◀ ▲ ▶Branches / Analysis / Part: Real Analysis of One Variable and Elements of Complex Analysis

Part: Real Analysis of One Variable and Elements of Complex Analysis

This part of BookofProofs deals with the analysis of real-valued (and whenever applicable also complex-valued) functions with one variable, i.e. the functions of the form $f:\mathbb R\mapsto \mathbb R$ or $f:\mathbb C\mapsto \mathbb C,$ $x\mapsto f(x)$. Using the axiomatic method, it systematically derives some basic concepts like sequences and limits, infinite series, and basic properties of functions.

Although there is another part dedicated solely to complex analysis, in which you can find deeper results applicable for complex numbers only, there are lots of concepts that are applicable for both, real ($\mathbb R$) and complex ($\mathbb C$) numbers, or even rational numbers ($\mathbb Q$), integers ($\mathbb Z$) or natural numbers ($\mathbb N$). In order to avoid unnecessary repetition of basic concepts, we will introduce them only once in this part. Whenever a theorem or definition can be used in the domain of the respective number system, we will use a marker

applicability: $\mathbb {N, Z, Q, R , C}$

listing all number system, it s applicable to.

Prerequisites to start learning calculus of one variable

Before you start further reading, you should be familiar / revise your knowledge about the following areas:

• Sets and their properties, including set manipulation (e.g. union, intersection, difference),
• Functions and their composition,
• Methods of mathematical proving, in particular, the proving principle of induction,
• Sums and methods of their manipulation,
• The sets of different number systems, including natural numbers, $\mathbb N$, integers $\mathbb Z$, rational numbers $\mathbb Q$, and, last but not least, real numbers $\mathbb R$, which are the most important mathematical objects we will be dealing with in this part of BookofProofs.

Theoretical minimum (in a nutshell)

Ideally, you should be already acquainted with the following facts:

• There are two basic operations defined for the real numbers $\mathbb R$, i.e. the addition "$+$" and the multiplication "$\cdot$".
• Together with these two operations, the real numbers form an algebraic structure $(\mathbb R, +,\cdot)$, called the field of real numbers. In a nutshell, this means that all calculations involving real numbers, let's call them $x,y,z\in\mathbb R$, obey the following rules:
  • The addition is associative $(x+y)+z=x+(y+z)$ and commutative $x+y=y+x$.
  • The multiplication is associative $(x\cdot y)\cdot z=x\cdot (y\cdot z)$ and commutative $x\cdot y=y\cdot x$.
  • There is a unique real number, called zero and denoted by $0$, which is neutral with respect to addition: $x+0=0+x=x$ for all $x\in\mathbb R$.
  • There is a unique real number, called one and denoted by $1$, which is neutral with respect to multiplication: $x\cdot 1=1\cdot x=x$ for all $x\in\mathbb R$.
  • Each real number $x$ has a unique additive inverse $-x$, also called its negative, for which we have that $x-x:=x+(-x)=0$. In fact, this is how the subtraction of real numbers "$-$" is defined.
  • Each real number $x\neq 0$ has a unique multiplicative inverse $x^{-1}=\frac 1x$, also called its reciprocal, for which we have that $x\div x:=x\cdot \frac 1x=1$. In fact, this is how the division of real numbers "$\div `$"is defined.
  • When calculations involve both, the multiplication and addition, then the distributivity laws hold: $x(y+z)=xy+xz$ and $(x+y)z=xz+yz.$¹
• You should know some basic results following from the field properties of real numbers, including:
  • $-0=0$ (proof).
  • $-(-x)=x$ for all $x\in\mathbb R$ (proof), and other rules for the multiplication of positive and negative numbers.
  • $x\cdot 0=0\cdot x=0$ for all $x\in\mathbb R$ (proof).
  • $xy=0$ holds if and only if $x=0$ or $y=0$ (or both, proof).
  • $-(x+y)=-x-y$ (proof).
  • The equality $a+x = b$ with the variable $x$ has exatly one solution: $b-a$ (proof).
  • The equality $ax = b$ with the variable $x$ and $a\neq 0$ has exatly one solution: $\frac ba$ (proof).
  • $\frac 1{xy}=\frac 1x\cdot \frac 1y$ for all $x,y\in\mathbb R$ (proof)
• Basically, the same rules apply for complex numbers $\mathbb C$ with the operations addition "$+$" and the multiplication "$\cdot$".
• Any two real numbers $x,y\in\mathbb R$ can be ordered using the order relation "$\le$" . This relation is strict and total, i.e. only one of the following situations can be true (this is known as the trichotomy property): Either $x < y$, or $x=y$, or $x > y$.
• The above order relation is transitive, i.e. if $x \le y$ and $y \le z$, then $x \le z$.
• You should know some basic results for the calculation with inequalities.
• You should know that complex numbers cannot be ordered.

Concepts you will learn in this part of BookofProofs

• Why are real numbers not countable while rational numbers are?
• What are real intervals and is the difference between open and closed intervals?
• Why we need yet another axiom (the Archimedean axiom) to fully describe the set $\mathbb R$ of real numbers?
• Absolute values of real and complex numbers and their properties
• What are sequences of numbers and what does it mean to converge and to absolutely converge against a limit?
• What are Cauchy sequences and why the sets of real numbers $\mathbb R$ and complex numbers $\mathbb C$ are said to be complete?
• You will learn about different types of real (and sometimes also complex) functions, e.g. including polynomials, rational functions, nth powers, the logarithm, and the exponential function, trigonometric functions
• You will learn which properties real functions might have, including being bounded, continuous, monotonic, differentiable, integrable, etc.
• What are the different types of real infinite series and their properties, including their convergence behaviour, the power, Taylor, and Fourier series
• You will learn the fundamental theorem of calculus and its various applications.

Table of Contents

1. Chapter: Basics of Real Analysis of One Variable
2. Chapter: Real-valued Sequences and Limits of Sequences and Functions
3. Chapter: Completeness of Real Numbers
4. Chapter: Criteria for Convergence of Sequences
5. Theorem: Defining Properties of the Field of Real Numbers
6. Chapter: Useful Inequalities
7. Chapter: Properties of Real Functions
8. Chapter: Types of Real Functions
9. Chapter: Infinite Series - Overview
10. Chapter: Infinite Products - Overview
11. Chapter: Representation of Functions as Taylor Series
12. Chapter: Power Series Introduction
13. Chapter: Fourier Series

Mentioned in:

Chapters: 1
Parts: 2 3

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Footnotes