Mistakes are the best teacher when one is humble and honest enough to admit them and willing to learn and grow from them, Anawim, #justtothepoint.

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Introduction

Definition. Complex sequence A sequence of complex numbers is a function $a: \mathbb{N} \to \mathbb{C}$. We usually denote it by $(a_n)_{n \in \mathbb{N}}$ or simply $(a_n)$, where $a_n := a(n)$. The value $a_1$ is called the first term of the sequence, $a_2$ the second term, and in general $a_n$ the n-th term of the sequence.

Definition. Convergent complex sequence. A complex sequence $(a_n)_{n \in \mathbb{N}}$ is said to converge to a complex number $L \in \mathbb{C}$ if for every $\varepsilon > 0$ there exists an integer $N \in \mathbb{N}$ such that for all $n \geq N$ one has $|a_n - L| < \varepsilon$. In this case we write $\lim_{n \to \infty} a_n = L$ or $a_n \to L$ as $n \to \infty$, and L is called the limit of the sequence $(a_n)_{n \in \mathbb{N}}$.

Definition. Cauchy sequence. A complex sequence $(a_n)_{n \in \mathbb{N}}$ is called a Cauchy sequence if for every $\varepsilon > 0$ there exists an integer $N \in \mathbb{N}$ such that for all $n, m \geq N$ one has $|a_n - a_m| < \varepsilon$.

Definition. Series and partial sums.Let $(a_n)_{n \in \mathbb{N}}$ be a complex sequence. For each n $\in \mathbb{N}$, the finite sum $s_n := a_1 + a_2 + \cdots + a_n = \sum_{k=1}^n a_k$ is called the n-th partial sum of the (infinite) series $\sum_{k=1}^\infin a_k$ which we also denote simply by $\sum a_n$ when the index is clear from the context.

Definition. Convergent series. The series $\sum_{n=1}^{\infty} a_n$ is said to converge to the sum $s \in \mathbb{C}$ if the sequence of partial sums $(s_n)_{n \in \mathbb{N}}$ defined by $s_n = a_1 + a_2 + \cdots + a_n = \sum_{k=1}^n a_k$ converges to s, that is, $\lim_{n \to \infty} s_n = s$. In this case we write $s := \sum_{n=1}^\infin a_n$. If the sequence $(s_n)_{n \in \mathbb{N}}$ does not converge, we say that the series $\sum_{n=1}^{\infty} a_n$ diverges (or does not converge).

Definition. A complex power series centered at 0 in the variable z is a series of the form $a_0 + a_1z + a_2z^2 + \cdots = \sum_{n=0}^\infty a_n z^n$ with coefficients $a_i \in \mathbb{C}$

Definition. A complex power series centered at a complex number $a \in \mathbb{C} $ is an infinite series of the form: $\sum_{n=0}^\infty a_n (z - a)^n,$ where each $a_n \in \mathbb{C}$ is a coefficient, z is a complex variable, and $(z - a)^n$ is the nth power about the center.

Theorem. Given a power series $\sum_{n=0}^\infty a_n z^n$, there exists a unique value R, $0 \le R \le \infin$ (called the radius of convergence) such that:

For any z with |z| < R (inside the circle), the series $\sum_{n=0}^\infty a_n z^n$ converges absolutely (this is a “green light” zone).
For any z with |z| > R, the series diverges (this is a “red light” zone).
On the Circle (|z| = R), this theorem gives no information. This is the yellow light zone —the series could converge or diverge.

Differentiability of Power Series. If $f(z) = \sum_{n=0}^{\infty} a_nz^n$ for |z| < R (R > 0), then f is analytic on B(0; R) and $f'(z) = \sum_{n=1}^{\infty} na_nz^{n-1}$ for |z| < R.

Weierstrass M-test. Let $\{u_k(z)\}_{k=0}^\infty$ be a sequence of complex-valued functions defined on a set $\gamma^* \subseteq \mathbb{C}$. If there exists a sequence of non-negative real numbers $\{M_k\}_{k=0}^\infty$ such that:

Bounding Condition: $|u_k(z)| \leq M_k$ for all $z \in \gamma^*$ and all $k \in \mathbb{N}$
Convergence of Bound: The series $\sum_{k=0}^\infty M_k$ converges

Then, the original series $\sum_{k=0}^\infty u_k(z)$ converges uniformly on $\gamma^*$.

