Apostol mathematical analysis
===

[TOC]
# Chapter 1 - The Real and Complex Number Systems
## 1.1 Introduction
## 1.2 The Field Axioms
### Axiom 1
:::info
$x+y=y+x,\quad xy=yx$.
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### Axiom 2
:::info
$x+(y+z)=(x+y)+z$.
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### Axiom 3
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$x(y+z)=xy+xz$.
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### Axiom 4
:::info
Given any two real number $x$ and $y$, there exists a real number $z$ such that $x+z=y$.
This $z$ is denoted by $y-x$; the number $x-x$ is denoted by $\textbf{0}$.(It can be prove that $\textbf{0}$ is independent of $x$.)
We write $-x$ for $0-x$ and call $-x$ the negative of $x$.
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### Axiom 5
:::info
There exists at least one real number $x \ne 0$. If $x$ and $y$ are two real number with $x \ne 0$, then there exists a real number $z$ such that $xz=y$. This $z$ is denoted by $y/x$; the number $x/x$ is denoted by $1$ and can be shown to be independent of x.
We write $x^{-1}$ for $1/x$ and call $x^{-1}$ the *reciprocal* of x.
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## 1.3 The Order Axioms
### Axiom 6
:::info
Exactly one of relations $x=y,\ x<y,\ x>y$ holds.
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### Axiom 7
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If $x<y$, then for every $z$ we have $x+z<y+z$.
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### Axiom 8
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If $x>0$ and $y>0$, then $xy>0$.
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### Axiom 9
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If $x>y\quad and\quad y>z$, then $x>z$.
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### Theorem 1.1
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Given real number a and b such that $a \leq b+\epsilon$ for every $\epsilon > 0$.
Then $a \leq b$.
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- [x] Proof. Contrapositive with $\epsilon=(a+b)/2$
## 1.4 Geometric Representation of Real Numbers
## 1.5 Intervals

### Definition 1.2
Open interval and close interval
(omitted)

## 1.6 Integers
### Definition 1.3
:::info
A set of real numbers is called an *inductive set* if it has the following two properties:
a) The number ***1*** is in the set.
b) For every $x$in the set, the number $x+1$ is also in the set.
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### Definition 1.4
:::info
A real number is called a positive integer if it belongs to every inductive set. The set of positive integers is denoted by $\mathbf{Z}^+$.
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## 1.7 The Unique Factorization Theorem for Integers
### Theorem 1.5
:::info
Every integer is either a prime or a product of primes.
:::
Proof. We use induction on n. The theorem holds trivially for $n=2$. Assume it is true for every $k$ with $1<k<n$. If $n$ is not prime it has a positive divisor $d$ with $1<d<n$. Hence $n=dc$, where $1<c<n$. Since both c and d are $<n$, each is a prime or a product of primes; Hence n is a product of primes.

### Theorem 1.6
:::info
Every pair of integers a and b has a common divisor d of the form
$$d =ax+by
$$
where x and y are integers. Moreorver, every common divisor of a and b divides this $d$.
:::
Proof. Induction on n = a+b.

### Theorem 1.7 (Euclid's Lemma)
:::info
If $a|bc$ and $(a,b)=1$, then $a|c$.
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Since $(a,b)=1$ we can write $1=ax+by$.
Therefore, $c=acx+bcy$.
But $a|acx$ and $a|bcy$, so $a|c$.

### Theorem 1.8
:::info
If a prime $p$ divides $ab$, then $p|a$ or $p|b$.
More generally, if a prime p divides a product $a_1\cdot\cdot\cdot a_k$, then p divides at least one of the factors.
:::
Assume $p|ab$ and that $p\nmid a$. If we prove that $(p,a)=1$, then [The Euclid's Lemma](#Theorem-17-Euclid%E2%80%99s-Lemma) implies $p|b$.
Let $d=(p, a)$. Then $d=p$ or $d=1$. We cannot have $d=p$ because $d|a$ but $p\nmid a$ (by assumption). Hence $d=1$. To prove the more general statement we use induction on k, the number of factors.

