MATHEMATICAL
PREPARATION COURSE
before studying Physics
Accompanying Booklet to the Online Course:
www.thphys.uni-heidelberg.de/hefft/vk1
without Animations, Function Plotter
and Solutions of the Exercises
Klaus Hefft
Institute of Theoretical Physics
University of Heidelberg
Please send error messages to
k.hefft@thphys.uni-heidelberg.de
March 4, 2017
Contents
1 MEASURING:
Measured Value and Measuring Unit 5
1.1 The Empirical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Physical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Order of Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 SIGNS AND NUMBERS
and Their Linkages 13
2.1 Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.4 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 SEQUENCES AND SERIES
and Their Limits 27
3.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Monotony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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4 FUNCTIONS 39
4.1 The Function as Input-Output Relation or Mapping . . . . . . . . . . . . . 39
4.2 Basic Set of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.3 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.4 Functions with Kinks and Cracks . . . . . . . . . . . . . . . . . . . 52
4.3 Nested Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Mirror Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6 Monotony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.7 Bi-uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.8 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.8.1 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.8.2 Cyclometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.8.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.9 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.10 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 DIFFERENTIATION 77
5.1 Differential quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Differential Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Higher Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.5 The Technique of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5.1 Four Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5.2 Simple Differentiation Rules: Basic Set of Functions . . . . . . . . . 88
5.5.3 Chain and Inverse Function Rules . . . . . . . . . . . . . . . . . . . 92
5.6 Numerical Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.7 Preview of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 99
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6 TAYLOR SERIES 103
6.1 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Geometric Series as Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3 Form and Non-ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4 Examples from the Basic Set of Functions . . . . . . . . . . . . . . . . . . 107
6.4.1 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 108
6.4.3 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4.4 Further Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5 Convergence Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.6 Accurate Rules for Inaccurate Calculations . . . . . . . . . . . . . . . . . . 113
6.7 Quality of Convergence: the Remainder Term . . . . . . . . . . . . . . . . 116
6.8 Taylor Series around an Arbitrary Point . . . . . . . . . . . . . . . . . . . 117
7 INTEGRATION 121
7.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 Area under a Function over an Interval . . . . . . . . . . . . . . . . . . . . 123
7.3 Properties of the Riemann Integral . . . . . . . . . . . . . . . . . . . . . . 126
7.3.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.3.2 Interval Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.3.4 Mean Value Theorem of the Integral Calculus . . . . . . . . . . . . 129
7.4 Fundamental Theorem of Differential and Integral Calculus . . . . . . . . . 130
7.4.1 Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.4.2 Differentiation with Respect to the Upper Border . . . . . . . . . . 131
7.4.3 Integration of a Differential Quotient . . . . . . . . . . . . . . . . . 131
7.4.4 Primitive Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.5 The Art of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
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7.5.1 Differentiation Table Backwards . . . . . . . . . . . . . . . . . . . . 135
7.5.2 Linear Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.5.3 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.5.4 Partial Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.5.5 Further Integration Tricks . . . . . . . . . . . . . . . . . . . . . . . 143
7.5.6 Integral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.5.7 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.6 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.6.1 Infinite Integration Interval . . . . . . . . . . . . . . . . . . . . . . 148
7.6.2 Unbounded Integrand . . . . . . . . . . . . . . . . . . . . . . . . . 150
8 COMPLEX NUMBERS 155
8.1 Imaginary Unit and Illustrations . . . . . . . . . . . . . . . . . . . . . . . . 155
8.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.1.2 Imaginary Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.1.3 Definition of complex numbers . . . . . . . . . . . . . . . . . . . . . 157
8.1.4 Gauss Number Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.1.5 Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.1.6 Complex Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.2 Calculation Rules of Complex Numbers . . . . . . . . . . . . . . . . . . . . 164
8.2.1 Abelian Group of Addition . . . . . . . . . . . . . . . . . . . . . . . 164
8.2.2 Abelian Group of Multiplication . . . . . . . . . . . . . . . . . . . . 167
8.3 Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . 173
8.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.3.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.3.3 Graphic Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.3.4 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.3.5 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . 180
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8.3.6 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 181
8.3.7 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.3.8 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.3.9 General Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
9 VECTORS 195
9.1 Three-dimensional Euclidean Space . . . . . . . . . . . . . . . . . . . . . . 195
9.1.1 Three-dimensional Real Space . . . . . . . . . . . . . . . . . . . . . 195
9.1.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9.1.3 Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
9.1.4 Transformations of the Coordinate System . . . . . . . . . . . . . . 198
9.2 Vectors as Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
9.2.1 Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
9.2.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
9.2.3 Transformations of the Coordinate Systems . . . . . . . . . . . . . 207
9.3 Addition of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.3.1 Vector Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.3.2 Commutative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9.3.3 Associative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.3.4 Zero-vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
9.3.5 Negatives and Subtraction . . . . . . . . . . . . . . . . . . . . . . . 224
9.4 Multiplication with Real Numbers, Basis Vectors . . . . . . . . . . . . . . 225
9.4.1 Multiple of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.4.2 Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.4.3 Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.4.4 Linear Dependence, Basis Vectors . . . . . . . . . . . . . . . . . . . 227
9.4.5 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
9.5 Scalar Product and the Kronecker Symbol . . . . . . . . . . . . . . . . . . 230
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9.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9.5.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9.5.3 Commutative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.5.4 No Associative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.5.5 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.5.6 Distributive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.5.7 Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.5.8 Kronecker Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.5.9 Component Representation . . . . . . . . . . . . . . . . . . . . . . 235
9.5.10 Transverse Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
9.5.11 No Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
9.6 Vector Product and the Levi-Civita Symbol . . . . . . . . . . . . . . . . . 239
9.6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
9.6.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
9.6.3 Anticommutative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
9.6.4 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
9.6.5 Distributive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.6.6 With Transverse Parts . . . . . . . . . . . . . . . . . . . . . . . . . 245
9.6.7 Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
9.6.8 Levi-Civita Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
9.6.9 Component Representation . . . . . . . . . . . . . . . . . . . . . . 248
9.6.10 No Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.6.11 No Associative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.7 Multiple Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.7.1 Triple Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
9.7.2 Nested Vector Product . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.7.3 Scalar Product of Two Vector Products . . . . . . . . . . . . . . . . 258
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9.7.4 Vector Product of Two Vector Products . . . . . . . . . . . . . . . 259
9.8 Transformation Properties of the Products . . . . . . . . . . . . . . . . . . 262
9.8.1 Orthonormal Right-handed Bases . . . . . . . . . . . . . . . . . . . 262
9.8.2 Group of the Orthogonal Matrices . . . . . . . . . . . . . . . . . . . 263
9.8.3 Subgroup of Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 264
9.8.4 Transformation of the Products . . . . . . . . . . . . . . . . . . . . 265
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PREFACE
Johann Wolfgang von Goethe: FAUST, Part I
(transl. by Bayard Taylor)
WAGNER in Faust’s study to Faust:
How hard it is to compass the assistance
Whereby one rises to the source!
FAUST on his Easter walk to Wagner:
That which one does not know, one needs to use;
And what one knows, one uses never.
O happy he, who still renews
The hope, from Error’s deeps to rise forever!
From Knowledge to Skill
This course is intended to ease the transition from school studies to university
studies. It is intended to diminish or compensate for the sometimes pronounced differ-
ences in mathematical preparation among incoming students, resulting from the differing
standards of schools, courses and teachers. Forgotten and submerged material shall be
recalled and repeated, scattered knowledge collected and organized, known material re-
formulated, with the goal of developing common mathematical foundations. No new
mathematics is offered here, at any rate nothing that is not presented elsewhere, perhaps
even in a more detailed, more exact or more beautiful form.
The main features of this course to emphasize are its selection of material, its compact
presentation and modern format. Most of the material of an advanced mathematics
school course is selected less for the development of practical math skills, and more for
the purpose of intellectual training in logic and axiomatic theory. Here we shall organize
much of the same material in a way appropriate for university studies, in some places
supplementing and extending it a little.
It is well-known, that in the natural sciences you need to know mathematical terms and
operations. You must also be able to work effectively with them. For this reason the
many exercises are particularly important, because they allow you to determine your
own location in the crucial transition region “from knowledge to technique” We shall
especially stress practical aspects, even if thereby sometimes the mathematical sharpness
(and possibly also the elegance) is diminished. This online course is not a replacement for
mathematical lectures. It can, however, be a good preparation for these as well.