Coefficients of power series. Let f(z) = $\sum_{k=0}^\infty c_kz^k$ where this power series has radius of convergence R > 0. Then,the n-th coefficient of a power series $c_n$ can be extracted using the integral formula, $c_n = \frac{1}{2\pi i} \int_{C_r} \frac{f(z)}{z^{n+1}}dz, 0 \le r \lt R, n \ge 0$ where $C_r$ is a circle of radius r centered at 0 and oriented positively.

Taylor’s Theorem. If f is analytic on an open disk B(a; R) (a disk of radius R centered at a), then f(z) can be represented exactly by a unique power series within that disk: $f(z) = \sum_{n=0}^{\infty}a_n (z - a)^n, \forall z \in B(a; R)$

This theorem bridges the gap between differentiability and power series. It guarantees that if a function behaves well (it is analytic) in a disk, it must also be infinitely differentiable and expressed or representable by a power series (an infinite polynomial) within that disk.

Furthermore, there exist unique constants $a_n = \frac{f^{(n)}(a)}{n!} = \frac{1}{2\pi i}\int_{C_r} \frac{f(w)}{(w-a)^{n+1}}dw$ where $C_r$ is a circle of radius r < R centered at a and oriented in the counterclockwise direction (positively oriented).

Maximum Modulus, Winding Number, and the Argument Principle

Theorem. Local maximum modules theorem. Suppose that f is analytic on an open disk B(a; R). If $|f(z)| \le |f(a)|, \forall z \in \mathbb{B}(a; R)$ (meaning f has a local maximum at the center a), then f must be a constant function on that disk. In words, a non-constant analytic function cannot have a maximum at the center of a disk of analycity. Non-constant analytic functions cannot have a “local maximum” of their modulus inside the domain; maxima occur only on the boundary.

Think of the “graph” of the absolute value |f(z)| as a surface in 3D space. This theorem states that for an analytic function, this surface cannot have a “mountain peak” (a local maximum) at the center of a disk of analycity.

Proof.

First, we express the value at the center, f(a), as the average of the values surrounding it.

Fix a radius $r, \text{ such that } 0 \lt r \lt R$. By Cauchy Integral Formula, f(a) = $\frac{1}{2\pi i} \int_{C_r} \frac{f(z)}{z-a}dz$ where $C_r$ is a circle of radius “r” centered at “a” oriented positively.

We parametrize the circle $C_r, z = a + re^{i\theta}, 0 \le \theta \le 2\pi, dz = ire^{i\theta}d\theta$ and substitutes these into our integral:

f(a) = $\frac{1}{2\pi i} \int_{C_r} \frac{f(z)}{z-a}dz = \frac{1}{2\pi i} \int_0^{2\pi} \frac{f(a + re^{i\theta})}{re^{i\theta}}rie^{i\theta}d\theta = \frac{1}{2\pi} \int_0^{2\pi}f(a + re^{i\theta}) d\theta$ where the terms $i, r, e^{i\theta}$ cancel out perfectly.

Now, we look at the magnitude (modulus) of both sides:

$$ \begin{aligned} |f(a)| &=\left| \frac{1}{2\pi} \int_0^{2\pi} f(a + re^{i\theta}) d\theta \right| \\[2pt] &\text{Move the absolute value inside the integral. Recall that } |\int g| \le \int |g| \\[2pt] &\le \frac{1}{2\pi} \int_0^{2\pi} |f(a + re^{i\theta})| d\theta \\[2pt] &\text{Now, use the Theorem's Hypothesis.} |f(z)| \le |f(a)|, \forall z \in \mathbb{B}(a; R) \\[2pt] &\le \frac{1}{2\pi} \int_0^{2\pi} |f(a)|d\theta \\[2pt] &=\frac{1}{2\pi} |f(a)| \cdot (2\pi - 0) = |f(a)|. \end{aligned} $$

We have just sandwiched the integral between $|f(a)|$ and $|f(a)|, |f(a)| \le \frac{1}{2\pi} \int_0^{2\pi} |f(a + re^{i\theta})| d\theta \le |f(a)|$

Therefore, equality must hold everywhere: $|f(a)| = \frac{1}{2\pi} \int_0^{2\pi} |f(a + re^{i\theta})| d\theta$. Rearranging the equation, we get: $|f(a)| - \frac{1}{2\pi}\int_0^{2\pi} |f(a + re^{i\theta})| d\theta = 0$.