### Theorem 1.9 (Unique Factorization Theorem)
:::info
Every integer $n>1$ can be represented as a product of prime factors in only one way, apart from the order of the factors.
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Use induction on n. The theorem is true for $n=2$. Assume, then, that it is true for all integers greater than 1 and less than n. If $n$ is prime there is nothing more to prove. Therefore assume that n is composite and that n has two factorizations into prime factors, say
$$n=p_1p_2\cdot \cdot \cdot p_s=q_1q_2\cdot\cdot\cdot q_t \tag{2}
$$
We wish to show that $s=t$ and that each $p$ equals some $q$. Sine $p_1$ divides the product $q_1q_2\cdot\cdot\cdot q_t$, it divides at least one factor. Relabel the $q's$ if necessaryso that $p_1|q_1$. Then $p1=q1$ since both $p_1$ and $q_1$ are primes. In **(2)** we cancel $p_1$ on both sides to obtain
$$\frac{n}{p1}=p_2\cdot\cdot\cdot p_s=q_2\cdot\cdot\cdot q_t
$$
Since $n$ is compisite, $1<n/p_1<n$; so by induction hypothesis the two factorization of $n/p_1$ are identical, apart from the order of the factors. Therefore the same is true in **(2)** and the proof is complete.

## 1.8 Rational Numbers
## 1.9 Irrational Numbers

### Theorem 1.10
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If $n$ is a positive integer which is not a perfect square, then $\sqrt{n}$ is irrational.
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### Theorem 1.11
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If $e^x =1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdot\cdot\cdot+\frac{x^n}{n!}+\cdot\cdot\cdot$, then the number $e$ is irrational.
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## 1.10 Upper Bounds, Maximum Element, Least Upper Bound(Supremum)
### Definition 1.12
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Let $S$ be a set of real numbers. If there is a real number $b$ such that $x\leq b$ for every $x$ in $S$, then b is called an upper bound for $S$ and we say that $S$ is bounded above by $b$.
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### Definition 1.13
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Let $S$ be a set of real numbers bounded above. A real number $b$ is called a least upper bound for $S$ if it has the following properties:
a) b is an upper bound for S.
b) No number less than b is an upper bound for $S$.
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## 1.11 The Completeness Axiom
### Axiom 10
:::info
Every nonempty set $S$ of real numbers which is bounded above has a **supremum**;
that is, there is real number $b$ such that $b=supS$.
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## 1.12 Some Properties of The Supremum
### Theorem 1.14 (Approximation property)
:::info
Let $S$ be a nonempty set of real numbers with a supremum, say $b=supS$. Then for every $a < b$ there is some $x$ in $S$ such that
$$ a < x \leq b
$$
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### Theorem 1.15 (Addictive Property)
:::info
Given nonempty subsets $A$ and $B$ of $\mathbf{R}$, let C denote the set
$$C=\{x+y:x\in A, y\in B\}
$$
If each of $A$ and $B$ has a supremum, then C has a supremum and
$$ supC = supA+supB
$$
:::
### Theorem 1.16 (Comparison Property)
:::info
Given nonempty subsets $S$ and $T$ of $\mathbf{R}$ such that $s\leq t$ for every $s\in S\ and\ t\in T$. If $T$ has a supremum then $S$ has a supremum and
$$ supS \leq supT
$$
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## 1.13 Properties of the Integers Deduced from the Completeness Axiom
### Theorem 1.17
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The set $\mathbf{Z}^+$ of positive integers $1,2,3,...$ is unbounded above.
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### Theorem 1.18
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For every real $x$ there is a positive integer $n$ such that $n>x$.
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## 1.14 The Archimedean Property of The Real Number System
### Theorem 1.19
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If $x>0$ and *if* y is an arbitrary real number, there is a positive integer n such that $nx>y$
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## 1.15 Rational Number with Finite Decimal Representation
## 1.16 Finite Decimal Approximations to Real Numbers
### Theorem 1.20
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Assume $x\geq0$. Then for every integer $n\geq1$ there is a finite decimal $r_n=a_0\cdot a_1a_2\cdot\cdot\cdot a_n$ such that
$$r_n\leq x\leq r_n+\frac{1}{10^n}
$$
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## 1.17 Infinite Decimal Representations of Real Numbers
## 1.18 Absolute Values and The Triangle Inequality
### Theorem 1.21
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If $a\geq0$, then we have the inequality $|x|\leq a$ if, and only if, $-a\leq x\leq a$.
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### Theorem 1.22
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For arbitrary real $x$ and $y$ we have
$$|x+y|\leq|x|+|y| \qquad (the\ triangle\ inequality)
$$
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## 1.19 The Cauchy-Schwarz Inequality
### 1.23 (Cauchy-Schwarz Inequality)
:::info
If $a_1,...,a_n$ and $b_1,...,b_n$ are arbitrary real numbers, we have
$$\left(\sum_{k=1}^{n}a_kb_k\right)^2\leq\left(\sum_{k=1}^{n}a_k^2\right)\left(\sum_{k=1}^{n}b_k^2\right)
$$
Moreover, if some $a_i\neq0$ equality holds if and only if there is a real $x$ such that $a_kx+b_k=0$ for each $k=1,2,...,n$.
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## 1.20 Plus and Minus Infinity and The Extended Real Number System $\mathbf{R}^*$