1
Repeating and training basic knowledge must start as early as possible, before gaps in
this knowledge begin to impede the understanding of the basic lectures, and psychological
barriers can develop. Therefore in Heidelberg the physics faculty has offered to physics
beginners, since many years during the two weeks prior to the start of the first lectures,
a crash course in form of an all-day block course. I have given this course several times
since 84/85, with listeners also from other natural sciences and mathematics. We can
well imagine that this course can make the beginning considerably easier for engineering
students as well. In Heidelberg the online version shall by no means replace the proven
crash courses for the beginners prior to their first semester. But it will usefully support
and augment these courses. And perhaps it will help incoming students with their prepa-
ration, and later with solidifying their understanding of the material. This need might
be especially acute for students beginning in summer semester, in particular when Easter
holiday is unusually late. Over years this course may also help serve to standardize the
material.
This electronic form of the course, free of charge available on the net, seems ideally suited
for use during the relaxation time between school graduation and the beginning of lectures
at university. A thoughtful student will have time to prepare, to cushion the unfortunately
still frequent small shock of the first lectures, if not to avoid it altogether. It seems to
us appropriate and meaningful to present this electronic form of the course (which is
accessible to you always, and not only two weeks before the semester), to augment and
deepen the treatment beyond what is normally possible in our block courses in intensive
contact with the Heidelberg beginner students. Furthermore we have often noticed in
practice that small excursions in “higher mathematics”, historical reviews and physical
applications beyond school knowledge energize and awaken a desire to learn more about
what is coming. I shall therefore also address here some “higher things”, especially toward
the ends of the chapters. I will put these topics however in small or larger inserts or
special exercises, so that they can be passed over without hesitation.
As you have seen from the quotation at the beginning from Goethe’s (1749-1832) Faust
we have to deal with an old problem. But you are now in the fortunate situation of having
found this course, and you can hope. Don’t hesitate! Begin! And have a little fun, too!
Acknowledgements
First of all I want to thank Prof. Dr. J
¨
org H
¨
ufner for the suggeation and invitation to revise my old, proven “Vorkurs”
manuscript. The idea was to redesign and reformat it attractively, making it accessible online, or in the form of the new
medium of CD-ROM, to a larger number of interested people (before, during and after the actual preparation course).
Thank you for many discussions including detailed questions, tips or formulations, and last not least for the continuous
encouragement during the long labor on a project full of vicissitudes.
Then my special thanks go to Prof. Dr. Hans-Joachim Nastold who helped and encouraged me by answering a couple of
mathematical questions nearly 50 years ago when I - coming from a law oriented home and a grammar school concentrating
on classical languages, without knowing any scientist and lacking any access to mathematical textbooks or to a library, and
2
confronted with two young brilliant mathematics lecturers - was in a similar, but even more hopeless situation than you
could possibly be in now. At that time I decided to someday do something really effective to reduce the math shock, if not
to overcome it, if I survived this shock myself.
Prof. Dr. Dieter Heermann deserves my thanks for his competent advice, his influential support and active aid in an early
stage of the project. I thank Dr. Thomas Fuhrmann cordially for his enthusiasm for the multimedia ideas, the first work
on the electronic conversion of the manuscript, for the programming of the three Java applets, and in particular for the
function plotter. To his wife Dr. Andrea Schafferhans-Fuhrmann I owe the correction of a detail important for the users of
the plotter.
I also have to thank the following members of the Institute for numerous discussions, suggestions and help, especially Prof.
F. Wegner for the attentive correction of the last chapters of the Word script in an early stage, Dr. E. Thommes for
exceptionally careful aid during the error location in the HTML text, Prof. W. Wetzel for tireless advice and inestimable
help in all sorts of computer questions, Dr. Peter John for relief with some illustrations, Mr. Ting Wang for computational
assistance and many other members of the Institute for occasional support and ongoing encouragement.
My main thanks go to my immediate staff: firstly to Mrs. Melanie Steiert and then particularly to Mrs. Dipl.-Math.
Katharina Schmock for the excellent transcription of the text into LATEX, Mrs. Birgitta Schiedt and Mr. Bernhard
Zielbauer for their enthusiasm and skill in transferring the TEX formulae into the HTML version and finally to Olsen
Technologies for the conception of the navigation and the fine organization of the HTML version. To the board of directors
of the Institute, in particular Prof. C. Wetterich and Prof. F. Wegner, I owe a great dept of gratitude for providing the
funds for this team in the decisive stage.
Furthermore I would like to thank the large number of interested students over the years who, through their rousing
collaboration and their questions during the course, or via even later feedback, have contributed decisively to the quality
and optimization of the compact form of my lecture script “Mathematical Methods of Physicists” of which the “Vorkurs” is
the first part. As a representative of the many whose faces and voices I remember better than their names I want to name
Bj
¨
orn Seidel. Many thanks also to all those users of the online course who spared no effort in reporting actual transference
problems, or remaining typing and other errors to me, and thus helped me to asymptotically approach the ideal of a faultless
text. My thanks go also to Prof. Dr. rer.nat.habil. L. Paditz for critical hints and suggestions for changes of the limits for
the arguments of complex numbers.
My thanks especially go to my former student tutors Peter Nalbach, Rainer Tafelmayer, Steffen Weinstock and Carola von
Saldern for their help in welcoming the beginners from near and far cheerfully, and motivating and encouraging them. They
raised the course above the dull routine of everyday and helped to make it an experience which one may remember with
pleasure even years later.
Finally I am full of sincere gratitude to my three children and my son-in-law Christoph L
¨
ubbe. Without their perpetual
encouragement and untiring help at all times of the day or night I never would have been been able to get so deeply into
the world of modern media. To them and to my grandchildren I want to dedicate this future-oriented project:
to ANGELIKA, JOHANNES, BETTINA and CHRISTOPH
as well as CAROLINE, TOBIAS, FABIAN, NIKLAS and HENRI.
After the online course resulted in a doubling of the number of German speaking beginners at the physics faculty in
Heidelberg within two years, an English version was suggested by Prof. Dr. Karlheinz Meier. I am deeply grateful to
cand. phil. transl. Aleksandra Ewa Dastych for her very careful, patient and indispensable help in composing this English
version. Also I owe thanks to Prof. K. Meier, Prof. Dr. J. Kornelius and Mr. Andrew Jenkins, B.A. for managing contact
to her. My thanks go also to the directors of the Institute, Prof. Dr. C. Wetterich and Prof. Dr. O. Nachtmann, for
providing financial support for this task. Many special thanks go to my friend Prof. Dr. Alfred Actor (from Pennsylvania
State University) for a very careful and critical expert reading of the English translation.
For the rapid and competent transfer of the English text to the LaTeX format in oder to allow easy printing my whole-
hearted thanks go to cand. phys. Lisa Speyer. The support for this work was kindly provided by the relevant commission
of our faculty under the chairman Prof. Dr. H.-Ch. Schultz-Coulon.
3
4
Chapter 1
MEASURING:
Measured Value and Measuring Unit
1.1 The Empirical Method
All scientific insight begins when a curious and attentive person wonders about some
phenomenon, and begins a detailed qualitative observation of this aspect of nature. This
observing process then can become more and more quantitative, the object of interest
increasingly idealized, until it becomes an experiment asking a well-defined question.
The answers to this experiment, the measured data, are organized into tables, and can
be graphically visualized in diagram form to facilitate the search for correlations and
dependencies. After calculating or estimating the precision of the measurement, the so-
called experimental error, one can interpolate and search for a description or at least
an approximation in terms of known mathematical curves or formulae From
such empirical connections, conformities to known laws may be discovered. These are
mostly formulated in mathematical language (e.g. as differential equations). Once one
has found such a connection, one wants to “understand” it. This means either one finds
a theory (e.g. some known physical laws) from which one can derive the experimentally
obtained data, or one tries using a “hypothesis” to guess the equation which underlies the
phenomenon. Obviously also for doing this task a lot of mathematics is necessary. Finally
mathematics is needed once again to make predictions which are intended to be checked
against experiments, and so on. In such an upward spiral science is progressing.
5
1.2 Physical Quantities
In the development of physics it turned out again and again how difficult, but also impor-
tant it was to develop the most suitable concepts and find the relevant quantities (e.g.
force or energy) in terms of which nature can be described both simply and comprehen-
sively.
Insert: History: It took more than 100 years for the discussion among the “nat-
ural philosophers” (especially D´Alembert, Bruno, Newton, Leibniz, Boskovic and
Kant) to create our modern concepts of force and action from the old terms prin-
cipium, substantia, materia, causa efficiente, causa formale, causa finale, effectum,
actio, vis viva and vis insita.
Every physical quantity consists of a a measured value and a measuring unit, i.e.
a pure number and a dimension. All difficulties in conversations are avoided, if we treat
both parts like a product “value times dimension”.
Example: Velocity: In residential districts often a speed limit v = 30
km
h
is imposed, which
means 30 kilometers per hour. How many meters is that per second?
One kilometer contains 1000 meters: 1km = 1000m, thus v = 30 · 1000
m
h
.
Every hour consists of 60 minutes: 1h = 60min, consequently v = 30 · 1000
m
60min
.
