merge?

pages/Math4121/Math4121_L4.md

@@ -1 +1,137 @@
 # Lecture 4
+
+## Chapter 5. Differentiation
+
+### The continuity of the derivative
+
+#### Theorem 5.12
+
+Suppose $f$ is differentiable on $[a,b]$, Then $f'$ attains intermediate values between $f'(a)$ and $f'(b)$.
+
+Proof:
+
+Let $\lambda\in (f'(a),f'(b))$. We need to show that there exists $x\in (a,b)$ such that $f'(x)=\lambda$.
+
+Let $g(x)=f(x)-\lambda x$. Then $g$ is differentiable on $(a,b)$ and
+
+$$
+g'(x)=f'(x)-\lambda.
+$$
+
+So $g'(a)=f'(a)-\lambda<0$ and $g'(b)=f'(b)-\lambda>0$.
+
+We need to show that $g'(x)=0$ for some $x\in (a,b)$.
+
+Since $g'(a)<0$, $\exists t_1\in (a,b)$ such that $g'(t_1)<g(a)$.
+
+If not, then $g(t)\geq g(a)$ for all $t\in (a,b)$. But then $g'(a)\gets \frac{g(t)-g(a)}{t-a}\geq 0$, which contradicts $g'(a)<0$.
+
+With the loss of generality, since $g'(b)>0$, $\exists t_2\in (a,b)$ such that $g'(t_2)<g(b)$.
+
+Hence, $g$ attains its infimum on $[a,b]$ at some $x\in (a,b)$. Then this $x$ is a local minimum of $g$ on $(a,b)$.
+
+So $g'(x)=0$ and $f'(x)=\lambda$.
+
+EOP
+
+### L'Hôpital's Rule
+
+#### Theorem 5.13
+
+Suppose $f$ and $g$ are differentiable on $(a,b)$ and $g'(x)\neq 0$ for all $x\in (a,b)$, where $-\infty\leq a<b\leq \infty$. Suppose
+
+$$
+\frac{f'(x)}{g'(x)}\to A \text{ as } x\to a\dots
+$$
+
+If
+
+$$
+f(x)\to 0 \text{ and } g(x)\to 0 \text{ as } x\to a,
+$$
+
+or
+
+$$
+g(x)\to \infty \text{ as } x\to a,
+$$
+
+then
+
+$$
+\frac{f(x)}{g(x)}\to A \text{ as } x\to a.
+$$
+
+Note that all these numbers $A$ can be $\infty$ or $-\infty$ (on extended real line).
+
+We're using the open neighborhood definition of $\to$ here. An open neighborhood of $\infty$ is an interval of the form $(c,\infty)$ for some $c\in \mathbb{R}$.
+
+> Recall the [Definition 3.1](https://notenextra.trance-0.com/Math4111/Math4111_L13#definition-31).
+
+Proof:
+
+Main step:
+
+Suppose $-\infty\leq A\leq \infty$, and let $q>A$ with neighborhood $(-,\infty,q)$. Then $\exists c\in \mathbb{R}$ such that $\frac{f(x)}{g(x)}<q,\forall x\in (a,c)$.
+
+Proof of the main step:
+
+Fix $A<r<q$. Then $\exists c\in (a,b)$ such that $\frac{f'(x)}{g'(x)}<r,\forall x\in (a,c)$.
+
+Now, for any $a<x<y<c$, by generalized mean value theorem, $\exists t\in (x,y)$ such that
+
+$$
+\frac{f(x)-f(y)}{g(x)-g(y)}=\frac{f'(t)}{g'(t)}
+$$
+
+Since $t\in (a,c)$, $\frac{f'(t)}{g'(t)}<r$.
+
+Case 1: $f(x)\to 0$ and $g(x)\to 0$ as $x\to a$.
+
+As $x\to a$, $f(x)\to 0$ and $g(x)\to 0$. So
+
+$$\begin{aligned}
+\lim_{x\to a}\frac{f(x)-f(y)}{g(x)-g(y)}&=\lim_{x\to a}\frac{0-f(y)}{0-g(y)}\\
+&=\lim_{x\to a}\frac{f(y)}{g(y)}\\
+&=\frac{f'(y)}{g'(y)}\\
+&\leq r<q
+\end{aligned}$$
+
+$\forall y\in (a,c)$, $\frac{f(y)}{g(y)}<q$.
+
+Case 2: $g(x)\to \infty$ as $x\to a$.
+
+We can find $c_1\in (a,y)$ such that $g(x)>g(y)$ for all $x\in (a,c_1)$.
+
+Therefore,
+
+$$
+\begin{aligned}
+\frac{f(x)-f(y)}{g(x)}&<\frac{r[g(x)-g(y)]}{g(x)}\\
+\frac{f(x)}{g(x)}&<r-\frac{rg(y)}{g(x)}+\frac{f(y)}{g(x)}
+\end{aligned}
+$$
+
+To make the right side less than $q$, we need
+
+$$
+\frac{|rg(y)|+|f(y)|}{|g(x)|}<q-r
+$$
+
+so,
+
+$$
+|g(x)|>\frac{|rg(y)|+|f(y)|}{q-r}
+$$
+
+There exists $c_2\in (a,c_1)$ such that $|g(x)|>\frac{|rg(y)|+|f(y)|}{q-r},\forall x\in (a,c_2)$.
+
+So $\forall x\in (a,c_2)$,
+
+$$
+\frac{f(x)}{g(x)}<\frac{rg(y)+f(y)}{g(x)}<r+(q-r)=q
+$$
+
+$\forall x\in (a,c_2)$, $\frac{f(x)}{g(x)}<q$.
+
+EOP

