update notes

pages/Math416/Math416_L6.md

@@ -50,7 +50,7 @@ Which is in the form of circle equation.
 
 EOP
 
-## Chapter 4 Elements of functions
+## Chapter 4 Elementary functions
 
 > $e^t=\sum_{n=0}^{\infty}\frac{t^n}{n!}$
 

pages/Math416/Math416_L7.md

@@ -0,0 +1,150 @@
+# Lecture 7
+
+## Review
+
+### Exponential function
+
+$$
+e^z=e^{x+iy}=e^x(\cos y+i\sin y)
+$$
+
+### Logarithm
+
+#### Definition 4.9 Logarithm
+
+A logarithm of $a$ is any $b$ such that $e^b=a$.
+
+### Branch of Logarithm
+
+A branch of logarithm is a continuous function $f$ on a domain $D$ such that $e^{f(z)}=\exp(f(z))=z$ for all $z\in D$.
+
+## Continue on Chapter 4 Elementary functions
+
+### Logarithm
+
+#### Theorem 4.11
+
+$\log(\zeta)$ is holomorphic on $\mathbb{C}\setminus\{0\}$.
+
+Proof:
+
+We proved that $\frac{\partial}{\partial\overline{z}}e^{\zeta}=0$ on $\mathbb{C}\setminus\{0\}$.
+
+Then $\frac{d}{dz}e^{\zeta}=\frac{\partial}{\partial x}e^{\zeta}=0$ if we know that $e^{\zeta}$ is holomorphic.
+
+Since $\frac{d}{dz}e^{\zeta}=e^{\zeta}$, we know that $e^{\zeta}$ is conformal, so any branch of logarithm is also conformal.
+
+Since $\exp(\log(\zeta))=\zeta$, we know that $\log(\zeta)$ is the inverse of $\exp(\zeta)$, so $\frac{d}{dz}\log(\zeta)=\frac{1}{e^{\log(\zeta)}}=\frac{1}{\zeta}$.
+
+EOP
+
+We call $\frac{f'}{f}$ the logarithmic derivative of $f$.
+
+## Chapter 5. Power series
+
+If $\sum_{n=0}^{\infty}c_n$ converges, then $\lim_{n\to\infty}c_n=0$ exists.
+
+### Geometric series
+
+$$
+\sum_{n=0}^{N}c^n=\frac{1-c^{N+1}}{1-c}
+$$
+
+If $|c|<1$, then $\lim_{N\to\infty}\sum_{n=0}^{N}c^n=\frac{1}{1-c}$.
+
+otherwise, the series diverges.
+
+Proof:
+
+The geometric series converges if $\frac{c^{N+1}}{1-c}$ converges.
+
+$$
+(1-c)(1+c+c^2+\cdots+c^N)=1-c^{N+1}
+$$
+
+If $|c|<1$, then $\lim_{N\to\infty}c^{N+1}=0$, so $\lim_{N\to\infty}(1-c)(1+c+c^2+\cdots+c^N)=1$.
+
+If $|c|\geq 1$, then $c^{N+1}$ does not converge to 0, so the series diverges.
+
+EOP
+
+### Convergence
+
+#### Definition 5.4
+
+$$
+\sum_{n=0}^{\infty}c_n
+$$
+
+converges absolutely if $\sum_{n=0}^{\infty}|c_n|$ converges.
+
+Note: _Some other properties of converging series covered in Math4111, bad, very bad._
+
+#### Definition 5.6 Convergence of sequence of functions
+
+A sequence of functions $f_n$ **converges pointwise** to $f$ on a set $G$ if for every $z\in G$, $\forall\epsilon>0$, $\exists N$ such that for all $n\geq N$, $|f_n(z)-f(z)|<\epsilon$.
+
+(choose $N$ based on $z$)
+
+A sequence of functions $f_n$ **converges uniformly** to $f$ on a set $G$ if for every $\epsilon>0$, there exists a positive integer $N$ such that for all $n\geq N$ and all $z\in G$, $|f_n(z)-f(z)|<\epsilon$.
+
+(choose $N$ based on $\epsilon$)
+
+A sequence of functions $f_n$ **converges locally uniformly** to $f$ on a set $G$ if for every $z\in G$, $\forall\epsilon>0$, $\exists r>0$ such that for all $z\in B(z,r)$, $\forall n\geq N$, $|f_n(z)-f(z)|<\epsilon$.
+
+(choose $N$ based on $z$ and $\epsilon$)
+
+A sequence of functions $f_n$ **converges uniformly on compacta** to $f$ on a set $G$ if it converges uniformly on every compact subset of $G$.
+
+#### Theorem 5.?
+
+If the subsequence of a converging sequence of functions converges (a), then the original sequence converges (a).
+
+You can replace (a) with locally uniform convergence, uniform convergence, pointwise convergence, etc.
+
+#### UNKNOWN
+
+We defined $a^b=\{e^{b\log a}\}$ if $b$ is real, then $a^b$ is unique, if $b$ is complex, then $a^b=e^{b\log a}\{e^{2k\pi ik b}\},k\in\mathbb{Z}$.
+
+### Power series
+
+#### Definition 5.8
+
+A power series is a series of the form $\sum_{n=0}^{\infty}c_n(\zeta-\zeta_0)^n$.
+
+#### Theorem 5.10
+
+For every power series, there exists a radius of convergence $R$ such that the series converges absolutely and locally uniformly on $B(\zeta_0,R)$.
+
+And it diverges pointwise outside $B(\zeta_0,R)$.
+
+Proof:
+
+Without loss of generality, we can assume that $\zeta_0=0$.
+
+Suppose that the power series is $\sum_{n=0}^{\infty}c_n (\zeta)^n$ converges at $\zeta=re^{i\theta}$.
+
+We want to show that the series converges absolutely and uniformly on $\overline{B(0,r)}$ (_closed disk, I prefer to use this notation, although they use $\mathbb{D}$ for the disk (open disk)_).
+
+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 $\sum_{n=0}^{\infty}|c_n\zeta^n|$ converges absolutely.
+
+So the series converges absolutely and uniformly on $\overline{B(0,r)}$.
+
+_Some steps are omitted._
+
+EOP
+
+There are few cases for the convergence of the power series.
+
+Let $E=\{\zeta\in\mathbb{C}: \sum_{n=0}^{\infty}c_n(\zeta-\zeta_0)^n\text{ converges}\}$.
+
+1. It cloud only converge at $\zeta=0$.
+2. It could converge everywhere.
+
+Continue next time.

pages/Math416/_meta.js

@@ -9,4 +9,5 @@ export default {
     Math416_L4: "Complex Variables (Lecture 4)",
     Math416_L5: "Complex Variables (Lecture 5)",
     Math416_L6: "Complex Variables (Lecture 6)",
+    Math416_L7: "Complex Variables (Lecture 7)",
 }