One minute has 60 seconds: 1 min = 60 s , therefore v = 30 ·1000
m
60·60s
= 8.33
m
s
.
Even that may be too fast for a ball playing child.
Insert: Denotations: It is an accepted thing in international physics for long
time past to abbreviate as many of the physical quantities as possible by the first
letter of the corresponding English word, e.g. s(pace), t(ime), m(ass), v(elocity),
a(cceleration), F(orce), E(nergy), p(ressure), R(esistance), C(apacity), V(oltage),
T(emperature), etc..
Of course there are some exceptions from this rule: e.g. momentum p, angular
momentum l, electric current I or potential V
Whenever the Latin alphabet is not sufficient, we use the Greek one:
alpha α A
beta β B
gamma γ Γ
delta δ
epsilon E
zeta ζ Z
eta η H
theta θ Θ
iota ι I
kappa κ K
lambda λ Λ
my µ M
ny ν N
xi ξ Ξ
omikron o O
pi π Π
rho ρ P
sigma σ Σ
tau τ T
ypsilon υ Y
phi φ Φ
chi χ X
psi ψ Ψ
omega ω
In addition the Gothic alphabet is at our disposal.
6
1.3 Units
The units are defined in terms of yardsticks. The search for suitable yardsticks and their
definition, by as international a convention as possible, is an important part of science.
Insert: Standard units: What can be used as a standard unit? - The an-
swers to this question have changed greatly through the centuries. Originally people
everywhere used easily available comparative quantities like cubit or foot as units of
length, and the human pulse beat as unit of time. (The Latin word tempora initially
meant temple!) But not every foot has equal length, and the pulse can beat more
quickly or slowly. Alone in Germany there have been more than 100 different cubit
and foot units in use.
Therefore, since 1795 people referred to the ten millionth part of the earth meridian
quadrant as the “meter” and represented this length by the well-known rod made out
of an alloy of platinum and iridium. The measurement of time was referred to the
earth’s rotation: for a long time the second was defined as the 86400th part of an
average solar day.
In the meantime more exact atomic standards have been introduced: One meter is
now the distance light travels within the 1/299 792 485 part of a second. One second
is now defined in terms of the period of a certain oscillation of cesium 133 atoms in
“atomic clocks”. Perhaps some day these standards will also be improved.
Today, these questions are solved after many error ways by the conventions of the SI-units
(Syst`eme International d’Unit´es) The following fundamental quantities are specified:
length measured in meters: m
time in seconds: s
mass in kilograms: kg
electric current in ampere: A
temperature in kelvin: K
luminous intensity in candelas: cd
even angle in radiant: rad
solid angle in steradiant: sr
amount of material in mol: mol
All remaining physical quantities are to be regarded as derived, thus by laws, definitions
or measuring regulations traced back to the fundamental quantities: e.g.
7
frequency measured in hertz: Hz := 1/s
force in newton: N := kg m/s
2
energy in joule: J := Nm
power in watt: W := J/s
pressure in pascal: Pa := N/m
2
electric charge in coulomb: C := As
electric potential in volt: V := J/C
electric resistance in ohm: := V/A
capacitance in farad: F := C/V
magnetic flux in weber: Wb := Vs
Exercise 1.1 SI-units
a) What is the SI-unit of momentum?
b) From which law can we deduce the unit of force?
c) Who formulated this law first?
d) What is the dimension of work?
e) What is the unit of the electric field strength?
Insert: Old units: Some examples of units which are still widely in use in spite
of the SI-convention:
grad:
= (π/180)rad = 0.01745 rad
kilometer per hour: km/h = 0.277 m/s
horse-power: PS = 735.499 W
calorie: cal ' 4.185 J
kilowatt-hour: kW h = 3.6 · 10
6
J
elektron volt: eV ' 1.6 · 10
19
J
Many non-metric units are still used especially in England and the USA:
inch = Zoll: in = ” = 2.54 cm
foot: ft = 12 in ' 0.30 m
yard: yd = 3 ft ' 0.9144 m
(amer.) mile: mil = 1760 yd ' 1609 m
ounce: oz ' 28.35 g
(engl.) pound: lb = 16 oz ' 0.454 kg
(amer.) gallon: gal ' 3.785 l
(amer.) barrel: bbl = 42 gal ' 158.984 l
8
Exercise 1.2 Conversion of units
a) You are familiar with the conversion of angles from degrees to radiants using your
pocket calculator: Calculate 30
, 45
, 60
, and 180
in radiant and 1 rad and 2 rad in
degrees.
b) How many seconds make up one sidereal year with 12 months, 5 days, 6 hours, 9
minutes and 9.5 seconds?
c) How much does it cost with an “electricity tariff of 0.112 /kWh, if you burn one night
long a 60-Watt bulb for six hours and your PC runs needing approximately 200 watts?
d) Maria and Lucas measure their training distance with a stick, which is 5 feet and 2
inches long. The stick fits in 254 times. What is the run called in Europe?
How many rounds do Maria and Lucas have to run, until they put a mile back?
e) Bill Gates said: “If General Motors had kept up with technology like the computer
industry has, we would all be driving twenty-five dollar cars that go 1000 miles per gallon.”
Did he mean the “3-litre car”?
1.4 Order of Magnitude
Natural phenomena are so various and cover so many orders of magnitude, that in
relation to a standard unit, e.g. meter, tiny or enormous numbers often result. Just think
of the diameter of an atom or the size of our Milky Way expressed in meters. In both
cases “useless” zeros arise. One has therefore introduced powers of ten and as well as
abbreviations and easily remembered names: e.g. the kilogram 1000 g = 103 g = kg. The
decimal prefixes, too, are today internationally standardized. We indicate the most
important ones:
tenth 10
1
= d dezi- ten 10
1
= D deka-
hundredth 10
2
= c centi- hundred 10
2
= h hecto-
thousandth 10
3
= m milli- thousand 10
3
= k kilo-
millionth 10
6
= µ mikro- million 10
6
= M mega-
billionth 10
9
= n nano- billion 10
9
= G giga-
trillionth 10
12
= p pico- trillion 10
12
= T tera-
quadrillionth 10
15
= f femto- quadrillion 10
15
= P peta-
Examples: In order to give you an idea of orders of magnitude, we give some examples
9
from the field of length measurement:
The diameter of the range, within which scattered electrons feel a proton, amounts
to about 1.4 fm, atomic nuclei are between 3 and 20 fm thick.
The wavelengths of gamma-rays lie within the range of pm. Atomic diameters reach
from 100 pm to 1 nm.
Important molecules are about 10 nm thick. 100 nm is the order of magnitude of
viruses, and also the wavelengths of visible light lie between 300 and 800 nm.
Bacteria have typical diameters of µm, our blood corpuscles of 10µm, and protozoan
measure some 100µm.
Thus we already come to your everyday life scale of pinheads: 1 mm, hazel-nuts: 1
cm and grapefruits: 1 dm.
Electromagnetic short waves are 10 to 100 m long, medium waves 100 m to 1 km
and oscillate with 1 MHz. The distance e.g. of the bridges over the Neckar river in
Heidelberg amounts to 1 km. Flight altitudes of the large airliners are about 10 km.
The diameter of the earth is to 12.7 Mm and that of the Jupiter is about 144 Mm.
The sun’s diameter is with 1.4 Gm, the average distance of the earth from the sun
is approximately 150 Gm, and Saturn circles at a distance of approximately 1.4 Tm
around the sun.
Finally, light travels 9.46 Pm in one year.
Insert: Billion: While these prefixes of the SI system are internationally fixed,
this is by no means so with our familiar number words . The Anglo-American
and also French expression “billion in the above table means the German “Mil-
liarde” = 10
9
and is different from the German Billion = 10
12
. “The origin of our
sun system 4,6 billion years ago...” must be translated as “die Entstehung unseres
Sonnensystems vor 4,6 Milliarden Jahren...”. Similar things apply to the Anglo-
American “trillion = 10
12
, while the German “Trillion= 10
18
.
Insert: Other unit names: Special names are also still used for some metric
units: You know perhaps 10
2
m
2
as are, 10
4
m
2
as hectare, 10
3
m
3
as litre, 10
2
kg
as quintal and 10
3
kg as ton.
Do you also know 10
5
P a as bar, 10
28
m
2
= bn as barn, 10
5
N = dyn, 10
7
J = erg,
10
15
m = fm under the name of Fermi, 10
10
m = 1
˚
A after
˚
Angstr
¨
om or 10
8
W b
under the name of Maxwell?
10
Exercise 1.3 Decimal prefixes
a) Express the length of a stellar year (365 d + 6 h + 9 min + 9.5 s) in megaseconds.
b) The ideal duration of a scientific seminar talk amounts to one microcentury.
c) How long does a photon need, in order to fly with the speed of light
c = 2.997 924 58 ·10
8
m/s 21 m far through the lecture-room?
d) With the Planck energy of E
p
= 1.22 · 10
16
TeV gravitation effects for the elementary
particles are expected. Express the appropriate Planck mass M
P
in grams.