Move |f(a)| inside the integral (as a constant, it equals $\frac{1}{2\pi}\int_0^{2\pi} |f(a)| d\theta$): $\frac{1}{2\pi} \int_0^{2\pi} \left( |f(a)| - |f(a + re^{i\theta})|\right) d\theta = 0$

f is analytic, so |f| is continuous. By hypothesis, $|f(a)|$ is the maximum, so $|f(a)| \ge |f(z)|$. Therefore, $(|f(a)| - |f(z)|) \ge 0$. If you integrate a continuous, non-negative function and the result is 0, the function itself must be identically 0 everywhere. $|f(a)| - |f(a + re^{i\theta})| = 0 \implies |f(a + re^{i\theta})| = |f(a)|$

Since this was true for any circle of radius r inside the open disk B(a; r), the modulus $|f(z)|$ is constant throughout the entire disk.

We have proved that $|f(z)| = c$ (a constant). Does this mean $f(z)$ is constant?

If c = 0, then $f(z)=0$ everywhere (constant).
If $c \ne 0$, we use the Cauchy-Riemann equations. Write f(z) = u(x, y) + iv(x, y). Since $|f|^2 = u^2 + v^2 = c^2$, taking partial derivatives with respect to x and $y$ leads to a system of equations $2uu_x+2vv_x=0 \Rightarrow uu_x+vv_x=0 \text{ (1) }, 2uu_y+2vv_y=0 \Rightarrow uu_y+vv_y=0$ (2).

Apply Cauchy–Riemann equations: $u_x=v_y, u_y=-v_x$, we get: From (1): $uu_x + vv_x = 0 \implies uu_x - vu_y = 0$ (using $v_x = -u_y$). From (2): $uu_y + vv_y = 0 \implies uu_y + vu_x = 0$ (using $v_y = u_x$)

This gives us the system: $\begin{pmatrix} u & -v \\\ v & u \end{pmatrix} \begin{pmatrix} u_x \\\ u_y \end{pmatrix} = \begin{pmatrix} 0 \\\ 0 \end{pmatrix}$

The determinant of this matrix is $u^2 + v^2 = c^2 \neq 0$, so the matrix is invertible and the only solution is $u_x = u_y = 0$. By the Cauchy–Riemann equations, this forces $v_x=v_y=0$ as well. All partial derivatives vanish, so both u and v are constant functions. Therefore f(z) is constant. Q.E.D.

Maximum modulus principle. Let G a bounded region. Let f be analytic in G and continous on the closure of G, $\bar{G}$. Then, |f| attains its maximum on the boundary of G, $\partial G := \bar{G} \setminus G$.

Lemma. If $\gamma$ is a piecewise differentiable closed curve (not necessarily simple) not passing through the point a, $a \in \mathbb{C}$, then the value of the contour integral $\int_{\gamma} \frac{dz}{z-a}$ is an integer multiple of $2\pi i$.

Recall that the antiderivative of $\frac{1}{z-a}$ is $\log(z-a)$. However, the complex logarithm is multi-valued, so the integral measures the net change in the argument (angle) of z - a as z traces the curve $\gamma$. Every time you make a full circle around a, the angle (argument) of the logarithm increases by $2\pi$. If you loop once, the value changes by $2\pi i$. If you loop twice, it changes by $4\pi i$, and so on. Therefore, the total change must be $k \cdot 2\pi i$ for some integer k.

We normalize the integral by dividing by $2\pi i$ to get that integer k.

Definition. The index or the winding number of a closed curve $\gamma$ with respect a point a is the integer defined as, $n(\gamma; a) := \frac{1}{2\pi i} \int_{\gamma} \frac{dz}{z-a}$.

The winding number counts how many times a closed curve $\gamma$ wraps counterclockwise around a specific point a. $n(\gamma; a) = 1$: The curve wraps once counterclockwise. $n(\gamma; a) = -1$: The curve wraps once clockwise. $n(\gamma; a) = 0$: The point a is “outside” the loop.

Lemma. Let $\gamma: [a, b] \to \mathbb{C}$ be a closed curve, i.e., a continuous function with $\gamma(a) = \gamma(b)$. Then, its trace $\gamma^* = \gamma$([a, b]) is a compact subset of $\mathbb{C}$ (hence, closed and bounded set). Consequently, the complement $\mathbb{C} \setminus \gamma^*$ is an open subset of $\mathbb{C}$ (it ensures that every point not on the curve has a small neighborhood entirely contained in the complement).

