The Principle of Argument and the Open Mapping Theorem

The problem is not the problem. The problem is your attitude about the problem, Captain Jack Sparrow

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:

1. 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).
2. 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:

1. Bounding Condition: $|u_k(z)| \leq M_k$ for all $z \in \gamma^*$ and all $k \in \mathbb{N}$
2. 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).

The Principle of Argument

The Argument Principle is a profound theorem that bridges complex calculus and topology. It allows us to count the exact number of zeroes a function has inside a region simply by looking at how the function behaves on the boundary of that region.

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

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: $\int_{\gamma} \frac{f'(z)}{f(z)}dz = \int_{\Gamma} \frac{dw}{w} = 2\pi i N [\text{ where }] N = \frac{1}{2\pi i}\int_{\Gamma} \frac{dw}{w} = n(\Gamma; 0)$

Interpretation: The number of zeroes of $f(z)$ (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.

Generalizing to $f(z) = a$

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 shifted function, namely 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 in the w-plane.

The Constancy of Solutions: The winding number $n(\Gamma; a)$ is constant on any connected component of $\mathbb{C} \setminus \Gamma^*$. Imagine drawing the loop $\Gamma$ on a piece of paper; it divides the paper into distinct regions. As long as you choose target values $a$ and $b$ inside the same region (without crossing the ink of $\Gamma$), $n(\Gamma; a) = n(\Gamma; b)$. 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.

Proof of the Principle of Argument

Let f be analytic on and inside a positively oriented simple closed contour $\gamma$, an assume f has no zeros on the boundary. Let the zeros of f inside $\gamma$ be $\{ a_1, \cdots, a_k \}$, with multiplicities $\{ m_1, \cdots, m_k \}$. Then, $N=m_1+\cdots+m_k.$

Step 1. Factorization of f.

Let $a_j$ be a zero of multiplicity $m_j$
for each zero $a_j$ with multiplicity $m_j$, there exists a neighborhood of $a_j$ where $f(z) = (z-a_j)^{m_j}h_j(z)$ with $h_j$ analytic and $h_j(a_j) \ne 0$.
Because zeros of analytic functions are isolated, we can choose disjoint small disks $D_j= \{ z:|z-a_j| < \varepsilon_j \}$ such that:
* each disk contains exactly one zero $a_j$,
* $f(z)\neq 0$ on each $\partial D_j$,
So inside each $D_j$, the factorization $f(z)=(z-a_j)^{m_j}h_j(z)$ is valid.
Remove all the zeros at once. Define $P(z)=\prod _{j=1}^k(z-a_j)^{m_j}, g(z)=\frac{f(z)}{P(z)}.$
* Is g analytic? Each factor $(z-a_j)^{m_j}$ cancels exactly at $a_j$. Inside each disk $D_j, g(z)=h_j(z),$ which is analytic.
Outside all the disks, f and P are analytic and $P\neq 0$, so the quotient is analytic. Thus g is analytic everywhere inside and on $\gamma$.
* Does g vanish anywhere? At each zero $a_j, g(a_j)=h_j(a_j) \neq 0$.
Outside the zeros, $P(z)\neq 0$, so $g(z)=\frac{f(z)}{P(z)}\neq 0$ because f has no other zeros.
Thus, g has no zeros inside or on $\gamma$ (the boundary that is).
* Is log g(z) analytic?. Yes, because g is analytic, g never vanishes, the interior of $\gamma$ is simply connected.
Therefore, we can choose a branch of log(g) that is analytic on the entire region.
$f(z) = \prod _{j=1}^k(z-a_j)^{m_j} \cdot g(z)$ where g is analytic on and inside $\gamma, g(z) \ne 0$ on and inside $\gamma$, hence log(g(z)) is analytic.

Step 2. Take the logarithmic derivative: $\frac{f'(z)}{f(z)} = \sum_{j=1}^k \frac{m_j}{z-a_j}+\frac{g'(z)}{g(z)}.$

Step 3. Integrate around $\gamma$.

Integrate termwise (justified because everything is analytic except for simple poles at the zeros): $\oint_\gamma \frac{f'(z)}{f(z)}\,dz = \sum_{j=1}^k m_j \oint_\gamma \frac{dz}{z-a_j} + \oint_\gamma \frac{g'(z)}{g(z)}\,dz.$

Step 4. Evaluate each term.

Since g has no zeros in the region, $\frac{g'}{g}$ is analytic there, so by Cauchy’s theorem, $\oint_\gamma \frac{g'(z)}{g(z)}\,dz=0.$

Also, each $a_j$ lies inside $\gamma$, so by the standard residue computation $\oint_\gamma \frac{dz}{z-a_j}=2\pi i.$

Therefore, $\oint_\gamma \frac{f'(z)}{f(z)}\,dz = \sum_{j=1}^k m_j (2\pi i) = 2\pi i\,N.$

Dividing by $2\pi i$, $\frac{1}{2\pi i}\oint_\gamma \frac{f'(z)}{f(z)}\,dz=N. \qquad \blacksquare$

Open mapping theorem

Using the Argument Principle, we can deduce exactly how an analytic function behaves near any point.