In the following we are only concerned with the numerical values of the examined
physical quantities, which we read off usually in the form of lengths or angles from our
measuring apparatuses, these being calibrated for the desired measuring range in appro-
priate units of the measured quantities.
11
12
Chapter 2
SIGNS AND NUMBERS
and Their Linkages
The laws of numbers and their linkages are the main objects of mathematics. Although
numbers have developed from basic needs of human social interaction, and natural sci-
ence has inspired mathematics again and again, e.g. for differential and integral calculus,
mathematics actually does not belong to natural sciences, but rather to humanities. Math-
ematics does not start from empirical (i.e. measured) facts. Instead, it investigates the
logical structure of numbers and their generalizations within the human ability of thought.
In many cases empirical facts can be well represented in terms of these logical structures.
In this way mathematics became an indispensable tool for natural scientists and engineers.
2.1 Signs
Mathematics like every other science has developed its own language. This language
includes among other things some mathematical and logical signs, which we would like
to list here for quick, clear reference, because we will be using them continually:
Question game: Some mathematical signs
The meaning of the following mathematical signs is known to most of you. ONLINE you
can challenge yourself and click directly on the symbols to check if you are right. If your
browser does not support this, you will find a complete list of answers here:
13
+: plus -: minus ± : plus or minus
· : times /: divided by : is perpendicular to
<: is smaller than : is smaller or equal to : is much smaller than
=: is equal to 6=: is unequal to : is identically equal to
>: is bigger than : is bigger or equal to : is much bigger than
: angle between ': is approximately equal to : bigger than every number
Insert: Infinity: Physicists often use the sign , known as “infinity”, rather
casually. Assuming the meaning “bigger than every number” we avoid the problems
mathematicians warn us about: thus a < means a is a finite number. Shortly
we will use the combination of symbpols whenever we mean that a quantity is
“growing beyond all limits”.
In addition, we use the
Sum Sign
P
: for example
3
P
n=1
a
n
:= a
1
+ a
2
+ a
3
A famous example is the sum of the first m natural numbers:
m
X
n=1
n := 1 + 2 + . . . + (m 1) + m =
m
2
(m + 1),
just as the young Gauss has proved by skillful composition and clever use of brackets:
m
P
n=1
n = (1 + m) + (2 + (m 1)) + (3 + (m 2)) + . . . =
m
2
(m + 1).
Another example is the sum of the first m squares of natural numbers:
m
X
n=1
n
2
:= 1 + 4 + . . . + (m 1)
2
+ m
2
=
m
6
(m + 1)(2m + 1),
a formula we will later need for the calculation of integrals.
A further example is the sum of the first m powers of a number q:
m
X
n=0
q
n
:= 1 + q + q
2
+ . . . + q
m1
+ q
m
=
1 q
m+1
1 q
for q 6= 1,
which is known as the “geometrical” sum.
14
Insert: Geometric sum: Just as an exception, we want to prove the formula
for the geometric series which we will need several times. To do this we define the
sum
s
m
:= 1+ q + q
2
+ . . . + q
m1
+ q
m
,
then we subtract from this q · s
m
= q + q
2
+ q
3
+ . . . + q
m
+ q
m+1
and obtain (since nearly everything cancels)
s
m
q · s
m
= s
m
(1 q) = 1 q
m+1
, from which we easily get for q 6= 1 dividing by
(1 q) the above formula for s
m
.
Much more important than the product sign
Q
, defined analogously to the sum sign:
for instance
3
Q
n=1
a
n
:= a
1
· a
2
· a
3
is for us the
factorial sign ! : m! := 1 · 2 · 3 · . . . · (m 1) · m =
m
Q
n=1
n
(speak: “m factorial”), e.g. 3! = 1 ·2 ·3 = 6 or 5! = 120, augmented by the convention
0! = 1.
Question game: Some logical signs
From the logical symbols which most of you are familiar with from math class, we use the
following symbols to display logical connections in a simpler, more concise, and memorable
way, as well as to make it easier for us to memorize them. ONLINE you can click directly
on the symbols to get the answer. If your browser does not support this, you will find a
complete list of answers :
: is an element of 3: contains as element /: is no element of
: is a subset of or equal : contains as a subset or is equal := : is defined by
: there exists !: there exists exactly one : for all
: union of sets : intersection of sets : empty set
: from this it follows that, : this holds when, : this holds exactly when,
is a sufficient condition for is a necessary condition for is a nec. and suff. cond. for
These symbols will be explained once more when they occur for the first time in the text.
15
2.2 Numbers
In order to display our measured data we need the numbers which you have been familiar
with for a long time. In order to get an overview, we shall put together here their properties
as a reminder. In addition we recall some selected concepts which mathematicians have
formulated as rules for the combination of numbers, so that we can later on compare those
rules with the ones for more complicated mathematical quantities.
2.2.1 Natural Numbers
We begin with the set of natural numbers {1, 2, 3, . . .}, given the name N by number
theoreticians and called “natural” because they have been used by mankind to count
within living memory. Physicists think for instance of particle numbers, e.g. the number
L of atoms or molecules in a mole.
Insert: Avogadro’s Number: Avogadro’s Number L is usually named after
Amedeo Avogadro (only in Germany after Joseph Loschmidt). It is a number with
24 digits from which only the first six ( 602213 ) are reliably known, while the next
two (67) are possibly affected by an error of (±36) Physicists use the following way
of writing: N = 6.022 136 7(36) · 10
23
.
For long time now there have been two different linkages: the operation of addition and
multiplication, assigning a new natural number to each pair of natural numbers a, b N
(“the numbers a and b are elements of the set N) and therefore called internal linkages:
the ADDITION:
internal linkage: a + b = x N with the
Commutative Law: a + b = b + a and the
Associative Law: a + (b + c) = (a + b) + c and
the MULTIPLICATION:
internal linkage: a ·b or ab = x N also with a
Commutative Law: ab = ba and a
Associative Law: a(bc) = (ab)c and furthermore a
Neutral element: the one: 1a = a
16
Both linkages, addition and multiplication, are connected through the
Distributive Law: (a + b)c = ac + bc
with each other.
Insert: Shorthand: If we want to express that in the set of natural numbers
(n N) there exists only exactly one (!) element one which for all () natural
numbers a fulfils the equation 1a = a , we could express this using the logical signs
in the following manner: ! 1 N : a N 1a = a. Please appreciate this compact
logical writing.
Insert: Counter-examples: As an example of a linkage that leads out of a set,
we will soon deal with the well known scalar product of two vectors, which combines
their components into a simple number.
Non-communicative are for example the rotations of the match box shown in Figure
9.10 in a Cartesian coordinate system: First turn it clockwise around the longitu-
dinal symmetry axis parallel to the 3-axis and then around the shortest transversal
axis parallel to the 1-axis and compare the result with the position of the box after
you have performed the two rotations in the reversed order!
Counter-examples of the bracket law for three elements of a set are very hard to
find: From the home chemistry sector we remember the three ingredients for non-fat
whipped cream for children: ( sugar + egg-white ) + juice = cream. If you try to
whip first the egg-white together with juice as suggested by the instruction: sugar +
(egg-white + juice) you will never get the cream whipped.
We can clearly imagine the natural numbers as equally spaced points on a half line as
shown in the next figure:
For physicists it is sometimes convenient to add the zero 0 as you would with a ruler, and
so to extend N to N
0
:= N {0}. Through this, the addition operation also obtains a
uniquely defined
17
Neutral element: the zero: 0 + a = a
In “logical shorthand”: ! 0 N
0
: a N
0
: 0 + a = a in full analogy to the neutral
element of multiplication.
Insert: History: Even the ancient Greeks and Romans did not know numbers
other than the natural ones: N = {I, II, III, IV, . . .}. The Chinese knew zero as
“empty place” already in the 4th century BC. Not before the 12th century AD did
the Arabs bring the number zero to Europe.
2.2.2 Integers
Along with the progress in civilization and human culture it became necessary to extend
the numbers. For example, when talking about money it is not sufficient to know the
amount (e.g. the number of coins) we also need to be able to express whether we have or
owe that amount. Sometimes this is expressed through the colour of the number (“black
and red numbers”) or through a preceding + or sign. In the natural sciences such signs
have been established.
Physicists can shift a marking on their ruler by an arbitrary number of points to the right,
they will however encounter difficulties if they want to move it to the left. Mathematically
speaking does not have for all natural numbers a and b the equation a+x = b a solution x
which is itself a natural number: e.g. the equation 2+x = 1. Such equations can then only
be solved if we extend the natural numbers through the negative numbers {−a| a N}
to form the set of all integers:
To every positive element a there exists exactly one
Negative element a with: a + (a) = 0
Even for 1 we get a -1, meaning owing a pound in contrast to possessing a pound. In
“logical shorthand” : a Z ! a : a + (a) = 0.
