The Jordan Curve Theorem

You can only grow if you’re willing to feel awkward and uncomfortable when you try something new, Brian Tracy

Introduction

A complex function $f(z)$ maps $z = x + iy \in \mathbb{C}$ to another complex number. For example: $f(z) = z^2 = (x + iy)^2 = x^2 - y^2 + 2ixy, f(z) = \frac{1}{z}, f(z) = \sqrt{z^2 + 7}$.

A contour is a continuous, piecewise-smooth curve defined parametrically as: $z(t) = x(t) + iy(t), \quad a \leq t \leq b$.

Definition (Smooth Contour Integral). Let ᵞ be a smooth contour (a continuously differentiable path in the complex plane), $\gamma: [a, b] \to \mathbb{C}$. Let $f: \gamma^* \to \mathbb{C}$ be a continuous complex-valued function defined on the trace $\gamma^*$ of the contour (i.e. along the image of $\gamma$). Then, the contour integral of f along $\gamma$ is defined as $\int_{\gamma} f(z)dz := \int_{a}^{b} f(\gamma(t)) \gamma^{'}(t)dt$.

Properties

• Cauchy Integral Formula. If a function f is analytic in a simply connected domain D and γ is a simply closed contour (positive orientated) in D. Then, for any point $z_0$ inside γ we have $f(z_0) = \frac{1}{2\pi i}\cdot \int_{\gamma} \frac{f(z)}{z-z_0}dz$ (Figure D).

• The Residue Theorem states that the integral of a function f(z) around a simple (doesn't cross itself), and positively oriented (counter-clockwise) contour γ is equal to 2πi times the sum of the residues of all singularities inside the contour, $\int_{\gamma} f(z)dz = 2\pi i \sum Res(f, z_k)$ where $z_k$ is a reside inside γ. f must be analytic inside the contour γ, except for a finite number of isolated singularities.

1. For a simple pole at a point $z_k$ (i.e., the denominator has a simple root) is calculated using the formula $Res(f, z_k) = \lim_{z \to z_k}(z-z_k)f(z)$.
2. For a pole of order m, $Res(f, z_k) = \frac{1}{(m-1)!}\lim_{z \to z_k} \frac{d^{m-1}}{dz^{m-1}}[(z-z_k)^mf(z)]$

• Proposition. Linearity of the contour integral, $\int_{\gamma} (af(z) + bg(z))dz = a\int_{\gamma} f(z)dz + b\int_{\gamma} g(z)dz$
• Proposition. Properties of contour integrals under path manipulation. Let γ be a contour, defined by a function $\gamma: [a, b] \to \mathbb{C}$ and let f be a continuous functions $f: \gamma^* \to \mathbb{C}$ defined along the image of ᵞ. Then,

1. Reversal of orientation. $\int_{-\gamma} f(z)dz = -\int_{\gamma} f(z)dz$ where $-\gamma:[a,b] \to \mathbb{C}$ is the reverse of the contour (traversing the same path in the opposite direction) defined by $(- \gamma)(t)=\gamma(a+b-t)$. This property states that reversing the direction of a contour changes or flips the sign of the integral. Intuitively, this is analogous to how reversing the limits of integration in real analysis changes the sign: $\int_a^b f(x)dx = -\int_b^a f(x)dx$. In complex analysis, the orientation of the contour matters because the integral depends on the direction in which we traverse the path.
2. Additivity under subdivision. Suppose a < c < b, let split $\gamma$ into two sub-contours $\gamma_1 = \gamma|_{[a, \gamma_1]} \text{ and } \gamma_2 = \gamma|_{[c, b]}$, then $\int_{\gamma} f(z)dz = \int_{\gamma_1} f(z)dz + \int_{\gamma_2} f(z)dz$. This property states that integrating over the whole contour is the same as integrating over the pieces successively. This is the complex analogue of the additive property of definite integrals: $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx $ for a < c < b.
3. Invariance of Contour Integrals Under Reparameterization. Let $\tilde{\gamma}$ be a contour, defined by a function $\tilde{\gamma}: [c, d] \to \mathbb{C}$. If $\tilde{\gamma}$ is another parameterization of the same oriented path $\gamma$ meaning there exists a one-to-one, continuously differentiable map $\psi: [c,d]\to[a,b]$ with a positive derivative $\psi'(t)>0$ such that $\tilde{\gamma}(t) = \gamma(\psi(t))$, then $\int_{\gamma} f(z)dz = \int_{\tilde{\gamma}} f(z)dz$

• Deformation of Contours. If two contours $\gamma_1$ and $\gamma_2 $ are homotopic (i.e., one can be continuously deformed into the other without crossing any singularities of f) in a domain where f(z) is analytic, then: $\int_{\gamma_1} f(z)dz = \int_{\gamma_2} f(z)dz.$

• Fundamental Theorem of Calculus for Contours. Suppose $\gamma$ is a contour (piecewise smooth path) from a to b, f is defined on a domain D containing $\gamma^*$ (the image of $\gamma$) and admits a primitive (antiderivative) F on D (i.e., $F'(z) = f(z)$), then $\int_{\gamma} f(z)dz = F(\gamma(b)) - F(\gamma(a))$. In particular, if $\gamma$ is a closed contour (i.e., $\gamma(a)=\gamma(b)$), this integral evaluates to zero, $\int_{\gamma} f(z)dz = 0.$

• Estimation Theorem or the Triangle Inequality for Integrals. The triangle inequality for integrals in complex analysis states thatfor any continuous complex function $f:[a,b] \to \mathbb{C}$ on a closed real interval [a,b] (f(t) = u(t) + iv(t), t a real parameter), the following holds: $∣\int_a^b f(t)dt| \leq \int_a^b |f(t)|dt$.