### Definition 1.24
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By the extended real number system $\mathbf{R}^*$ we shall mean the set of real numbers $\mathbf{R}$ together with two symbols $+\infty$ and $-\infty$ which satisfy the following properties:
a) If $x\in\mathbf{R}$, then we have
|$x+(+\infty)=+\infty$|$x+(-\infty)=-\infty$ |
|---|---|
|$x-(+\infty)=-\infty$|$x-(-\infty)=+\infty$|
|$x/(+\infty)=x/(-\infty)=0$
b) If $x>0$, then we have
|$x(+\infty)=+\infty$|$x(-\infty)=-\infty$ |
|---|---|
c) If $x<0$, then we have
|$x(+\infty)=-\infty$|$x(-\infty)=+\infty$|
|---|---|

d)
|$(+\infty)+(+\infty)=(+\infty)(+\infty)=(-\infty)(-\infty)=(+\infty),$|
|---|
|$(-\infty)+(-\infty)=(+\infty)(-\infty)=-\infty$|
e) if $x \in \mathbf{R}$, then we have $-\infty<x<+\infty$
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### Definition 1.25
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Every open interval $(a, \infty)$ is called a neighborhood of $\infty$ or a ball with center $+\infty$. Every open interval $(-\infty, a)$ is called a neighborhood of $-\infty$ or a ball with center $-\infty$.
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## 1.21 Complex Numbers
### Definition 1.26
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By a complex number we shall mean an ordered pair of real numbers which we denote by $(x_1, x_2)$. The first member, $x_1$, is called the real part of the complex number; the second member,$x_2$, is called the imaginary part.
Two complex nubmer $x=(x_1, x_2)\ and\ y=(y_1, y_2)$ are called equal, and we write $x=y$, if and only if, $x_1=y_1\ and\ x_2=y_2$. We define the sum $x+y$ and the product of $xy$ by the equations
$$ x+y=(x_1+y_1, x_2+y_2), \qquad xy=(x_1y_1-x_2y_2,\ x_1y_2+x_2y_1)
$$
Note. The set of all complex number will be denoted by $\mathbf{C}$
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### Theorem 1.27
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The operations of addition and multiplication just defined satisfy the commutative, associative, distributive laws.
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### Theorem 1.28
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|$(x_1,x_2)+(0,0)=(x_1,x_2)$|$(x_1,x_2)(0,0)=(0,0)$|
|---|---|
|$(x_1,x_2)(1,0)=(x_1,x_2)$|$(x_1,x_2)+(-x_1,-x_2)=(0,0)$|
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### Theorem 1.29
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Given two complex numbers $x=(x_1,x_2)$ and $y=(y_1,y_2)$, there exists a complex number $z$ such that $x+z=y$. In fact, $z=(y_1-x_1,y_2-x_2)$. This $z$ denoted by $y-x$. The complex number $(-x,-y)$ is denoted by $y-x$.
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### Theorem 1.30
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For any two complex number $x$ and $y$, we have
$$(-x)y=x(-y)=-(xy)=(-1,0)(xy)
$$
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### Definition 1.31
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If $x=(x_1,x_2)\neq(0,0)$ and $y$ are complex numbers, we define $x^{-1}=[\frac{x_1}{x_1^2+x_2^2},\frac{-x_2}{x_1^2+x_2^2}]$, and $\frac{y}{x}=yx^{-1}$.
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### Theorem 1.32
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If $x$ and $y$ are complex numbers with $x\neq(0,0)$, there exists a complex number $z$ such that $xz=y$, namely, $z=yx^{-1}$.