Mathematicians refer to the set of integers, which consist of all natural numbers a N,
their negative partners a (N) and zero as Z := N {0} {−a| a N}.
With this extension, the above equation a + x = b has now, as desired, always a solution
for all pairs of integers, namely the difference x = b a which once again is an integer
18
x Z. We also say that Z is closed concerning addition: i.e. addition does not lead out
of the set. This brings us to a central concept in mathematics (and in physics), namely
that of a group :
We call a set of objects (e.g. the integers) a group, if
1. it is closed concerning an internal linkage (like e.g. addition),
2. an Associative Law holds (like e.g.: a + (b + c) = (a + b) + c),
3. it encloses exactly one neutral element (like e.g. the number 0) and
4. if there exists exactly one reversal for each element (like e.g. the negative element).
If moreover the Commutative Law (like e.g. a + b = b + a) holds, mathematicians call the
group Abelian.
Insert: Groups: Later on you will learn that groups play a very important role
in the search for symmetries in physics, e.g. for crystals or the classification of
elementary particles. The elements of a group are often operations, like e.g. rota-
tions: the result of two rotations performed one after the other can also be reached
by one single rotation. In performing three rotations the result does not depend on
the brackets. The operation no rotation leaves the body unchanged. Each rotation
can be cancelled. Usually these groups are not Abelian, e.g. two rotations performed
in different order yield different results. Therefore mathematicians did not incor-
porate the Commutative Law into the properties of groups. The more specialized
commutative groups are given the name Abelian after the Norwegian mathematician
Niels Henrik Abel (1802-1829).
We can imagine the integers geometrically as equidistant points on a whole straight line.
Insert: Absolute value: If we, while viewing a number decide to ignore its
sign, we use the term
19
absolute value: |a| := a for a 0 and |a| := a for a < 0,
so that |a| 0 a Z.
For Instance for the number 5 : | 5| = 5 and for the number 3 : |3| = 3 = 3.
The multiplication rule for the product of absolute values:
|a · b| = |a| · |b|
can easily be verified. For the absolute values of the sum and difference of integers
there hold only inequalities which we will meet later.
||a| |b|| |a ± b| |a| + |b|.
The second part is known as “Triangle Inequality.
The term |a b| then gives the distance between the numbers a and b on the line of
numbers.
All points a in the neighbourhood interval of a point a
0
, having a distance from a
0
which is smaller than a positive number ε is called a ε-neighbourhood U
ε
(a
0
) of a
0
:
Insert: ε-neighbourhood: You will often encounter the term of an ε-neighbourhood
you will often meet in future mathematics lectures:
a U
ε
(a
0
) |a a
0
| < ε with ε > 0.
We will use it only a few times here.
The Figure shows the ε-neighbourhood of the number 1 for ε = 1/2. It contains all numbers x
with 0.5 < x < 1.5. Realize that the borders (here 0.5 and 1.5) do not belong to the
neighbourhood.
20
2.2.3 Rational Numbers
Whenever people have been forced to do division, they have noticed that integers are not
enough. Mathematically speaking: to solve the equation a · x = b for a 6= 0 within a
number set we are forced to extend the integers to rational numbers Q by adding the
inverse numbers {
1
a
or a
1
|a Z}. We use the notation Z \ {0} for the set of integers
without the zero. Then we have for each integer a different from 0 exactly one
inverse element a
1
with: a · a
1
= 1
In “logical shorthand”: a Z \ {0} ! a
1
: a · a
1
= 1.
We are familiar with this concept. The inverse to the number 3 is
1
3
, the inverse number
to 7 is
1
7
.
This way the fraction x =
b
a
for a 6= 0 solves our starting equation ax = b as desired. In
general, a rational number is the quotient of two integers, consisting out of a numerator
and a denominator (different from 0). Rational numbers are therefore mathematically
speaking, ordered pairs of integers: x = (b, a).
Insert: Class: Strictly speaking one rational number is always represented by a
whole class of ordered pairs of integers, e.g. (1, 2) = (2, 4) = (3, 6) = (1a, 2a) for
a Q and a 6= 0 should be taken as one single number: 1/2 = 2/4 = 3/6 = 1a/2a :
Cancelling should not change the number, as we know.
When they are divided out, the rational numbers become finite, meaning breaking off
or periodic decimal fractions: for example
1
5
= 0.2 ,
1
3
= 0.3333333... = 0, 3 and
1
11
=
0.09090909... = 0.09, where the line over the last digits indicates the period.
With this definition of the inverse elements the rational numbers form a group not only
relative to addition, but also, relative to multiplication (with the Associative Law, the
one and the inverse elements). This group is, due to the Commutative Law of the factors
ab = ba Abelian.
Insert: Field: For sets which form groups subject to two internal linkages
connected by a Distributive Law mathematicians have created a special name because
of their importance: They call such a set a field.
21
The rational numbers lie densely on our number line, meaning in every interval we can
find countable infinity of them:
Because of the finite accuracy of every physical measurement the rational numbers are
in every practical aspect the working numbers of physics as well as in every other
natural science. This is why we had paid such an attention to their rules.
By stating results as rational numbers, mostly in the form of decimal fractions, scientists
worldwide have agreed on indicating only as many decimal digits as they have measured.
Along with every measured value the uncertainty should also be indicated. This for
example is what we find in a table for Planck’s quantum of action
~ = 1.054 571 68(18) · 10
34
Js.
This statement can also be written in the following way:
~ = (1.054 571 68 ± 0.000 000 18) · 10
34
Js
meaning that the value of ~ (with a probability of 68 %) lies between the following two
borders:
1.054 571 50 · 10
34
Js ~ 1.054 571 86 · 10
34
Js.
Exercise 2.1
a) Show with the above indicated prescription of Gauss for even m, that the formula for
the sum of the first m natural numbers
m
P
n=1
n =
m
2
(m + 1) holds also for odd m gilt.
b) Prove the above stated formula for the first m squares of natural numbers
m
P
n=1
n
2
=
m
6
(m + 1)(2m + 1) by considering
m
P
n=1
(n + 1)
3
c) What do the following statements out of the “particle properties data booklet” mean:
e = 1.602 176 53(14) · 10
19
Cb and m
e
= 9.109 382 6(16) · 10
31
kg?
22
Insert: Powers: Repeated application of the same factor we describe usually as
power with the number of factors as
exponent: b
n
:= b · b · b ···b in case of n factors b,
where the known
calculation rules b
n
b
m
= b
n+m
, (b
n
)
m
= b
n·m
and (ab)
n
= a
n
b
n
for n, m N
hold true. With the definitions b
0
:= 1 and b
n
:= 1/b
n
these calculation rules can
be extended to all integer exponents: n, m Z. Later we will generalize yet further.
As a first application of powers we mention the Pythagoras Theorem: In a right-
angled triangle the square over the hypotenuse c equals the sum of the squares over
both catheti a and b:
Pythagoras Theorem: a
2
+ b
2
= c
2
Figure 2.5 illustrates the Pythagoras Theorem, ONLY ONLINE with coloured
parallelograms indicating the geometrical proof.
23
Very frequently we need the so-called
binomial formulas: (a ± b)
2
= a
2
± 2ab + b
2
and (a + b)(a b) = a
2
b
2
,
which can be easily derived, but need to be memorized.
The binomial formulas are a special case (for n = 2) of the more general formula
(a ±b)
n
=
n
X
k=0
n!
k!(n k)!
a
nk
(±b)
k
,
where
n!
k!(nk)!
=:
n
k
are the so-called binomial coefficients. We can calculate them
either directly from the definition of the factorial, e.g.
5
3
=
5!
3!(5 3)!
=
1 ·2 · 3 · 4 · 5
1 ·2 · 3 · 1 · 2
= 10
or find them in the Pascal Triangle. This triangle is constructed in the following way:
n = 0 : 1
n = 1 : 1 1
n = 2 : 1 2 1
n = 3 : 1 3 3 1
n = 4 : 1 4 6 4 1
n = 5 : 1 5 10 10 5 1
n = 6 : 1 6 15 20 15 6 1
We start with the number 1 in the line n = 0. In the next line (n = 1) we write two ones,
one on each side. Then for n = 2 we add two ones to the left and right side once again,
and in the gap between them a 2 = 1 + 1 as the sum of the left and right “front man”
(in each case a 1). In the framed box, we once again recognize the formation rule. The
required binomial coefficient
5
3
is then found in line n = 5 on position 3.
