update

pages/Math416/Math416_L7.md

@@ -130,21 +130,23 @@ We know $c_n r^ne^{in\theta}\to 0$ as $n\to\infty$.
 
 So there exists $M\geq|c_n r^ne^{in\theta}|$ for all $n\in\mathbb{N}$.
 
-So $\forall \zeta\in\overline{B(0,r)}$, $|c_n\zeta^n|\leq |c_n|e^n\leq M(\frac{e}{r})^n$.
+So $\forall \zeta\in\overline{B(0,r)}$, $|c_n\zeta^n|\leq |c_n| |\zeta|^n \leq M \left(\frac{|\zeta|}{r}\right)^n$.
 
 So $\sum_{n=0}^{\infty}|c_n\zeta^n|$ converges absolutely.
 
 So the series converges absolutely and uniformly on $\overline{B(0,r)}$.
 
-_Some steps are omitted._
+If $|\zeta| > r$, then $|c_n \zeta^n|$ does not tend to zero, and the series diverges.
 
 EOP
 
-There are few cases for the convergence of the power series.
+#### Possible Cases for the Convergence of Power Series
 
-Let $E=\{\zeta\in\mathbb{C}: \sum_{n=0}^{\infty}c_n(\zeta-\zeta_0)^n\text{ converges}\}$.
+1. **Convergence Only at $\zeta = 0$**:
+   - **Proof**: If the power series $\sum_{n=0}^{\infty} c_n (\zeta - \zeta_0)^n$ converges only at $\zeta = 0$, it means that the radius of convergence $R = 0$. This occurs when the terms $c_n (\zeta - \zeta_0)^n$ do not tend to zero for any $\zeta \neq 0$. The series diverges for all $\zeta \neq 0$ because the terms grow without bound.
 
-1. It cloud only converge at $\zeta=0$.
-2. It could converge everywhere.
+2. **Convergence Everywhere**:
+   - **Proof**: If the power series converges for all $\zeta \in \mathbb{C}$, the radius of convergence $R = \infty$. This implies that the terms $c_n (\zeta - \zeta_0)^n$ tend to zero for all $\zeta$. This can happen if the coefficients $c_n$ decrease rapidly enough, such as in the exponential series.
 
-Continue next time.
+3. **Convergence Within a Finite Radius**:
+   - **Proof**: For a power series with a finite radius of convergence $R$, the series converges absolutely and uniformly for $|\zeta - \zeta_0| < R$ and diverges for $|\zeta - \zeta_0| > R$. On the boundary $|\zeta - \zeta_0| = R$, the series may converge or diverge depending on the specific series. This is determined by the behavior of the terms on the boundary.

