝑠, 1) = 𝛾_{1} (1) = 𝛾_{2} (1)$ for any $𝑠 \in [0, 1]$ .

Definition 13.3.2.

Let

Ω \subseteq ℂ

be a region and let

𝑈 \subseteq Ω

be an open disk centered at

𝑃 \in Ω

. Suppose

(𝑓, 𝑈)

is an analytic function element. If there is an analytic continuation of

(𝑓, 𝑈)

along any curve

𝛾 \subset Ω

from

𝑃

, then

(𝑓, 𝑈)

has unrestricted continuation in

Ω

As for our question in interest:

Let $𝛾_{1} : [0, 1] \to ℂ$ and $𝛾_{2} \to ℂ$ be two curves with the same endpoints ( $𝛾_{1} (0) = 𝛾_{2} (0)$ and $𝛾_{1} (1) = 𝛾_{2} (1)$ ), and let ${(𝑓_{𝑡}, 𝑈_{𝑡})}_{0 \leq 𝑡 \leq 1}$ and ${({\tilde{𝑓}}_{𝑡}, {\tilde{𝑈}}_{𝑡})}_{0 \leq 𝑡 \leq 1}$ be the unique analytic continuations along $𝛾_{1}$ and $𝛾_{2}$ respectively. Under what conditions will the terminal analytic function elements be equivalent (when will it be satisfied that $𝑓_{1} \equiv {\tilde{𝑓}}_{1}$ on $𝑈_{1} \cap {\tilde{𝑈}}_{1}$ )?

This question is not necessarily an affirmative. The principal branch logarithm on $𝐷 (1, 1)$ , when analytically continued on the unit upper semicircle ${e^{π i 𝜃}}_{0 \leq 𝑡 \leq 1}$ and unit lower semicircle ${e^{- π i 𝜃}}_{0 \leq 𝑡 \leq 1}$ , yields inequivalent terminal analytic function elements (differing by $2 π i$ ). The general answer to the question is given below:

Theorem 13.3.3.

Let

Ω \subseteq ℂ

be a region, and suppose

𝑈 \subseteq Ω

is a disk centered at

𝑃

, and let

𝛾_{1}

and

𝛾_{2}

be two curves in

Ω

with the same endpoints (

𝑃 = 𝛾_{1} (0) = 𝛾_{2} (0)

and

𝑄 = 𝛾_{1} (1) = 𝛾_{2} (1)

). Let

{(𝑓_{𝑡}, 𝑈_{𝑡})}_{0 \leq 𝑡 \leq 1}

and

{({\tilde{𝑓}}_{𝑡}, {\tilde{𝑈}}_{𝑡})}_{0 \leq 𝑡 \leq 1}

be the unique analytic continuations along

𝛾_{1}

and

𝛾_{2}

respectively. If

𝛾_{1}

and

𝛾_{2}

are homotopic in

Ω

and

(𝑓, 𝑈)

has unrestricted continuation in

Ω

, then

𝑓_{1} \equiv {\tilde{𝑓}}_{1}

𝑈_{1} \cap {\tilde{𝑈}}_{1}

Proof.

Let $𝑠$ be a fixed value in $[0, 1]$ and consider the curve defined by $𝑡 \mapsto 𝐻 (𝑠, 𝑡)$ from $𝑃$ to $𝑄$ . By the unrestricted continuation assumption, there exists an analytic continuation ${(𝑓_{𝑠, 𝑡}, 𝑈_{𝑠, 𝑡})}_{0 \leq 𝑡 \leq 1}$ of $(𝑓, 𝑈)$ along this curve. In this form, we aim to show that $𝑓_{0, 1} \equiv 𝑓_{1, 1}$ on $𝑈_{0, 1} \cap 𝑈_{1, 1}$ (we have taken the liberties to denote $𝑓_{1}$ by $𝑓_{0, 1}$ and ${\tilde{𝑓}}_{1}$ by $𝑓_{1, 1}$ , with similar notions for $𝑈_{1}$ and ${\tilde{𝑈}}_{1}$ ).