The Local Mapping Theorem. Suppose that f is analytic at $z_0$ with $f(z_0) = w_0$ and that the function $f(z)-w_0$ has a zero of order n at $z_0$. Then, for all sufficiently small $\varepsilon \gt 0$, there exists a corresponding $\delta \gt 0$ such that $\forall a: |a - w_0| \lt \delta ↔ a \in B(w_0; \delta) \setminus {w_0}$, the equation f(z) = a has exactly n roots inside the disk $B(z_0; \varepsilon) ↔ |z - z_0| \lt \varepsilon$.

Every point $a$ near the center (except the center itself $w_0$) comes from exactly $n$ points in the domain. Essentially, it shows that an analytic function behaves like a polynomial $w = z^n$ near a point, mapping a small neighborhood to another small neighborhood in an n-to-1 fashion. 

Proof.

Notice that analytic at a point $z_0$ means that the function is differentiable at $z_0$ and also at every point in some neighborhood (disk) surrounding $z_0$.

We are looking at the function $h(z) = f(z) - w_0$. We know $h(z_0) = 0$. Since zeroes of non-zero analytic functions are isolated, there is a small disk around $z_0$ where no other zeroes exist. In the closed disk $\overline{B(z_0; \varepsilon)}, f(z) = w_0$ only at the center $z_0$ ($z_0$ is the only zero in this closed disk).

f is analytic on the whole disk $|z-z_0| < \varepsilon$

Let $\gamma$ be the boundary circle of this disk: $|z - z_0| = \varepsilon$, a circle of center $z_0$ and radius $\varepsilon$ orientated positively. On this boundary $\gamma$, we know $f(z) \neq w_0$ (because the zero is only at the center).

Let $\Gamma$ be the image of this boundary $\Gamma = f(\gamma)$.

Because $f(z) \neq w_0$ on the boundary $\gamma$, the image curve $\Gamma$ does not pass through $w_0$. Therefore, $w_0$ is in the “open complement” $\mathbb{C} \setminus \Gamma^*$.

Since the curve $\Gamma$ is a closed set and does not touch $w_0$, there must be some “breathing room” around $w_0$ ($\mathbb{C} \setminus \Gamma^*$ is open). There exists a neighborhood around $w_0$ (a small radius $\delta \gt 0$) such that the disk $B(w_0; \delta)$ does not intersect $\Gamma$. This means the entire little disk $B(w_0; \delta)$ lies inside the same connected component as $w_0$.

Now, pick any target value inside this small disk (neighbor) $a \in B(w_0, \delta)$.

1. For the center $w_0$: We know the equation $f(z) = w_0$ has n solutions inside $\gamma$ (by assumption, it was a zero of order n). Then, $n(\Gamma; w_0) = n$
2. For the neighbor a: Since a and $w_0$ are in the same component (they are close neighbors and didn’t cross the boundary $\Gamma$), their winding numbers must be equal, $n(\Gamma; a) = n(\Gamma; w_0) = n$

The winding number $n(\Gamma; a)$ represents the number of solutions to f(z) = a. Since the winding number is $n$, there are exactly $n$ roots or solutions to the equation f(z) = a.

Why are the $n$ roots distinct?

If $f(z) - w_0$ has a zero of order $n$ at $z_0$, its derivative $f'(z)$ has a zero of order $n-1$ at $z_0$.

We can write $f(z)-w_0=(z-z_0)^ng(z)$ where g is analytic in a neighborhood of $z_0$ and $g(z_0)\neq 0 \leadsto f'(z) = n (z - z_0)^{n-1} g(z) + (z - z_0)^n g'(z) = (z - z_0)^{n-1} \bigl[ n g(z) + (z - z_0) g'(z) \bigr]$. Define $h(z) = n g(z) + (z - z_0) g'(z)$. Since g is analytic at $z_0$, h is also analytic at $z_0$. Moreover, $h(z_0) = n g(z_0) \neq 0$. $f'(z) = (z - z_0)^{n-1} h(z)$, which means that f’(z) has a zero of order n - 1 at $z_0$.

Zeroes of analytic functions are isolated. Therefore, in a sufficiently small disk around $z_0$, the derivative $f'(z)$ is never zero except at the center $z_0$. So in that disk the only critical point is $z_0$.

If we pick a target $a \neq w_0$ close enough to $w_0$, the roots of $f(z) = a$ lie in this small disk where $f'(z) \neq 0$. Suppose $z_k$ is a solution, $f(z_k) = a$ and (since $z_k \ne z_0$ because $f(z_0) = w_0 \ne a$) $f'(z_k) \neq 0$, then $z_k$ is a simple root (multiplicity 1). Since the sum of multiplicities must be n and each root counts for 1, there must be exactly $n$ distinct roots (geometric points mapped to $a$).