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:

Theoretical minimum (in a nutshell)

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

Concepts you will learn in this part of BookofProofs

  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

Chapters: 1
Parts: 2 3

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  1. Here, we have used a short notation for multiplication, omitting the dot "$\cdot$". Note that the distributity laws cover subtraction as a special case of addition: $x(y-z)=xy-xz$ and $(x-y)z=xz-yz$. However, they only sometimes cover division as a special case of multiplication: Although we have $(x+y)\div z=x\div z+y\div z$, for $z\neq 0$, in general we have $x\div (y+z)\neq x\div y+x\div z$ for $(y+z)\neq 0$. This is because the multiplicative inverse element of $(y+z)$ equals $\frac 1{y+z}$, and does not equal $\frac 1 y+\frac 1z$ (which is $\frac{z+y}{yz}$ rather than $\frac 1{y+z}$).