Let

𝑆 = {𝑠 \in [0, 1] : \forall 0 \leq 𝜆 \leq 𝑠, 𝑓_{𝜆, 1} \equiv 𝑓_{0, 1} on 𝑈_{𝜆, 1} \cap 𝑈_{0, 1} ∋ 𝑄} .

Let us fix $𝑠 \in 𝑆$ . The analytic continuation ${(𝑓_{𝑠, 𝑡}, 𝑈_{𝑠, 𝑡})}_{0 \leq 𝑡 \leq 1}$ along the curve $𝑡 \mapsto 𝐻 (𝑠, 𝑡)$ generates the cover ${𝑈_{𝑠, 𝑡}}_{0 \leq 𝑡 \leq 1}$ of the compact curve ${𝐻 (𝑠, 𝑡)}_{0 \leq 𝑡 \leq 1}$ . By Heine–Borel (Theorem 1.1.9 $Theorem 1.1.9 Theorem 1.1.9$ ), there exists a finite subcover ${𝑈_{𝑠, 𝑡_{𝑘}}}_{𝑘 = 1}^{𝑛}$ . Then $\exists 𝜀 > 0$ such that each $𝐷 (𝐻 (𝑠, 𝑡), 𝜀) \subseteq ⋃_{𝑘 = 1}^{𝑛} 𝑈_{𝑠, 𝑡_{𝑘}}$ for all $𝑡 \in [0, 1]$ . In fact, we can choose $𝜀$ to be

𝜀 = {min}_{𝑘 = 1}^{𝑛} [dist (𝐻 (𝑠, [0, 1]), 𝜕 𝑈_{𝑠, 𝑡_{𝑘}} ∖ ⋃_{𝑗 = 1}^{𝑛} 𝑈_{𝑠, 𝑡_{𝑗}})] .

It can be verified that $dist$ here is positive since $𝐻 (𝑠, [0, 1]) \subset ⋃_{𝑗 = 1}^{𝑛} 𝑈_{𝑠, 𝑡_{𝑗}}$ (thus both sets are disjoint) and both are compact sets. By continuity, $\forall 𝑡 \in [0, 1]$ , $\exists 𝛿 > 0$ such that $\forall 𝑠^{'} \in [0, 1] \cap (𝑠 - 𝛿, 𝑠 + 𝛿)$ , $| 𝐻 (𝑠, 𝑡) - ℎ (𝑠^{'}, 𝑡) | < 𝜀$ . By the Heine–Cantor Theorem (Theorem 1.2.26 $Theorem 1.2.26 Theorem 1.2.26$ ), $𝛿$ attains a positive infimum, and thus $\exists 𝛿 > 0$ such that $\forall 𝑠^{'} \in [0, 1] \cap (𝑠 - 𝛿, 𝑠 + 𝛿)$ , $\forall 𝑡 \in [0, 1]$ , $| 𝐻 (𝑠, 𝑡) - 𝐻 (𝑠^{'}, 𝑡) | < 𝜀$ .

Fix $𝑠^{'}$ and $𝑡$ . By the derived inequality, we have that $𝐻 (𝑠^{'}, 𝑡) \in 𝑈_{𝑠, 𝑡}$ and therefore, ${\tilde{𝑈}}_{𝑠^{'}, 𝑡} = 𝐷 (𝐻 (𝑠^{'}, 𝑡), 𝜀 - | 𝐻 (𝑠, 𝑡) - 𝐻 (𝑠^{'}, 𝑡) |) \subseteq 𝑈_{𝑠, 𝑡}$ . Thus, the analytic function element $(𝑓_{𝑠, 𝑡}, {\tilde{𝑈}}_{𝑠^{'}, 𝑡})$ is a direct analytic continuation of $(𝑓_{𝑠, 𝑡}, 𝑈_{𝑠, 𝑡})$ . By constructing analytic function elements similarly for all $𝑡$ , we obtain an analytic continuation of $(𝑓_{𝑠, 0}, {\tilde{𝑈}}_{𝑠^{'}, 0})$ along $𝐻 (𝑠^{'}, [0, 1])$ . Because $(𝑓_{𝑠, 0}, 𝑈_{𝑠, 0})$ and $(𝑓_{𝑠, 0}, {\tilde{𝑈}}_{𝑠^{'}, 0})$ are direct analytic continuations of each other, by Lemma 13.2.2 $Lemma 13.2.2 Lemma 13.2.2$ , the continuations ${(𝑓_{𝑠, 𝑡}, {\tilde{𝑈}}_{𝑠^{'}, 𝑡})}_{0 \leq 𝑡 \leq 1}$ and ${(𝑓_{𝑠^{'}, 𝑡}, 𝑈_{𝑠^{'}, 𝑡})}_{0 \leq 𝑡 \leq 1}$ are equivalent. Thus, all elements of $(𝑠 - 𝛿, 𝑠 + 𝛿) \cap [0, 1]$ belong in $𝑆$ , and thus $𝑆$ is relatively open in $[0, 1]$ .

