NoteNextra-origin/content/Math416/Math416_L1.md

Math416 Lecture 1

Chapter 1: Complex Numbers

Preface

I don't know what happened to the first class. I will try to rewrite the notes from my classmates here.

Rigidity

Integral is preserved for any closed path.

Group

A set with a multiplication operator. (G,\cdot) such that: for all a,b,c\in G:

1. a\cdot b\in G
2. a\cdot (b\cdot c)=(a\cdot b)\cdot c
3. a\cdot 1=a
4. a\cdot a^{-1}=1

Ring

A group with two operations: addition and multiplication. (R,+,\cdot) such that: for all a,b,c\in R:

1. Commutative under addition: a+b=b+a
2. Associative under multiplication: (a\cdot b)\cdot c=a\cdot (b\cdot c)
3. Distributive under addition: a\cdot (b+c)=a\cdot b+a\cdot c

Example:

\{a+\sqrt{6}b\mid a,b\in \mathbb{Z}\} is a ring

Definition 1.1

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:

• Addition: (x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)
• Multiplication: (x_1,y_1)(x_2,y_2)=(x_1x_2-y_1y_2,x_1y_2+x_2y_1)

Axiom 1.2

The operation of addition and multiplication on \mathbb{C} satisfies the following conditions (The field axioms):

For all z_1,z_2,z_3\in \mathbb{C}:

1. z_1+z_2=z_2+z_1 (commutative law of addition)
2. (z_1+z_2)+z_3=z_1+(z_2+z_3) (associative law of addition)
3. z_1\cdot z_2=z_2\cdot z_1 (commutative law of multiplication)
4. (z_1\cdot z_2)\cdot z_3=z_1\cdot (z_2\cdot z_3) (associative law of multiplication)
5. z_1\cdot (z_2+z_3)=z_1\cdot z_2+z_1\cdot z_3 (distributive law)
6. There exists an additive identity element 0=(0,0) such that z+0=z for all z\in \mathbb{C}.
7. There exists a multiplicative identity element 1=(1,0) such that z\cdot 1=z for all z\in \mathbb{C}.
8. There exists an additive inverse -z=(-x,-y) for all z=(x,y)\in \mathbb{C} such that z+(-z)=0.
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.

(a,b)^{-1}=\left(\frac{a}{a^2+b^2},-\frac{b}{a^2+b^2}\right)

Embedding of \mathbb{R} in \mathbb{C} 1.3

Let z=x+iy\in \mathbb{C} where a,b\in \mathbb{R}.

• x is called the real part of z and
• y is called the imaginary part of z.
• |z|=\sqrt{x^2+y^2} is called the absolute value or modulus of z.
• 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).
• \overline{z}=x-iy is called the conjugate of z. (z\cdot \overline{z}=|z|^2)
• z_1+z_2=(x_1+x_2,y_1+y_2) (vector addition)

Lemma 1.3

|z_1z_2|=|z_1||z_2|

Theorem 1.5 (Triangle Inequality)

|z_1+z_2|\leq |z_1|+|z_2|

Proof

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.

Algebraically,

\begin{aligned}
(|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})\\
&=|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)\\
&=2|z_1||z_2|-2Re(\overline{z_1}z_2)\\
&=2(|z_1||z_2|-|z_1z_2|)\\
&\geq 0
\end{aligned}

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|...

What is the use of this?

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).

z_1z_2=r_1r_2[cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)]

(Define \text{cis}(\theta)=\cos\theta+i\sin\theta)

Theorem 1.6 Parallelogram Equality

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.

Proof

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.

\begin{aligned}
|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 \\
&= 2x_1^2+2x_2^2+2y_1^2+2y_2^2 \\
&= 2(|z_1|^2+|z_2|^2)
\end{aligned}

Definition 1.9

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.

De Moivre's Formula

Theorem 1.10 De Moivre's Formula

Let z=r\text{cis}(\theta), then

\forall n\in \mathbb{Z}:

z^n=r^n\text{cis}(n\theta)

Proof

For n=0, z^0=1=1\text{cis}(0).

For n=-1, z^{-1}=\frac{1}{z}=\frac{1}{r}\text{cis}(-\theta)=\frac{1}{r}(cos(-\theta)+i\sin(-\theta)).

Application:

\begin{aligned}
(\text{cis}(\theta))^3&=\text{cis}(3\theta)\\
&=\cos(3\theta)+i\sin(3\theta)\\
&=cos^3(\theta)-3cos(\theta)sin^2(\theta)+i(3cos^2(\theta)sin(\theta)-sin^3(\theta))\\
\end{aligned}