Exercise 2.2
a) Determine the length of the space diagonal in a cube with side length a.
b) Calculate (a
4
b
4
)/(a b).
c) Calculate
n
0
and
n
n
.
d) Calculate
7
4
and
8
3
.
e) Show that
n
nk
=
n
k
holds true.
f) Prove the formation rule for the Pascal Triangle:
n
k1
+
n
k
=
n+1
k
.
24
2.2.4 Real Numbers
Mathematicians were however not fully satisfied with the rational numbers, seeing how
for example something as important as the circumference π of a circle with the diameter
of 1 is not a rational number: π / Q. They also wanted the solution of the equation
x
2
= a at least for a 6= 0, as well as the roots x = a
1/2
=:
a to be included. This is why
the rational numbers (by addition of infinite decimal fractions) have been extended to the
real numbers R which can be mapped one-to-one onto a straight line R
1
(meaning every
point on the line corresponds to exactly one real number).
Insert: History: Already in antiquity some mathematicians knew that there are
numbers which cannot be represented as fractions. They showed this with a so-called
indirect proof:
If e.g. the diagonal of a square with side length 1 were a rational number, like
2 = b/a, two natural numbers b, a N would exist with b
2
= 2a
2
. Think now of
the prime factor decompositions of b and a. On the left hand side of the equation
there stands an even number of these factors, because of the square each factor
appears twice. On the right hand side, however, an odd number of factors shows up,
because in addition the factor 2 appears. Since the prime factor decomposition is
unique, the equation cannot be right.
With this it is shown that the assumption,
2 can be represented as a fraction, leads
to a contradiction and thus must be wrong.
With the real numbers, which have the same calculation rules of a field as the rational
numbers, both solutions of the general
quadratic equation: x
2
+ ax + b = 0, x
1,2
=
a
2
±
r
a
2
4
b
will then be real numbers, as long as the discriminant under the root is not negative:
a
2
4b.
Insert: Preview: complex numbers: Later in Chapter 8 we will go one
step further by introducing the complex numbers C for which e.g. also x
2
= a
for a < 0 is always solvable and, amazingly enough, many other beautiful laws hold.
25
26
Chapter 3
SEQUENCES AND SERIES
and Their Limits
Direct mathematical study of sequences and series are, for natural scientists, less im-
portant than the fact that they greatly help us to understand and perform the limiting
procedures which are of fundamental importance in physics. For this reason, we have
combined in this chapter the most important facts of this part of mathematics. Later you
will deal in greater detail with these things in your future mathematics lectures.
3.1 Sequences
The first important mathematical concept we have to inspect is that of a sequence.
With this physicists think for instance of the sequence of the bounce heights of a steel
ball on a plate, which due to the inevitable dissipation of energy decrease with time and
tend more or less quickly to zero. After a while, the ball remains still. The resulting
physical sequence of the jump heights has only a finite number of non-vanishing members
in contrast to the ones that are of interest to mathematicians: Mathematically, a sequence
is an infinite set of numbers which can be numbered consecutively, i.e. labelled by the
set of the natural numbers: (a
n
)
nN
. Because it is impossible to list all infinite many
members (a
1
, a
2
, a
3
, a
4
, a
5
, a
6
, . . .), a sequence is mostly defined by the “general member”
a
n
, which is a law stating how to calculate the individual members of the sequence. Let
us look at the following typical examples which already enable us to display all important
concepts:
27
(F1) 1, 2, 3, 4, 5, 6, 7, . . . = (n)
nN
the natural numbers themselves
(F2) 1, 1, 1, 1, 1, 1, . . . = ((1)
n+1
)
nN
a simple “alternating” sequence,
(F3) 1,
1
2
,
1
3
,
1
4
,
1
5
, . . . =
1
n
nN
the inverse natural numbers,
the so-called “harmonic” sequence,
(F4) 1,
1
2
,
1
6
,
1
24
, . . . =
1
n!
nN
the inverse factorials,
(F5)
1
2
,
2
3
,
3
4
,
4
5
, . . . =
n
n+1
nN
a sequence of proper fractions and
(F6) q, q
2
, q
3
, q
4
, q
5
. . . = (q
n
)
nN
, q R the “geometric” sequence.
Insert: Compound interest: Many of you know the geometrical sequence
from school because it causes a capital K
0
at p% compound interest after n years to
increase to K
n
= K
0
q
n
with q = 1 +
p
100
.
In order to give us a first clear idea of these sample sequences, we have plotted the
sequence members a
n
(in the 2-direction) over the equidistant natural numbers n (in the
1-direction) in the following Cartesian coordinate system in a plane:
Figure 3.1: Visualization of our sample sequences over the natural numbers, in case of the
geometrical sequence (F6) for q = 2 and q =
1
2
.
Also the sum, the difference or the product of two sequences are again a sequence. For
example, the sample sequence (F5) with a
n
=
n
n+1
=
n+11
n+1
= 1
1
n+1
is the difference
of the trivial sequence (1)
nN
= 1, 1, 1, . . ., consisting purely of ones, and the harmonic
sequence (F3) except for the first member.
The termwise product of the sample sequences (F2) and (F3) makes up a new sequence:
(F7) 1,
1
2
,
1
3
,
1
4
, ... =
(1)
n+1
n
nN
the “alternating” harmonic sequence.
Similarly the termwise product of the harmonic sequence (F3) with itself is once again a
sequence:
28
(F8) 1,
1
4
,
1
9
,
1
16
, ... =
1
n
2
nN
the sequence of the inverse natural squares.
The termwise product of the sample sequences (F1) and (F6), too, gives a new sequence:
(F9) q, 2q
2
, 3q
3
, 4q
4
, 5q
5
. . . = (nq
n
)
nN
, q R a modified geometric sequence.
An other more complicated combined sequence will attract our attention later:
(F10) 2, (
3
2
)
2
, (
4
3
)
3
, . . . =
(1 +
1
n
)
n
nN
the so-called exponential sequence.
Exercise 3.1 Illustrate these additional sample sequences graphically. Project the
points on the 2-axis.
There are three characteristics that are of special interest to us as far as sequences are
concerned: boundedness, monotony and convergence:
3.2 Boundedness
A sequence is called bounded above, if there is an upper bound B for the members of
the sequence: a
n
B: in shorthand notation this means:
(a
n
)
nN
bounded above B : a
n
B n N
Bounded below is defined in full analogy with a lower lower bound A:
A : A a
n
n N.
For example, our first sample sequence (F1) consisting of the natural numbers is bounded
only from below e.g. by 1: A = 1. The alternating sequence (F2) is obviously bounded
from above and from below, e.g. by A = 1 and B = 1, respectively. For the harmonic
sequence (F3) the first member, the 1, is an upper bound: B = 1
1
n
n N and the
zero a lower one: A = 0. The sample sequence (F4) of the inverse factorials has the lower
bound A = 0 and the upper one B = 1.
Exercise 3.2 Investigate the boundedness of the other two of our sample sequences.
29
3.3 Monotony
A sequence is said to be monotonically increasing, if the successive members increase
with increasing number: To memorize:
(a
n
)
nN
monotonically increasing a
n
a
n+1
n N.
If the stronger condition a
n
a
n+1
holds true, one calls the sequence strictly monotonic
increasing.
In full analogy, monotonically decreasing is defined with a
n
a
n+1
.
For example, the sequence (F1) of the natural numbers is strictly monotonic increasing,
the alternating harmonic sequence (F2) is not monotonic at all and the harmonic sequence
(F3) as well as the sequence (F4) of the inverse factorials are strictly monotonic decreasing.
Exercise 3.3 Monotonic sequences
Investigate the monotony of the other two of our sample sequences.
3.4 Convergence
Now we come to the central topic of the whole chapter: As you may have seen from the
projection of the visualizing points onto the 2-axis there are sequences, whose members
a
n
accumulate around a number a on the number line, so that infinitely many members
of the sequence lie in every ε-neighbourhood U
ε
(a) of this number a, which by the way
needs not necessarily to be itself a member of the sequence. We call a in such a case a
cluster point of the sequence.
In our examples we immediately realize that the sequence (F1) of the natural numbers
has none and the harmonic sequence (F3) has one cluster point, namely the zero. The
alternating sequence (F2) has even two cluster points: one at +1 and one at 1.
The Theorem of Bolzano and Weierstrass guarantees, that every sequence which is
bounded above and below has to have at least one cluster point.
In the case that a sequence has only one single cluster point, it may occur that all sequence
members from a certain number on, lie in the neighbourhood of that point. We then call
this point the limit of the sequence and this situation turns out to be the central concept
30
of analysis: Therefore mathematicians have several terms for it: They also say that the
sequence converges or is convergent to a and write: lim
n→∞
a
n
= a, or sometimes more
casually: a
n
n→∞
a.