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### Theorem 1.33
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$(x_1, 0)+(y_1,0) = (x_1+y_1,0)$
$(x_1, 0)(y_1,0) = (x_1y_1,0)$
$(x_1, 0)/(y_1,0) = (x_1/y_1,0)$
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## The Imaginary Unit
### Definition 1.34
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The complex number $(0,1)$ is denoted by $i$ and is called the imaginary unit.
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### Theorem 1.35
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Every complex number $x=(x_1,x_2)$ can be represented in the form $x=x_1+ix_2$
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### Theorem 1.36
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$i^2=-1$
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## 1.24 Absolute Value of A Complex Number
### Definition 1.37
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If $x=(x_1,x_2)$, we define the modulus, or absolute value, of $x$ to be the nonnegative real number $|x|$ given by
$$|x|=\sqrt{x_1^2+x_2^2}
$$
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### Theorem 1.38
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$i)|(0,0)|=0, and |x|>0, if\ x\neq0$
$ii)|xy|=|x||y|$
$iii)|x/y|=|x|/|y|\ if\ y\neq0$
$iv)|(x_1,0)|=|x|$
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### Theorem 1.39
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If $x$ and $y$ are complex numbers, then we have
$$|x+y| \leq|x|+|y| \tag{triangle inequality}
$$
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## 1.25 Impossibility Of Ordering The Complex Number
## 1.26 Complex Exponentials
### Definition 1.40
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If $z=x+iy$, we define $e^z=e^{x+iy}$ to be the complex number $e^z = e^x(cosy+isiny)$
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### Theorem 1.41
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If $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ are two complex numbers, then we have
$$e^{z_1}e^{z_2} = e^{z_1+z_2}
$$
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## 1.27 Further Properties of Complex Exponentials
In the folloing theorem, $z_1,z_2,z_3$ denote complex numbers.

### Theorem 1.42
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$e^x$ is never zero.
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### Theorem 1.43
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If $x$ is real, then $e^{ix}=1$
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### Theorem 1.44
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$e^x=1$ if, and only if $z$ is an integral mutiple of $2\pi i$
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### Theorem 1.45
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if $e^{z_1}=e^{z_2}$ if, and only if, $z_1-z_2=2\pi n$ (where n is integer)
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## 1.28 The Argument of A Complex Number
### Definition 1.46
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Let $z=x+iy$ be a nonzero complex number. The unique real number $\theta$ which satisfies the conditions
$$x=|z|cos\theta,\quad y=|z|sin\theta,\qquad -\pi<\theta\leq+\pi
$$
is called *the principle argument of $z$*, denoted by $\theta=arg(z)$
:::

