168 lines
5.0 KiB
Markdown
168 lines
5.0 KiB
Markdown
# Math416 Lecture 1
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## Chapter 1: Complex Numbers
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### Preface
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I don't know what happened to the first class. I will try to rewrite the notes from my classmates here.
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#### Rigidity
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Integral is preserved for any closed path.
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#### Group
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A set with a multiplication operator. $(G,\cdot)$ such that: for all $a,b,c\in G$:
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1. $a\cdot b\in G$
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2. $a\cdot (b\cdot c)=(a\cdot b)\cdot c$
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3. $a\cdot 1=a$
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4. $a\cdot a^{-1}=1$
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#### Ring
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A group with two operations: addition and multiplication. $(R,+,\cdot)$ such that: for all $a,b,c\in R$:
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1. Commutative under addition: $a+b=b+a$
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2. Associative under multiplication: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$
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3. Distributive under addition: $a\cdot (b+c)=a\cdot b+a\cdot c$
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Example:
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$\{a+\sqrt{6}b\mid a,b\in \mathbb{Z}\}$ is a ring
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#### Definition 1.1
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the complex number is defined to be the set $\mathbb{C}$ of ordered pairs $(x,y)$ with $x,y\in \mathbb{R}$ and the operations:
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- Addition: $(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$
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- Multiplication: $(x_1,y_1)(x_2,y_2)=(x_1x_2-y_1y_2,x_1y_2+x_2y_1)$
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#### Axiom 1.2
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The operation of addition and multiplication on $\mathbb{C}$ satisfies the following conditions (The field axioms):
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For all $z_1,z_2,z_3\in \mathbb{C}$:
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1. $z_1+z_2=z_2+z_1$ (commutative law of addition)
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2. $(z_1+z_2)+z_3=z_1+(z_2+z_3)$ (associative law of addition)
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3. $z_1\cdot z_2=z_2\cdot z_1$ (commutative law of multiplication)
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4. $(z_1\cdot z_2)\cdot z_3=z_1\cdot (z_2\cdot z_3)$ (associative law of multiplication)
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5. $z_1\cdot (z_2+z_3)=z_1\cdot z_2+z_1\cdot z_3$ (distributive law)
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6. There exists an additive identity element $0=(0,0)$ such that $z+0=z$ for all $z\in \mathbb{C}$.
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7. There exists a multiplicative identity element $1=(1,0)$ such that $z\cdot 1=z$ for all $z\in \mathbb{C}$.
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8. There exists an additive inverse $-z=(-x,-y)$ for all $z=(x,y)\in \mathbb{C}$ such that $z+(-z)=0$.
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9. There exists a multiplicative inverse $z^{-1}=\left(\frac{x}{x^2+y^2},-\frac{y}{x^2+y^2}\right)$ for all $z=(x,y)\in \mathbb{C}$ such that $z\cdot z^{-1}=1$.
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$$
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(a,b)^{-1}=\left(\frac{a}{a^2+b^2},-\frac{b}{a^2+b^2}\right)
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$$
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#### Embedding of $\mathbb{R}$ in $\mathbb{C}$ 1.3
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Let $z=x+iy\in \mathbb{C}$ where $a,b\in \mathbb{R}$.
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- $x$ is called the real part of $z$ and
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- $y$ is called the imaginary part of $z$.
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- $|z|=\sqrt{x^2+y^2}$ is called the absolute value or modulus of $z$.
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- The angle between the positive real axis and the line segment from $0$ to $z$ is called the argument of $z$ and is denoted by $\theta$ (argument of $z$).
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- $\overline{z}=x-iy$ is called the conjugate of $z$. ($z\cdot \overline{z}=|z|^2$)
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- $z_1+z_2=(x_1+x_2,y_1+y_2)$ (vector addition)
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#### Lemma 1.3
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$$
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|z_1z_2|=|z_1||z_2|
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$$
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#### Theorem 1.5 (Triangle Inequality)
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$$
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|z_1+z_2|\leq |z_1|+|z_2|
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$$
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<details>
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<summary>Proof</summary>
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Geometrically, the triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
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Algebraically,
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$$
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\begin{aligned}
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(|z_1+z_2|)^2-|z_1+z_2|^2&=|z_1+z_2|^2-2|z_1+z_2|-(z_1+z_2)(\overline{z_1}+\overline{z_2})\\
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&=|z_1|^2+|z_2|^2+2|z_1||z_2|-(|z_1|^2+|z_2|^2+\overline{z_1}z_2+\overline{z_2}z_1)\\
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&=2|z_1||z_2|-2Re(\overline{z_1}z_2)\\
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&=2(|z_1||z_2|-|z_1z_2|)\\
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&\geq 0
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\end{aligned}
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$$
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</details>
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Suppose $2(|z_1||z_2|-|z_1z_2|)=0$, and $\overline{z_1}z_2$ is a non-negative real number $c$, then $|z_1||z_2|=|z_1z_2|$...
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> What is the use of this?
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Let $\arg(z)=\theta\in (-\pi,\pi]$, $z_1=r_1(\cos\theta_1+i\sin\theta_1)$, $z_2=r_2(\cos\theta_2+i\sin\theta_2)$.
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$$
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z_1z_2=r_1r_2[cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)]
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$$
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(Define $\text{cis}(\theta)=\cos\theta+i\sin\theta$)
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#### Theorem 1.6 Parallelogram Equality
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The sum of the squares of the lengths of the diagonals of a parallelogram equals the sum of the squares of the lengths of the sides.
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<details>
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<summary>Proof</summary>
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Let $z_1,z_2$ be two complex numbers representing the two sides of the parallelogram, then the sum of the squares of the lengths of the diagonals of the parallelogram is $|z_1-z_2|^2+|z_1+z_2|^2$, and the sum of the squares of the lengths of the sides is $2|z_1|^2+2|z_2|^2$.
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$$
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\begin{aligned}
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|z_1-z_2|^2+|z_1+z_2|^2 &= (x_1-x_2)^2+(y_1-y_2)^2+(x_1+x_2)^2+(y_1+y_2)^2 \\
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&= 2x_1^2+2x_2^2+2y_1^2+2y_2^2 \\
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&= 2(|z_1|^2+|z_2|^2)
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\end{aligned}
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$$
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</details>
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#### Definition 1.9
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The argument of a complex number $z$ is defined as the angle $\theta$ between the positive real axis and the ray from the origin through $z$.
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### De Moivre's Formula
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#### Theorem 1.10 De Moivre's Formula
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Let $z=r\text{cis}(\theta)$, then
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$\forall n\in \mathbb{Z}$:
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$$
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z^n=r^n\text{cis}(n\theta)
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$$
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<details>
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<summary>Proof</summary>
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For $n=0$, $z^0=1=1\text{cis}(0)$.
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For $n=-1$, $z^{-1}=\frac{1}{z}=\frac{1}{r}\text{cis}(-\theta)=\frac{1}{r}(cos(-\theta)+i\sin(-\theta))$.
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</details>
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Application:
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$$
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\begin{aligned}
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(\text{cis}(\theta))^3&=\text{cis}(3\theta)\\
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&=\cos(3\theta)+i\sin(3\theta)\\
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&=cos^3(\theta)-3cos(\theta)sin^2(\theta)+i(3cos^2(\theta)sin(\theta)-sin^3(\theta))\\
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\end{aligned}
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$$
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