The real problem of humanity is the following: We have Paleolithic emotions, medieval institutions and godlike technology. And it is terrifically dangerous, and it is now approaching a point of crisis overall, Edward O. Wilson.

Topology and Limits

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

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).

As we have demonstrated earlier, if a function $f$ is analytic, it is automatically infinitely differentiable, i.e., the expression $f^{(n)}(a)$ (the $n$-th derivative at $a$) makes sense. The Cauchy Integral Formula for Derivatives gives us a way to calculate $f^{(n)}(a)$ using an integral: $f^{(n)}(a) = \frac{n!}{2\pi i} \oint_{C_r} \frac{f(w)}{(w-a)^{n+1}} dw$

Proof.

Fix a point z inside the disk B(a; R). We choose a radius r such that z sits inside the circle $C_r$, but $C_r$ is still inside the main disk, |z - a| < r < R.

According to Cauchy’s Integral Formula, the value of the function at z is determined by its values on the boundary circle, $C_r: f(z) = \frac{1}{2\pi i}\int_{C_r} \frac{f(w)}{w-z}dw$

We use an algebraic trick. $\frac{1}{w-z} = \frac{1}{(w-a) - (z-a)}$.

Factor out (w -a) from the denominator: $\frac{1}{w-z} = \frac{1}{w-a}\cdot \left[\frac{1}{1-\frac{z-a}{w-a}}\right]$

Remember the geometric series formula: $\frac{1}{1-q} = \sum_{n=0}^{\infty} q^n$, which is valid if $|q| < 1$. In our case, $q = \frac{z-a}{w-a}$. Because z is inside the circle and w is on the circle, $|z - a| < |w -a| = r, \forall w \in C_r, |\frac{z-a}{w-a}| \lt 1$.

We can now expand the bracketed term: $\frac{1}{w-z} = \frac{1}{w-a} \sum_{n=0}^{\infty} \frac{(z-a)^n}{(w-a)^n}$

Substitute this series back into our integral $f(z) = \frac{1}{2\pi i}\int_{C_r} \frac{f(w)}{w-z}dw = \frac{1}{2\pi i}\int_{C_r} \left( f(w)\cdot \frac{1}{w-a} \cdot \sum_{n=0}^{\infty} \frac{(z-a)^n}{(w-a)^n} \right) dw = \frac{1}{2\pi i}\oint_{C_r} \left( f(w) \sum_{n=0}^{\infty} \frac{(z-a)^n}{(w-a)^{n+1}} \right) dw$

$|w -a| = r, \forall w \in C_r$, Since f is analytic on an open disk B(a; R), it is continuous on the compact circle $C_r$, so it is bounded by some maximum value M, $\left| \frac{f(w)}{(w-a)^{n+1}} (z-a)^n \right| \le \frac{M}{r^{n+1}} |z-a|^n = \frac{M}{r} \left( \frac{|z-a|}{r} \right)^n$. Let’s define $M_n:= \frac{M}{r}(\frac{|z-a|}{r})^n$.

Since $\frac{|z-a|}{r} < 1$ (|z - a| < r), the sum of these bounds $\sum_{n=0}^{\infty} M_n$ is a convergent geometric series.

Lemma 1. Let $\gamma$ be a path, let $u_k(z)$ be a sequence of continuous function on $\gamma*$. If we can find a set of positive constants $M_k$ such that:

Bounding Condition. $|u_k(z)| \le M_k, \forall z \in \gamma*$ (each term is “dominated” by a constant).
Convergence of Bound. $\sum_{k=0}^\infin M_k$ converges.

Then, $\sum_{k=0}^\infin \int_\gamma u_k(z)dz$ converges uniformly to a continuous function U(z), and we can legally swap the integral and the sum: $\sum_{k=0}^\infin \int_\gamma u_k(z)dz = \int_\gamma \sum_{k=0}^\infin u_k(z)dz = \int_\gamma U(z)dz$.

By lemma 1, because the series is dominated by a convergent geometric series, it converges uniformly. Therefore, we are allowed to swap the integral and the sum, $f(z) = \sum_{n=0}^\infin \left[ \frac{1}{2\pi i} \int_{C_r} \frac{f(w)}{(w-a)^{n+1}}dw \right] (z-a)^n =[\text{By the Cauchy Integral Formula for Derivatives}] \sum_{n=0}^\infin \frac{f^{(n)}(a)}{n!} (z-a)^n$

Taylor's Theorem in Complex Analysis

Introduction

Taylor’s Theorem