### Theorem 1.47
:::info
Every complex number $z\neq0$ can be represented in the form $z=re^{i\theta}$, where $r=|z|$ and $\theta=arg(z)+2\pi n$, $n$ being any integer
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### Theorem 1.48
:::info
If $z_1z_2\neq0$, then $arg(z_1z_2)=arg(z_1)+arg(z_2)+2\pi n(z_1,z_2)$, where
$$n(z_1, z_2)= \\
0\qquad ,\ if -\pi<arg(z_1)+arg(z_2)\leq+\pi \\
+1\qquad ,\ if -2\pi<arg(z_1)+arg(z_2)\leq-\pi \\
-1\qquad ,\ if \pi<arg(z_1)+arg(z_2)\leq+2\pi
$$
:::
## 1.29 Integral Powers and Roots of Complex Numbers
### Definition 1.49
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Given a complex number $z$ and an integer $n$, we define the $nth$ power of $z$ as follows:
$$z^0=1,\quad z^{n+1}=z^nz,\qquad if\ n\geq0,\\
\qquad \qquad z^{-n}=(z^{-1})^n, \qquad\qquad if\ z\neq0\ and\ n>0
$$
:::
### Theorem 1.50
:::info
Given two integers $m$ and $n$, we have, for $z\neq0$,
$$z^nz^m=z^{n+m} \qquad (z_1z_2)^n=z_1^nz_2^n
$$
:::

### Theorem 1.51

## Complex Logorithm
### Theorem 1.52
:::info
If $z$ is a complex number $\neq0$, then there exist complex numbers $w$ such that $e^w=z$. One such $w$ is the complex number
$$log|z|+iarg(z),
$$
and any other such w must have the form
$$log|z|+iarg(z)+2n\pi i,
$$
where $n$ is an integer.
:::
[Theorem 1.45](#Theorem-145)

### Definition 1.53
:::info
Let $z\neq0$ be a given complex number, if $w$ is a complex number such that $e^w=z$, then w called a logarithm of $z$. The particular value of w given by
$$w=log|z|+iarg(z)
$$
is called the principle logarithm of $z$, and for this w we write
$$w=Log\ z
$$
:::

#### Example
1.Since $|i|=1$ and $arg(i)=\frac{\pi}{2}, Log(i)=i\pi/2$
2.Since $|-i|=1$ and $arg(-i)=-\frac{\pi}{2}, Log(-i)=-i\pi/2$
3.Since $|-1|=1$ and $arg(-1)=\pi, Log(-1)=\pi i$
4.If $x>0, Log(x)=logx, since|x|=x$ and $arg(x)=0$
5.Since $|1+i|=\sqrt2$ and $arg(1+i)=\pi/4, Log(1+i)=log{\sqrt2}+i\pi/4$

### Theorem 1.54
:::info
If $z_1z_2\neq0$, then
$$Log(z_1z_2)=Logz_1+Logz_2+2\pi in(z_1,z_2)
$$
where $n(z_1,z_2)$ is the integer defined in [Theorem 1.48](#Theorem-148)
:::

## 1.31 Complex Powers
### Definition 1.55
:::info
$If z\neq0$ and if $w$ is any complex number, we define
$$z^w=e^{wLogz}
$$
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#### Exmaple:
1.$i^i=e^{iLogi}=e^{i(i\pi/2)}=e^{-\pi/2}$
2.$(-1)^i=e^{iLog(-1)}=e^{i(i\pi)}=e^{-\pi}$
3.If n is an integer, then $z^{n+1}=e^{(n+1)Log(z)}=e^{nLogz}e^{Logz}=z^nz$, so Definition 1.55 does not confict with Definition 1.49

### Theorem 1.56
:::info
$z^{w_1}z^{w_2}=z^{w_1+w_2}$ if $z\neq0$
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### 1.57 Complex Sine and Cosine