Real Functions in One Variable

Most students already have assigned textbooks when studying Calculus I, therefore this book takes a different approach to Calculus. It focuses primarily on explaining the central theories and warns students of the areas where mistakes are traditionally made.

The publisher recently asked me to write an overview of the most common subjects in a first course of Calculus at university level. I have been very pleased by this request, although the task has been far from easy.

Since most students already have their recommended textbook, I decided instead to write this contribution in a totally different style, not bothering too much with rigoristic assumptions and proofs. The purpose was to explain the main ideas and to give some warnings at places where students traditionally make errors.

By rereading traditional textbooks from the first course of Calculus I realized that since I was not bound to a strict logical structure of the contents, always thinking of the students’ ability at that particular stage of the text, I could give some additional results which may be useful for the reader. These extra results cannot be given in normal textbooks without violating their general idea. This has actually been great fun to me, and I hope that the reader will find these additions useful. At the same time most of the usual stuff in these initial courses in Calculus has been treated.

When emphasizing formulæ I had the choice of putting them into a box, or just give them a number. I have chose the latter, because too many boxes would overwhelm the reader. On the other hand, I had sometimes also to number less important formulæ because there are local references to them. I hope that the reader can distinguish between these two applications of the numbering.

In the Appendix I have collected some useful formulæ, which the reader may use for references.

It should be emphasized that this is not an ordinary textbook, but instead a supplement to existing ones, hopefully giving some new ideas in how problems in Calculus can be solved.

It is impossible to avoid errors in any book, so even if I have done my best to correct them, I would not dare to claim that I have got rid of all of them. If the reader unfortunately should use a formula or result which has been wrongly put here (misprint or something missing) I do hope that my sins will be forgiven.

Leif Mejlbro

Preface

1. Complex Numbers
1.1 Introduction
1.2 Definition
1.3 Rectangular description in the Euclidean plane
1.4 Description of complex numbers in polar coordinates
1.5 Algebraic operations in rectangular coordinates
1.6 The complex exponential function
1.7 Algebraic operations in polar coordinates
1.8 Roots in polynomials

2. The Elementary Functions
2.1 Introduction
2.2 Inverse functions
2.3 Logarithms and exponentials
2.4 Power functions
2.5 Trigonometric functions
2.6 Hyperbolic functions
2.7 Area functions
2.8 Arcus functions
2.9 Magnitude of functions

3. Differentiation
3.1 Introduction
3.2 Definition and geometrical interpretation
3.3 A catalogue of known derivatives
3.4 The simple rules of calculation
3.5 Differentiation of composite functions
3.6 Differentiation of an implicit given function
3.7 Differentiation of an inverse function

4. Integration
4.1 Introduction
4.2 A catalogue of standard antiderivatives
4.3 Simple rules of integration
4.4 Integration by substitution
4.5 Complex decomposition of fractions of polynomials
4.6 Integration of a fraction of two polynomials
4.7 Integration of trigonometric polynomials

5. Simple Differential Equations
5.1 Introduction
5.2 Differential equations which can be solved by separation
5.3 The linear differential equation of first order
5.4 Linear differential equations of constant coefficients
5.5 Euler’s differential equation
5.6 Linear differential equations of second order with variable coefficients

6. Approximations of Functions
6.1 Introduction
6.2 ε - functions
6.3 Taylor’s formula
6.4 Taylor expansions of standard functions
6.5 Limits
6.6 Asymptotes
6.7 Approximations of integrals
6.8 Miscellaneous applications

A. Formulæ
A.1 Squares etc.
A.2 Powers etc.
A.3 Differentiation
A.4 Special derivatives
A.5 Integration
A.6 Special antiderivatives
A.7 Trigonometric formulæ
A.8 Hyperbolic formulæ
A.9 Complex transformation formulæ
A.10 Taylor expansions
A.11 Magnitudes of functions

Leif Mejlbro was educated as a mathematician at the University of Copenhagen, where he wrote his thesis on Linear Partial Differential Operators and Distributions. Shortly after he obtained a position at the Technical University of Denmark, where he remained until his retirement in 2003. He has twice been on leave, first time one year at the Swedish Academy, Stockholm, and second time at the Copenhagen Telephone Company, now part of the Danish Telecommunication Company, in both places doing research.

At the Technical University of Denmark he has during more than three decades given lectures in such various mathematical subjects as Elementary Calculus, Complex Functions Theory, Functional Analysis, Laplace Transform, Special Functions, Probability Theory and Distribution Theory, as well as some courses where Calculus and various Engineering Sciences were merged into a bigger course, where the lecturers had to cooperate in spite of their different background. He has written textbooks to many of the above courses.

His research in Measure Theory and Complex Functions Theory is too advanced to be of interest for more than just a few specialist, so it is not mentioned here. It must, however, be admitted that the philosophy of Measure Theory has deeply in uenced his thinking also in all the other mathematical topics mentioned above.

After he retired he has been working as a consultant for engineering companies { at the latest for the Femern Belt Consortium, setting up some models for chloride penetration into concrete and giving some easy solution procedures for these models which can be applied straightforward without being an expert in Mathematics. Also, he has written a series of books on some of the topics mentioned above for the publisher Ventus/Bookboon.