pages/Math416/Math416_L1.md

@@ -109,7 +109,7 @@ $$
 
 (Define $\text{cis}(\theta)=\cos\theta+i\sin\theta$)
 
-### De Howtes' Formula
+### De Moivre's Formula
 
 Let $z=r\text{cis}(\theta)$, then
 

pages/Math416/Math416_L3.md

@@ -0,0 +1,142 @@
+# Lecture 3
+
+## Differentiation of functions in complex variables
+
+### Differentiability
+
+#### Definition of differentiability in complex variables
+
+Suppose $G$ is an open subset of $\mathbb{C}$.
+
+A function $f:G\to \mathbb{C}$ is differentiable at $\zeta_0\in G$ if
+
+$$
+\lim_{\zeta\to \zeta_0}\frac{f(\zeta)-f(\zeta_0)}{\zeta-\zeta_0}
+$$
+
+exists.
+
+Or equivalently,
+
+We can also express the $f$ as $f=u+iv$, where $u,v:G\to \mathbb{R}$ are real-valued functions.
+
+Recall that $u:G\to \mathbb{R}$ is differentiable at $\zeta_0\in G$ if and only if there exists a complex number $(x,y)\in \mathbb{C}$ such that a function
+
+$$
+R(x,y)=u(x,y)-\left(u(x_0,y_0)+\frac{\partial u}{\partial x}(x_0,y_0)(x-x_0)+\frac{\partial u}{\partial y}(x_0,y_0)(y-y_0)\right)
+$$
+
+satisfies
+
+$$
+\lim_{(x,y)\to (x_0,y_0)}\frac{|R(x,y)|}{|(x,y)-(x_0,y_0)|}=\lim_{(x,y)\to (x_0,y_0)}\frac{|R(x,y)|}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=0.
+$$
+
+> Theorem from 4111?
+> 
+> If $u$ is differentiable at $(x_0,y_0)$, then $\frac{\partial u}{\partial x}(x_0,y_0)$ and $\frac{\partial u}{\partial y}(x_0,y_0)$ exist.
+> 
+> If $\frac{\partial u}{\partial x}(x_0,y_0)$ and $\frac{\partial u}{\partial y}(x_0,y_0)$ exist and one of them is continuous at $(x_0,y_0)$, then $u$ is differentiable at $(x_0,y_0)$.
+
+$$
+\begin{aligned}
+\lim_{(x,y)\to (x_0,y_0)}\frac{|R(x,y)|}{|(x,y)-(x_0,y_0)|}&=\lim_{(x,y)\to (x_0,y_0)}\frac{|u(x,y)-u(x_0,y_0)-\frac{\partial u}{\partial x}(x_0,y_0)(x-x_0)-\frac{\partial u}{\partial y}(x_0,y_0)(y-y_0)|}{\sqrt{(x-x_0)^2+(y-y_0)^2}}\\
+&=\lim_{(x,y)\to (x_0,y_0)}\frac{|u(x,y)-u(x_0,y_0)-\frac{\partial u}{\partial x}(x_0,y_0)(x-x_0)-\frac{\partial u}{\partial y}(x_0,y_0)(y-y_0)|}{\sqrt{(x-x_0)^2+(y-y_0)^2}}\\
+\end{aligned}
+$$
+
+Let $a(x,y)=\frac{\partial u}{\partial x}(x,y)$ and $b(x,y)=\frac{\partial u}{\partial y}(x,y)$.
+
+We can write $R(x,y)$ as
+
+$$
+R(x,y)=u(x,y)-u(x_0,y_0)-a(x,y)(x-x_0)-b(x,y)(y-y_0).
+$$
+
+So $\lim_{(x,y)\to (x_0,y_0)}\frac{|R(x,y)|}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=0$ if and only if $\lim_{(x,y)\to (x_0,y_0)}\frac{a(x-x_0)}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=0$ and $\lim_{(x,y)\to (x_0,y_0)}\frac{b(y-y_0)}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=0$.
+
+On the imaginary part, we have
+
+...
+
+Conclusion (The Cauchy-Riemann equations):
+
+If $f=u+iv$ is complex differentiable at $\zeta_0\in G$, then $u$ and $v$ are real differentiable at $(x_0,y_0)$ and
+
+$$
+\frac{\partial u}{\partial x}(x_0,y_0)=\frac{\partial v}{\partial y}(x_0,y_0),\quad \frac{\partial u}{\partial y}(x_0,y_0)=-\frac{\partial v}{\partial x}(x_0,y_0).
+$$
+
+And $u$ and $v$ have continuous partial derivatives at $(x_0,y_0)$.
+
+And let $c=\frac{\partial u}{\partial x}(x_0,y_0)$ and $d=\frac{\partial v}{\partial x}(x_0,y_0)$.
+
+Then $f'(\zeta_0)=c+id$.
+
+### Holomorphic Functions
+