Let ${𝑠_{𝑛}}_{𝑛 \in ℕ} \subset 𝑆$ be an arbitrarily chosen convergent sequence accumulating at $𝑠_{\infty}$ . Since there exists an analytic continuation of $(𝑓_{𝑠_{\infty}, 0}, 𝑈_{𝑠_{\infty}, 0})$ along the curve $𝐻 (𝑠_{\infty}, [0, 1])$ , we can use the same argument as before to construct $𝜀$ such that each $𝐷 (𝐻 (𝑠_{\infty}, 𝑡), 𝜀) \subseteq 𝑈_{𝑠_{\infty}, 𝑡}$ and $𝛿$ such that $\forall 𝑠 \in [0, 1] \cap (𝑠_{\infty} - 𝛿, 𝑠_{\infty} + 𝛿)$ , $\forall 𝑡 \in [0, 1]$ , $| 𝐻 (𝑠, 𝑡) - 𝐻 (𝑠_{\infty}, 𝑡) | < 𝜀$ . By convergence, for sufficiently large $𝑛$ , $| 𝑠_{𝑛} - 𝑠_{\infty} | < 𝛿$ . Hence, there is an analytic continuation

{(𝑓_{𝑠_{\infty}, 𝑡}, 𝐷 (𝐻 (𝑠_{𝑛}, 𝑡), 𝜀 - | 𝐻 (𝑠_{𝑛}, 𝑡) - 𝐻 (𝑠_{\infty}, 𝑡) |))}_{0 \leq 𝑡 \leq 1}

along $𝐻 (𝑠_{𝑛}, [0, 1])$ , which is equivalent to ${(𝑓_{𝑠_{𝑛}, 𝑡}, 𝑈_{𝑠_{𝑛}, 𝑡})}_{0 \leq 𝑡 \leq 1}$ by Lemma 13.2.2 $Lemma 13.2.2 Lemma 13.2.2$ under the same preliminary assumptions as shown previously. By the following logic, $𝑓_{𝑠_{\infty}, 1} \equiv 𝑓_{𝑠_{𝑛}, 1}$ on the intersection, and $𝑠_{\infty} \in 𝑆$ and $𝑆$ is closed. Hence, by Theorem 3.2.1.19 $Theorem 3.2.1.19 Theorem 3.2.1.19$ , $𝑆 = [0, 1]$ . Thus, $\forall 0 \leq 𝜆 \leq 1$ , $𝑓_{𝜆, 1} \equiv 𝑓_{0, 1}$ on $𝑈_{𝜆, 1} \cap 𝑈_{0, 1} ∋ 𝑄$ . ⁠ $□$

Corollary 13.3.3.1.

Let

Ω \subseteq ℂ

be simply connected, or that every closed curve in

Ω

is null-homotopic (homotopic to a constant curve, or a point) in

Ω

. Let

𝑈 \subseteq Ω

be a disk and suppose

(𝑓, 𝑈)

is an analytic function element with unrestricted continuation in

Ω

. Then

\exists! \tilde{𝑓}

that is holomorphic on

Ω

such that

\tilde{𝑓} \equiv 𝑓

𝑈

The problems arising from the complex logarithm are now understood specifically as the failure of the homotopy condition. In a set excluding albeit enclosing the origin, such as $ℂ ∖ {0}$ , curves enclosing the origin are not homotopic to the zero constant curve, and in the complex plane, the logarithm does not admit unrestricted continuation in $ℂ$ since it cannot be continued along any curve passing through the origin.