(a
n
)
nN
convergent: a : lim
n→∞
a
n
= a
ε > 0 N(ε) N : |a
n
a| < ε n > N(ε).
The last shorthand reads: for every pre-set positive number ε which may be as tiny as
you like, you can find a number N(ε) so that the distance from the cluster point a for all
sequence members with a number larger than N(ε) is smaller than the pre-given small ε.
For many sequences we can recognize the convergence or even the limit value with some
skill just by looking at it. But sometimes it is by no means easy to determine whether
a sequence is convergent. This is why the Theorem of Bolzano and Weierstrass is so
much appreciated: It shows us very generally when we can conclude the convergence of a
sequence:
Theorem of Bolzano and Weierstrass:
Every monotonically increasing sequence which is bounded above is
convergent, and
every monotonically decreasing sequence which is bounded below is
convergent, respectively.
In all cases where the limiting value is unknown or not easily identifiable mathematicians
often make also use of the necessary () and sufficient ()
Cauchy-Criterion: (a
n
) convergent
ε > 0 N(ε) N : |a
n
a
m
| < ε n, m > N(ε)
meaning, a sequence converges if and only if from a certain point onward the distances
between the members of the sequence decrease more and more, i.e. the corresponding
points on the number axis move closer and closer together. If that is not the case the
sequence diverges. In addition, it can be shown that every subsequence of a convergent
sequence and the sum and difference as well as the product and (provided the denominator
31
is different from zero) also the quotient of two convergent sequences are convergent as well.
This means that the limit is commutable with the rational arithmetic operations.
Many convergent sequences tend to zero as their cluster point, we call them zero sequences.
The harmonic sequence (F3) with a
n
=
1
n
is for example such a zero sequence.
Insert: Convergence proofs: For the sequence F3:(
1
n
)
nN
we want to test
all convergence criteria:
1. Most easily we check the Theorem of Bolzano and Weierstrass: the sequence
(F3)(
1
n
)
nN
is monotonically decreasing and bounded below: 0 <
1
n
, consequently it
converges.
2. The cluster point is apparently a = 0 : We pre-set an ε > 0 arbitrarily, e.g.
ε =
1
1000
and look for a number N(ε), so that |a
n
a| = |
1
n
0| = |
1
n
| =
1
n
< ε for
n > N(ε). That is surely the case if we choose N (ε) as the next natural number
larger than
1
ε
: N (ε) >
1
ε
(e.g. for ε = 0.001 we take N(ε) = 1001). Then there
holds for all n > N(ε) :
1
n
<
1
N(ε)
< ε.
3. Finally also the Cauchy Criterion can easily be checked here: If a certain ε > 0
is pre-given, it follows for the distance of two members a
n
and a
m
with n < m :
|a
n
a
m
| = |
1
n
1
m
| = |
mn
nm
| < |
m
nm
| =
1
n
< ε, if n > N(ε) =
1
ε
.
The sequences (F1) and (F2) obviously do not converge.
Exercise 3.4 Convergent sequences
a) Test the other three sample sequences for convergence.
b) Calculate - in order to become cautious - the first ten members of the sequence a
n
=
n · 0.9
n
, the product of (F1) with (F6) for q = 0.9, and compare with a
60
, as well as of
a
n
=
n!
10
n
, the quotient of (F6) for q =
1
10
and (F4), and compare with the corresponding
a
60
.
c) The sequence consisting alternately of the members of (F1) and (F3): i.e. 1,
1
2
,3,
1
4
,5,
1
6
,. . .
has only one single cluster point, namely 0. Does it converge to 0?
3.5 Series
After having studied the limits of number sequences, we can apply our newly acquired
knowledge to topics which occur more often in physics, for instance infinite sums s =
P
n=1
a
n
, called series:
32
These are often encountered sometimes in more interesting physical questions: For in-
stance if we want to sum up the electrostatic energy of infinitely many equidistant alter-
nating positive and negative point charges for one chain link (which gives a simple but
surprisingly good one-dimensional model of a ion crystal) we come across the infinite sum
over the members of the alternating harmonic sequence (F7): the series
P
n=1
(1)
n+1
n
. How
do we calculate this?
Series are sequences whose members are finite sums of real numbers: The definition of a
series
P
n=1
a
n
as sequence of partial sums s
m
=
m
P
n=1
a
n
mN
reduces the series to sequences which we have been dealing with just above.
Especially, a series is exactly then convergent and has the value s, if the sequence of its
partial sums s
m
(not that of its summands a
n
!!) converges: lim
m→∞
s
m
= s:
series s
m
=
m
P
n=1
a
n
convergent lim
m→∞
m
P
n=1
a
n
= s <
Also the multiple of a convergent series and the sum and difference of two convergent
series are again convergent.
The few sample series that we need, to see the most important concepts, we derive simply
through piecewise summing up our sample sequences:
(R1) The series of the partial sums of the sequence (F1) of the natural numbers:
s
m
=
m
P
n=1
n
mN
= 1, 3, 6, 10, 15, . . . is clearly divergent.
(R2) The series made out of the members of the alternating sequence (F2) always jumps
between 1 and 0 and has therefore two cluster points and consequently no limit.
(R3) Also the“harmonic series” summed up out of the members of the harmonic sequence
(F3), i.e. the sequence
s
m
=
m
P
n=1
1
n
mN
= 1,
3
2
,
11
6
,
25
12
,
137
60
, . . . is divergent. Because the
(also necessary) Cauchy Criterion is not fulfilled: If we for instance choose ε =
1
4
> 0 and
consider a piece of the sequence for n = 2m consisting of m terms: |s
2m
s
m
| =
2m
P
n=m+1
1
n
=
33
1
m+1
+
1
m+2
+ . . . +
1
2m
>
1
2m
+
1
2m
+ . . . +
1
2m
| {z }
m summands
=
1
2
> ε =
1
4
while for convergence < ε
would have been necessary.
(R7) Their alternating variant however, created out of the sequence (F7), our physical
example from above, converges
P
n=1
(1)
n+1
n
(= ln 2, as we will show later).
Because of this difference between series with purely positive summands and alternating
ones, it is appropriate to introduce a new term: A series is said to be absolutely convergent,
if already the series of the absolute values converges.
Series s
m
=
m
P
n=1
a
n
absolutely convergent lim
m→∞
m
P
n=1
|a
n
| <
We can easily understand that within an absolutely convergent series the summands can
be rearranged without any effect on the limiting value. Two absolutely convergent series
can be multiplied termwise to create a new absolutely convergent series.
For absolute convergence the mathematicians have developed various sufficient criteria,
the so-called majorant criteria which you will deal with more closely in the lecture about
analysis:
Insert: Majorants: If a convergent majorant sequence S = lim
m→∞
S
m
=
P
n=1
M
n
exists with positive M
n
> 0, whose members are larger than the corresponding
absolute values of the sequence under examination M
n
|a
n
|, then the series
lim
m→∞
s
m
=
P
n=1
a
n
is absolutely convergent, because from the Triangle Inequality
it follows
|s
m
| = |
m
P
n=1
a
n
|
m
P
n=1
|a
n
|
m
P
n=1
M
n
= S
m
.
Very often the “geometric series”
(R6):
P
n=0
q
n
, which follow from the geometric sequences (F6) (q
n
)
nN
, q R , serve as
majorants. To calculate them we benefit from the earlier for q 6= 1 derived geometric sum:
lim
m→∞
m
X
n=0
q
n
= lim
m→∞
1 q
m+1
1 q
=
1
1 q
< ,
meaning convergent for |q| < 1 and divergent for |q| 1.
34
Insert: Quotient criterion: We present here as example for a majorant
criterion only the quotient criterion which is obtained through comparison with
the geometric series:
If lim
n→∞
|
a
n+1
a
n
| < 1, is s
m
=
m
P
n=1
a
n
absolutely convergent.
As an example we prove the absolute convergence of the series (R9)
P
n=0
nq
n
for
|q| < 1, which can be obtained from the for |q| < 1 convergent geometric series
(R6) through termwise multiplication with the divergent sequence (F1) of the natural
numbers. We calculate therefore
lim
n→∞
a
n+1
a
n
= lim
n→∞
(n + 1)q
n+1
nq
n
= |q| lim
n→∞
n + 1
n
= |q| < 1.
That the criterion is not necessary can be seen from the series (R8), the summing
up of the sample sequence (F8):
P
n=1
1
n
2
=
π
2
6
, which is absolutely convergent, since all members are positive, but
lim
n→∞
n
2
(n+1)
2
= lim
n→∞
1
(1+n
1
)
2
= 1.
(R4) The series of the inverse natural factorials
P
n=1
1
n!
deserves to be examined in more
detail:
First we realize that the sequence of the partial sums
s
m
=
m
P
n=1
1
n!
mN
increases mono-
tonically: s
m+1
s
m
=
+1
(m+1)!