+#### Definition of holomorphic functions
+
+A function $f:G\to \mathbb{C}$ is holomorphic (or analytic) at $\zeta_0\in G$ if it is complex differentiable at $\zeta_0$.
+
+Example:
+
+Suppose $f:G\to \mathbb{C}$ where $f=u+iv$ and $\frac{\partial f}{\partial x}=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$, $\frac{\partial f}{\partial y}=\frac{\partial u}{\partial y}+i\frac{\partial v}{\partial y}$.
+
+Define $\frac{\partial}{\partial \zeta}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)$ and $\frac{\partial}{\partial \bar{\zeta}}=\frac{1}{2}\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)$.
+
+Suppose $f$ is holomorphic at $\bar{\zeta}_0\in G$ (Cauchy-Riemann equations hold at $\bar{\zeta}_0$).
+
+Then $\frac{\partial f}{\partial \bar{\zeta}}(\bar{\zeta}_0)=0$.
+
+Note that $\forall m\in \mathbb{Z}$, $\zeta^m$ is holomorphic on $\mathbb{C}$.
+
+i.e. $\forall a\in \mathbb{C}$, $\lim_{\zeta\to a}\frac{\zeta^m-a^m}{\zeta-a}=\frac{(\zeta-a)(\zeta^{m-1}+\zeta^{m-2}a+\cdots+a^{m-1})}{\zeta-a}=ma^{m-1}$.
+
+So polynomials are holomorphic on $\mathbb{C}$.
+
+So rational functions $p/q$ are holomorphic on $\mathbb{C}\setminus\{z\in \mathbb{C}:q(z)=0\}$.
+
+### Curves in $\mathbb{C}$
+
+#### Definition of curves in $\mathbb{C}$
+
+A curve $\gamma$ in $G\subset \mathbb{C}$ is a continuous map of an interval $I$ into $G$. We say $\gamma$ is differentiable if $\forall t_0\in I$, $\gamma'(t_0)=\lim_{t\to t_0}\frac{\gamma(t)-\gamma(t_0)}{t-t_0}$ exists.
+
+If $\gamma'(t_0)$ is a point in $\mathbb{C}$, then $\gamma'(t_0)$ is called the tangent vector to $\gamma$ at $t_0$.
+
+#### Definition of regular curves in $\mathbb{C}$
+
+A curve $\gamma$ is regular if $\gamma'(t)\neq 0$ for all $t\in I$.
+
+#### Definition of angle between two curves
+
+Let $\gamma_1,\gamma_2$ be two curves in $G\subset \mathbb{C}$ with $\gamma_1(t_0)=\gamma_2(t_0)=\zeta_0$ for some $t_0\in I_1\cap I_2$.
+
+The angle between $\gamma_1$ and $\gamma_2$ at $\zeta_0$ is the angle between the vectors $\gamma_1'(t_0)$ and $\gamma_2'(t_0)$. Denote as $\arg(\gamma_2'(t_0))-\arg(\gamma_1'(t_0))=\arg(\gamma_2'(t_0)\gamma_1'(t_0))$.
+
+#### Theorem of conformality
+
+Suppose $f:G\to \mathbb{C}$ is holomorphic function on open set $G\subset \mathbb{C}$ and $\gamma_1,\gamma_2$ are regular curves in $G$ with $\gamma_1(t_0)=\gamma_2(t_0)=\zeta_0$ for some $t_0\in I_1\cap I_2$.
+
+If $f'(\zeta_0)\neq 0$, then the angle between $\gamma_1$ and $\gamma_2$ at $\zeta_0$ is the same as the angle between the vectors $f'(\zeta_0)\gamma_1'(t_0)$ and $f'(\zeta_0)\gamma_2'(t_0)$.
+
+#### Lemma of function of a curve and angle
+
+If $f:G\to \mathbb{C}$ is holomorphic function on open set $G\subset \mathbb{C}$ and $\gamma$ is differentiable curve in $G$ with $\gamma(t_0)=\zeta_0$ for some $t_0\in I$.
+
+Then,
+
+$$
+(f\circ \gamma)'(t_0)=f'(\gamma(t_0))\gamma'(t_0).
+$$
+
+If Lemma of function of a curve and angle holds, then the angle between $f\circ \gamma_1$ and $f\circ \gamma_2$ at $\zeta_0$ is
+
+$$
+\begin{aligned}
+\arg\left[(f\circ \gamma_2)'(t_2)(f\circ \gamma_1)'(t_1)\right]&=\cdots
+\end{aligned}
+$$
+
+Continue on Thursday. (Applying the chain rules)