Imagine drawing a closed loop on a piece of paper without lifting your pen —this loop is the trace $\gamma^*$. Although the curve may intersect itself, the set of all points you’ve drawn is a closed, bounded figure in the plane. The complement $\mathbb{C} \setminus \gamma^*$ —the plane with the loop removed —breaks up into one or more connected components:

This line divides the paper into distinct regions, called components:

Bounded Components (the “interiors”): These are regions completely enclosed by the loop(s) (inside them). If you were standing inside one, you could not reach the edge of the paper without crossing the drawn line. You are trapped in them.
The unbounded component (the “exterior”): $\mathbb{C} \setminus \gamma^*$: This is the “outside.” This is the unique region that extends out to infinity. From any point in this component, you can travel arbitrarily far from the origin without ever touching (or crossing) the curve.

Thus, even though $\mathbb{C} \setminus \gamma^*$ is open, it is typically not connected —its connected components reflect the topological complexity of the curve.

Properties on $n(\gamma; a)$:

Reversing the Curve. $n(-\gamma; a) = -n(\gamma; a)$
The “Far Away” Point. If $\gamma$ is inside a circle (disk), then $n(\gamma; a) = 0$ for all points a outside of that circle.
$n(\gamma; a)$ is a constant function on each component of $\mathbb{C} \setminus \gamma^*$ and is the zero function on the unbounded component of $\mathbb{C} \setminus \gamma^*$.

Proof.

(1) $n(-\gamma; a) = \frac{1}{2\pi i} \int_{-\gamma} \frac{dz}{z-a} = -\frac{1}{2\pi i} \int_{\gamma} \frac{dz}{z-a} = -n(\gamma; a)$

If $-\gamma$ denotes the same geometric curve as $\gamma$ but traversed in the opposite direction, then: $\int_{-\gamma }f(z) dz = -\int_{\gamma }f(z) dz.$

(2) If $\gamma$ is inside a circle (disk) D, then for any point a outside of the circle, $\frac{1}{z-a}$ is an analytic function inside the circle (the denominator z - a is never zero), i.e., on the disk interior of the circle.

Cauchy’s Theorem states that the integral of an analytic function over a closed loop in a simply connected domain (e.g., our disk D) is 0, $\frac{1}{2\pi i} \int_{\gamma} \frac{dz}{z-a} = 0 \implies n(\gamma; a) = 0$.

(3) We pick two points, a and b, inside the same component (region) of $\mathbb{C} \setminus \gamma^*$. We connect them with a polygonal path connecting both points lying completely in the component of $\mathbb{C} \setminus \gamma^*$, i.e., this path does not intersect $\gamma^*$. If we can prove the winding number doesn’t change as we move slightly along this path, it is constant for the whole region.

In other words, it is enough to show that if a and b are joined by a straight line segment [a, b] connecting them which does not intersect (or touch) the curve $\gamma^*$, then $n(\gamma; a) = n(\gamma; b)$.

$\frac{z-a}{z-b}$ maps the segment joining a and b into the negative real axis. Therefore, outside the segment joining a and b, $\frac{z-a}{z-b}$ is never real and $\frac{z-a}{z-b} \le 0$ simultaneously.

Crucially, since the segment $[a, b]$ does not intersect $\gamma$, the argument of the log is never a negative real number, so the principal branch of $\log(\frac{z-a}{z-b})$ is analytic on the line segment joining a and b and $\frac{d}{dz}(log(\frac{z-a}{z-b})) = \frac{d}{dz} \left( \log(z-a) - \log(z-b) \right) = \frac{1}{z-a} - \frac{1}{z-b}, \forall z \in \mathbb{C} \setminus$ [a, b]. The function $F(z) = \log(\frac{z-a}{z-b})$ is a valid, single-valued analytic primitive (antiderivative) for points z on the curve $\gamma$.

Fundamental Theorem of Calculus: If a continuous function f(z) has an analytic primitive $F(z)$ on a closed curve $\gamma$, its integral is zero, $\oint_{\gamma} F'(z) dz = F(\text{end}) - F(\text{start}) = 0$ (Since start = end because $\gamma$ is a closed loop).

Therefore: $\int_{\gamma} (\frac{1}{z-a}-\frac{1}{z-b})dz = 0 \implies n(\gamma; a) = n(\gamma; b)$

We want to prove that for any point z in the unbounded component (let’s call this region $U_\infty$), the winding number is 0.

Because $\gamma$ is a closed path, it doesn’t stretch to infinity. $\gamma^*$ is indeed a compact set (closed and bounded). This means we can draw a giant circle $C_\rho$ (radius $\rho$) around the origin that completely swallows it, $\gamma \subset I(C_\rho)$

The unbounded component $U_\infty$ extends to infinity. This means we can find a point a in this component that is extremely far away. Specifically, we choose a point a such that $|a| > \rho$, i.e., a is outside the circle of radius $\rho$.