pages/Math416/Math416_L8.md

@@ -0,0 +1,125 @@
+# Lecture 8
+
+## Review
+
+### Sequences of Functions
+
+Let $f_n: G \to \mathbb{C}$ be a sequence of functions.
+
+#### Convergence Pointwise
+
+Definition:
+
+Let $\zeta\in G$, $\forall \epsilon > 0$, $\exists N \in \mathbb{N}$ such that $\forall n \geq N$, $|f_n(\zeta) - f(\zeta)| < \epsilon$.
+
+#### Convergence Uniformly
+
+Definition:
+
+$\forall \epsilon > 0$, $\forall \zeta\in G$, $\exists N \in \mathbb{N}$ such that $\forall n \geq N$, $|f_n(\zeta) - f(\zeta)| < \epsilon$.
+
+#### Convergence Locally Uniformly
+
+Definition:
+
+$\forall \epsilon > 0$, $\forall \zeta\in G$, $\exists N \in \mathbb{N}$ such that $\forall n \geq N$, $|f_n(\zeta) - f(\zeta)| < \epsilon$.
+
+#### Convergence Uniformly on Compact Sets
+
+Definition: $\forall C\subset G$ that is compact, $\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ s.t. } \forall n \geq N, \forall \zeta\in C, |f_n(\zeta) - f(\zeta)| < \epsilon$
+
+#### Power Series
+
+Definition:
+
+$$
+\sum_{n=0}^{\infty} c_n (\zeta - \zeta_0)^n
+$$
+
+$\zeta_0$ is the center of the power series.
+
+#### Theorem of Power Seriess
+
+If a power series converges at $\zeta_1$, then it converges absolutely at every point of $\overline{B(0,r)}$ that is strictly inside the disk of convergence.
+
+## Continue on Power Series
+
+### Limits of Power Series
+
+#### Theorem 5.12
+
+Cauchy-Hadamard Theorem:
+
+The radius of convergence of the power series is given by $\sum_{n=0}^{\infty} a_n (\zeta - \zeta_0)^n$ is given by
+
+$$
+\frac{1}{R} = \limsup_{n\to\infty} |a_n|^{1/n}
+$$
+
+Proof:
+
+Suppose $(b_n)^{\infty}_{n=0}$ is a sequence of real numbers such that $\lim_{n\to\infty} b_n$ may nor may not exists by $(-1)^n(1-\frac{1}{n})$.
+
+The limit superior of $(b_n)$ is defined as
+
+$$
+s_n = \sup_{k\geq n} b_k
+$$
+
+$s_n$ is a decreasing sequence, by completeness of $\mathbb{R}$, every bounded sequence has a limit in $\mathbb{R}$.
+
+So $s_n$ converges to some limit $s\in\mathbb{R}$.
+
+Without loss of generality, this also holds for infininum of $s_n$.
+
+Forward direction:
+
+We want to show that the radius of convergence of $\sum_{n=0}^{\infty} a_n (\zeta - \zeta_0)^n$ is greater than or equal to $\frac{1}{\limsup_{n\to\infty} |a_n|^{1/n}}$.
+
+Since $\sum_{n=0}^{\infty} 1\zeta^n=\frac{1}{1-\zeta}$ for $|\zeta|<1$. Assume $\limsup_{n\to\infty} |a_n|^{1/n}$ is finite, then $\sum_{n=0}^{\infty} a_n (\zeta - \zeta_0)^n$ converges absolutely at $\zeta_0$.
+
+Let $\rho>\limsup_{n\to\infty} |a_n|^{1/n}$, then $\exists N \in \mathbb{N}$ such that $\forall n \geq N$, $|a_n|^{1/n}\leq \rho$.
+
+So $\frac{1}{R}=\limsup_{n\to\infty} |a_n|^{1/n}<\rho$
+
+So $R>\frac{1}{\rho}$
+
+/*TRACK LOST*/
+
+Backward direction:
+
+Suppose $|\zeta|>R$, then $\exists$ number $|\zeta|$ such that $|\zeta|>\frac{1}{\rho}>R$.
+
+So $\rho<\limsup_{n\to\infty} |a_n|^{1/n}$
+
+This means that $\exists$ infinitely many $n_j$s such that $|a_{n_j}|^{1/n_j}>\rho$
+
+So $|a_{n_j}\zeta^{n_j}|>\rho^{n_j}|\zeta|^{n_j}$
+
+Series $\sum_{n=1}^{\infty} a_n\zeta^n$ diverges, each individual term is not going to $0$.
+
+So $\sum_{n=0}^{\infty} a_n (\zeta - \zeta_0)^n$ does not converge at $\zeta$
+
+EOP
+
+_What if $|\zeta-\zeta_0|=R$?_
+
+For $\sum_{n=0}^{\infty} \zeta^n$, the radius of convergence is $1$.
+
+It diverges eventually on the circle of convergence.
+
+For $\sum_{n=0}^{\infty} \frac{1}{(n+1)^2}\zeta^n$, the radius of convergence is $1$.
+
+This converges everywhere on the circle of convergence.
+
+For $\sum_{n=0}^{\infty} \frac{1}{n+1}\zeta^n$, the radius of convergence is $1$.
+
+This diverges at $\zeta=1$ (harmonic series) and converges at $\zeta=-1$ (alternating harmonic series).
+
+#### Theorem 5.15
+
+Suppose $\sum_{n=0}^{\infty} a_n (\zeta - \zeta_0)^n$ has a positive radius of convergence $R$. Define $f(\zeta)=\sum_{n=0}^{\infty} a_n (\zeta - \zeta_0)^n$, then $f$ is holomorphic on $B(0,R)$ and $f'(\zeta)=\sum_{n=1}^{\infty} n a_n (\zeta - \zeta_0)^{n-1}=\sum_{k=0}^{\infty} (k+1)a_{k+1} (\zeta - \zeta_0)^k$.
+
+Proof:
+
+/*TRACK LOST*/

pages/Math416/_meta.js

@@ -10,4 +10,5 @@ export default {
     Math416_L5: "Complex Variables (Lecture 5)",
     Math416_L6: "Complex Variables (Lecture 6)",
     Math416_L7: "Complex Variables (Lecture 7)",
+    Math416_L8: "Complex Variables (Lecture 8)",
 }