pages/Math416/_meta.js

@@ -4,4 +4,6 @@ export default {
         type: 'separator'
     },
     Math416_L1: "Complex Variables (Lecture 1)",
+    Math416_L2: "Complex Variables (Lecture 2)",
+    Math416_L3: "Complex Variables (Lecture 3)",
 }

pages/Swap/CSE361S_W1.md

@@ -0,0 +1,42 @@
+# CSE361S Week 1 (Compact Version)
+
+## Conservation of Bits
+
+What you get is what you assigned.
+
+### Signed and Unsigned
+
+- Signed: The most significant bit is the sign bit.
+  - Two's complement: Range is from -2^(w-1) to 2^(w-1) - 1.
+  - Created by `int i = -1;`
+- Unsigned: The most significant bit is the value bit.
+  - Range is from 0 to 2^w - 1.
+  - Created by `unsigned int i = 1;` or `int i = 1U;`
+
+During the conversion, the binary representation of the number is the same.
+
+For example, `-1` in signed is `0b11111111` in signed. and `0b11111111` in signed is `255` in unsigned.
+
+Any arithmetic operation in signed and unsigned follows the same rules. `0b00000000` is `0` in signed and unsigned. and `0-1=255` in unsigned but `-1` in signed. (The binary representation of `-1` in signed is `0b11111111` and `0b11111111` is `255` in unsigned.)
+
+### Shift Operations
+
+- Logical Shift: Shift the bits and fill in the new bits with 0. (used for unsigned numbers)
+- Arithmetic Shift: Shift the bits and fill in the new bits with the sign bit. (used for signed numbers)
+
+## Bytes
+
+### Byte Ordering
+
+For Example: `0x12345678`
+
+- Little Endian: The least significant byte is stored at the lowest address.  
+  - `0x78` is stored at the lowest address.
+- Big Endian: The most significant byte is stored at the lowest address.
+  - `0x12` is stored at the lowest address.
+
+### Representing Strings
+
+- `char *str = "12345";`
+
+In memory, it is stored as `0x31 0x32 0x33 0x34 0x35` with terminating null character `0x00`.