By the “Far away” 2 property (if $\gamma$ lies in the inside of a circle, then $n(\gamma; a) = 0$ for all points a outside of the circle), $n(\gamma; a) = 0$.

We have now proven that the winding number is 0 for one specific point a (the one really far away). But what about a point b that is in the unbounded component but close to the curve?

We proved before in Property (3a) that $n(\gamma; z)$ is a constant function on any connected component. Since $a$ and $b$ are in the same component (the unbounded one), they must have the same winding number, $n(\gamma; b) = n(\gamma; a) = 0$. Therefore, the winding number is 0 for every point in the unbounded component.

Theorem. The Principle of Argument (Counting zeroes). Let f be analytic inside and on a positively oriented closed contour γ. Assume f is never zero on the boundary $\gamma$. Let N be the total number of zeroes inside $\gamma$ including their multiplicities (orders). Then, $\frac{1}{2\pi i} \oint_{C_{\varepsilon_0}} \frac{f'(z)}{f(z)} dz = N$.

By counting zeroes theorem, $\int_{\gamma} \frac{f'(z)}{f(z)}dz = 2\pi i N$ where N is the number of zeroes of f inside $\gamma $ counting multiplicity. Let’s visualize what the integral $\frac{1}{2\pi i}\int_{\gamma} \frac{f'(z)}{f(z)}dz$ actually represents geometrically

Consider the mapping w = f(z). Consequently, using the chain rule, dw = f’(z)dz. Let $\Gamma$ be the map $f \circ \gamma$. $\gamma$ is a contour in the z-plane (the domain) and $\Gamma$ is the image of $\gamma$ in the w-plane (the range), $\Gamma = f(\gamma)$, $\Gamma$ is a closed curve in the image of f.

$\int_{\gamma} \frac{f'(z)}{f(z)} dz = \int_{\Gamma} \frac{1}{w} dw$

The integral $\frac{1}{2\pi i}\int_{\Gamma} \frac{dw}{w}$ is the definition of the Winding Number of the curve $\Gamma$ around the origin 0: $\int_{\gamma} \frac{f'(z)}{f(z)}dz = \int_{\Gamma} \frac{dw}{w} = 2\pi i N \leadsto N = \frac{1}{2\pi i}\int_{\Gamma} \frac{dw}{w} = n(\Gamma; 0)$

Interpretation: The number of zeroes of $f(z)$ (it is always counting multiplicity) inside the loop $\gamma$ is exactly equal to the number of times the image curve $\Gamma$ wraps around the origin in the $w$-plane.

Usually, we care about solving $f(z) = 0$. But what if we want to solve $f(z) = a, a \in \mathbb{C}$? This is equivalent to finding the zeroes of a new function: g(z) = f(z) - a. Let’s apply the Counting Zeroes Theorem to this new function g(z).

The zeroes of f(z) - a are the solutions to the equation f(z) = a. The number of solutions M is given by:

$$ \begin{aligned} M = \frac{1}{2\pi i} \int_{\gamma} \frac{g'(z)}{g(z)} dz &=\frac{1}{2\pi i} \int_{\gamma} \frac{f'(z)}{f(z) - a} dz \\[2pt] &\text{Substitution: Again, let w = f(z) and dw = f'(z)dz} \\[2pt] &=\frac{1}{2\pi i} \int_{\Gamma} \frac{dw}{w - a}\\[2pt] &\text{This integral is exactly the definition of the winding number of Γ around the point a} \\[2pt] &= n(\Gamma; a) \end{aligned} $$

Interpretation: The number of times f(z) hits the value a inside the domain $\gamma$ equals the number of times the image curve $\Gamma$ wraps around the point a.

If a and b belong to the same component determined by $\Gamma$, by property (3) we know that $n(\Gamma; a) = n(\Gamma; b)$, meaning the number of zeroes of f(z) - a inside $\Gamma$ is equal to the number of zeros of f(z) - b inside $\Gamma$ counting multiplicity. Therefore, the equation $f(z) = a$ and the equation $f(z) = b$ have the exact same number of solutions.

Property 3. The winding number $n(\Gamma; a)$ is constant on any connected component of $\mathbb{C} \setminus \Gamma^*$. Idea: if you draw a loop $\Gamma$ on a piece of paper, it divides the paper into regions. As long as the point a stays inside the same region (doesn’t cross the line $\Gamma$), the winding number stays the same.