• Estimation Lemma (ML Inequality) for contour integrals. For any continuous complex function $f:[a,b] \to \mathbb{C}$ on a closed real interval [a,b] (f(z) = u(x, y) + iv(x, y)) with f bounded by some constant M along the entire contour, |f(z)| ≤ M for all $z \in \gamma^*$ (the image/trace of the contour in the complex plane), the following holds: $∣\int_\gamma f(z)dz| \leq M \cdot l(\gamma)$ where l(γ) is the arc length of the contour γ given by $\int_a^b |\gamma^{'}(t)|dt = \int_a^b \sqrt{x'(t)^2 + y'(t)^2} \text{ where } \gamma(t) = x(t) + iy(t)$.

Jordan’s curve theorem

In the plane, closed loops that don’t cross themselves behave like “boundaries.” Jordan’s curve theorem makes this precise: every such loop splits the plane into an “inside” and an “outside,” and the loop is the common boundary. This simple-looking fact is surprisingly deep; its full proof is non-trivial and sits at the foundation of topology and complex analysis.

Definition. A simple closed curve (Jordan curve) is a continuous loop that does not intersect itself and starts and ends at the same point.

Alternative definition. A map $\gamma:[a,b]\to\mathbb{R}^2$ (or ($\mathbb{C})$) is a simple closed curve if:

1. Closed: The curve starts and ends at the same point, $\gamma(a)=\gamma(b)$.
2. Simple, meaning $\gamma$ is injective on [a,b). The curve does not intersect itself anywhere except at its starting and ending points (which are the same).
3. Continuous: The curve is continuous, meaning it can be drawn without lifting the pencil from the paper.

Examples and non-examples (Figures 1 & 2)

• Simple closed curves: circles, ellipses, polygons (triangle, square, arbitrary simple polygon), and wiggly non-self-intersecting loops.

• Not a simple closed curve: A figure-eight, curve with a cusp that crosses itself (it is not a simple closed curve due to self-intersection; it consists of two separate loops) or any open curve (e.g., a segment, a ray) — these fail injectivity on [a,b) or fail to be closed.

Figure 1 (left block): examples of simple closed curves. Figure 2 (right block): not simple.

Definition. A region or set $U\subset\mathbb{C}$ is connected if any two points of U can be joined by a continuous path lying entirely in the region or set U.

A domain D is simply connected if it is connected and every closed loop in D can be continuously shrunk to a point inside D.

Jordan’s curve theorem. Any simple closed curve (a continuous loop in the plane that does not intersect itself) separates the plane into two components or disjoint connected regions: one interior (bounded) and one exterior (unbounded). The curve itself is the boundary of both regions. In other words, it partitions the plane into exactly three disjoint sets:

1. The Curve itself ($\gamma$) is the boundary between these two regions or components.
2. The Interior or inside ($\mathcal{Int}(\gamma), \mathcal{I}(\gamma)$) is the finite area enclosed by the curve (e.g., if you draw a circle on a piece of paper, the interior region would be everything inside the circle). It is a bounded, simply connected region, inside of γ.
3. The Exterior or outside ($\mathcal{Ext}(\gamma), \mathcal{O}(\gamma)$) is the infinite area outside the curve (e.g., using the same circle example, the exterior region would include all points on the paper that are not inside the circle). It is an unbounded region.

The Jordan’s curve theorem leads directly to several critical concepts:

1. Every path from inside to outside crosses the curve.
2. The “Inside” and “Outside” are Well-Defined. For any simple closed curve, the concepts of “inside” and “outside” are mathematically rigorous and unambiguous. Every point lies either on the curve, strictly inside, or strictly outside.
3. The Complement is Disconnected: The plane minus the curve ($\mathbb{C}∖ \gamma$) is a disconnected space, with the interior and exterior being its two connected components, $\mathbb{C}∖ \gamma = Int(\gamma) \cup Ext(\gamma)$
4. It guarantees that the winding number is a powerful and well-defined tool for analyzing points relative to a curve.

Example. Let $\gamma: [0, 2\pi] \to \mathbb{C}$ be defined as $\gamma(t) = \cos(t) + i\sin(t), 0 \le t \le 2\pi$. This parameterization describes the unit circle in the complex plane. The interior of the curve, denoted as $\mathcal{Int}(\gamma)$, is given by $\mathcal{Int}(\gamma) = \{ z: |z| \lt 1 \}$ which represents all complex numbers whose magnitudes are less than 1.Conversely, the exterior of the curve, denoted as $\mathcal{Ext}(\gamma)$, is defined as $\mathcal{Ext}(\gamma) = \{ z: |z| \gt 1 \}$ encompassing all complex numbers with magnitudes greater than 1.

Definition. A simple closed contour γ is positively orientated if, as you traverse it, the bounded interior region (Int(γ)) is always to (stays on or appears to) your left (tracing the curve counterclockwise places the interior on the left). Conversely, it is negatively orientated if the interior is to your right (Figure 3).

Imagine walking along a circular path. If you are walking counterclockwise and the garden (the interior) is on your left, you are positively oriented. If the garden is on your right, you are walking clockwise (negatively oriented).

Definition. For a positively oriented contour, the “inside” is the bounded component (Int(γ)). For a negatively oriented contour, the “inside” is the unbounded component (Ext(γ)).

However, in most practical applications, especially in complex analysis (e.g., Cauchy’s Integral Theorem), we standardize our curves to be positively oriented. Therefore, unless stated otherwise, “the inside of a simple closed contour” universally refers to the bounded region.