> 0. To get an upper bound B we estimate through the
majorant geometric sum with q =
1
2
:
|s
m
| = 1 +
1
2!
+
1
3!
+ . . . +
1
m!
< 1 +
1
2
+
1
2
2
+ . . . +
1
2
m1
=
m1
X
n=0
(
1
2
)
n
=
1 (
1
2
)
m
1
1
2
<
1
1
1
2
= 2.
35
Since the monotonically increasing sequence of the partial sums s
m
is bounded from above
by B = 2 the Theorem of Bolzano and Weierstrass guarantees us convergence. We just
do not know the limiting value yet. This limit is indeed something fully new - namely
an irrational number. We call it = e 1, so that the number e after the supplementary
convention 0! = 1 is defined by the following series starting with n = 0:
Exponential series defined by: e :=
P
n=o
1
n!
.
Insert: The number e is irrational: we prove indirectly that the so defined
number e is irrational, meaning it cannot be presented as quotient of two integers g
and h:
If e were writable in the form e =
g
h
with integers g and h 2, then h!e = (h 1)!g
would be an integer:
However, from definition it holds
(h 1)!g = h!e = h!
X
n=0
1
n!
=
h
X
n=0
h!
n!
+
X
n=h+1
h!
n!
=
h! + h! +
h!
2!
+
h!
3!
+ . . . + 1
+
+ lim
n→∞
1
h + 1
+
1
(h + 1)(h + 2)
+ . . . +
1
(h + 1)(h + 2) . . . (h + n)
.
While the first bracket is an integer if h is, this cannot be true for the second bracket,
because
1
h + 1
+
1
(h + 1)(h + 2)
+ . . . +
1
(h + 1)(h + 2) . . . (h + n)
= ...
=
1
h + 1
1 +
1
h + 2
+ . . . +
1
(h + 2) . . . (h + n)
,
which can be estimated through the geometric series with q =
1
2
as follows,
<
1
h + 1
1 +
1
2
+ . . . +
1
2
n1
=
1
h + 1
·
1 (
1
2
)
n
1 (
1
2
)
<
1
h + 1
·
1
1 (
1
2
)
=
2
h + 1
2/3,
Because h should be h 2 there is a contradiction. Consequently e must be irra-
tional.
36
To get the numerical value of e we first calculate the members of the zero sequence
(F4) a
n
=
1
n!
:
a
1
=
1
1!
= 1, a
2
=
1
2!
=
1
2
= 0.50, a
3
=
1
3!
=
1
6
= 0.1666,
a
4
=
1
4!
=
1
24
= 0.041 666, a
5
=
1
5!
=
1
120
= 0.008 33,
a
6
=
1
6!
=
1
720
= 0.001 388, a
7
=
1
7!
=
1
5 040
= 0.000 198,
a
8
=
1
8!
=
1
40 320
= 0.000 024, a
9
=
1
9!
=
1
362 880
= 0.000 002, . . .
then we sum up the partial sums: s
m
=
m
P
n=1
1
n!
= 1 +
1
2!
+
1
3!
+
1
4!
+ . . . +
1
m!
s
1
= 1, s
2
= 1.50, s
3
= 1.666 666, s
4
= 1.708 333,
s
5
= 1.716 666, s
6
= 1.718 055, s
7
= 1.718 253,
s
8
= 1.718 278, s
9
= 1.718 281, . . ..
If we look at the rapid convergence, we can easily imagine that after a short calculation
we receive the following result for the limiting value: e = 2.718 281 828 459 045 . . .
Insert: A sequence converging to e: Besides this exponential series which
we used to define e there exists as earlier mentioned in addition a sequence, con-
verging to the number e, the exponential sequence(F10):
(1 +
1
n
)
n
nN
= 2, (
3
2
)
2
, (
4
3
)
3
, . . . , which we will shortly deal with for comparison:
According to the binomial formula we find firstly for the general sequence member:
a
n
= (1 +
1
n
)
n
=
n
X
k=0
n!
(n k)!k!n
k
= 1 +
n
n
+
n(n 1)
n
2
2!
+
n(n 1)(n 2)
n
3
3!
+ . . . +
n(n 1)(n 2) . . . (n (k 1))
n
k
k!
+
. . . +
n!
n
n
n!
= 1 + 1 +
(1
1
n
)
2!
+
(1
1
n
)(1
2
n
)
3!
+ . . . +
(1
1
n
)(1
2
n
) . . . (1
k1
n
)
k!
+ . . .
+
(1
1
n
)(1
2
n
) . . . (1
n1
n
)
n!
On the one hand we enlarge this expression for a
n
, by forgetting the subtraction
of the multiples of
1
n
within the brackets:
a
n
1 + 1 +
1
2!
+
1
3!
+ . . . +
1
n!
= 1 + s
n
and reach so (besides the term one) the corresponding partial sums of the exponential
series s
n
. Thus the exponential series is a majorant for the also monotonically
increasing exponential sequence and ensures the convergence of the sequence through
that of the series. For the limiting value we get:
37
lim
n→∞
a
n
e.
On the other hand we diminish the above expression for a
n
by keeping only
the first (k + 1) of the without exception positive summands and throwing away the
other ones:
a
n
1 + 1 +
(1
1
n
)
2!
+
(1
1
n
)(1
2
n
)
3!
+ . . . +
(1
1
n
)(1
2
n
) . . . (1
(k1)
n
)
k!
.
When we now first let the larger n, of the two natural numbers tend to infinity, we
get:
a := lim
n→∞
a
n
1 + 1 +
1
2!
+
1
3!
+ . . . +
1
k!
= 1 + s
k
and after letting also the smaller natural number k tend to infinity we reach:
a e.
Consequently the limit a := lim
n→∞
a
n
of the exponential sequence a
n
must be equal
to the number e defined by the exponential series:
lim
n→∞
(1 +
1
n
)
n
=
P
n=0
1
n!
= e
When you, however, calculate the members of the sequence and compare them with
the partial sums of the series, you will realize that the sequence converges much more
slowly than the series.
Through these considerations we now have got a first overview over the limiting procedures
and some of the sequences and series important for natural sciences with their limits, which
will be of great use for us in the future.
38
Chapter 4
FUNCTIONS
4.1 The Function as Input-Output Relation or Map-
ping
We would like to remind you of the empirical method of physics discussed in Chapter 1,
and take a look at the simplest, but common case: in an experiment we investigate the
mutual dependency of two physical quantities: y as a function of x or y = f(x): In our
experiment one quantity x, called the independent variable, is measurably changed and
the second quantity y, the dependant variable, is measured in each case. We may imagine
the measuring apparatus in the way depicted below as a black box, into which the x are
fed in as input, and from which the corresponding y come out as output.
Figure 4.1: Function as a black box with x as input and y as output
Physicists think for example of an electric circuit where the voltage is changed gradually by
a potentiometer and the electric current is measured with a mirror galvanometer in order
to investigate the characteristic curve. Also the time development of the amplitude of a
pendulum or a radioactively decaying material as function of time are further candidates
out of the huge number of physical examples.
39
The result of such a series of measurements is first of all a value table (x, y). The data can
also be displayed in a graphic illustration, as shown below in our samples. Illustration of
the functions as a picture, usually called by us graph, through plotting the measured values
in a plane with a Cartesian (meaning right-angled) coordinate system (with the abscissa
x on the 1-axis and the ordinate y on the 2-axis) is a matter of course for physicists.
In the following figures you will find examples for value tables, graphic illustrations and
interpolating functions for a swinging spiral spring
x
cm
F
mN
1 0.42
1.5 0.55
2 0.82
2.5 1.03
3 1.25
3.5 1.45
4 1.65
4.5 1.80
5 1.95
5.5 2.20
6 2.35
6.6 2.60
Figure 4.2 a: Reaction force F of the spring measured in mN in dependency on the
amplitude x in cm.
x
cm
E
mJ
1 0.6
1.5 1.0
2.5 2.8
2.9 3.9
3.1 4.8
3.5 6.1
Figure 4.2 b: Potential energy E stored in the spring measured in mJ in dependency on
the amplitude x in cm.
40
t
cs
x
cm
0.3 3.5
0.5 2.8
0.7 1.2
1.1 1.8
1.7 3.2
2.4 0.8
2.6 1.5
3.2 2.4
3.6 1.4
4.3 1.1
4.8 1.8
Figure 4.2 c: Deflection amplitude x of the spiral spring measured in cm in dependency
on the time t in s.
M
g
T
s
2.5 0.75
10 1.63
14 1.91
20 2.23
25 2.46
Figure 4.2 d: Oscillation time T of the spiral spring in s as a function of the mass M in
g with unaltered spring constant D.
D
Nm
1
T
s
3 3.25
4 2.72
5 2.16
7 1.75
8 1.71
10 1.59
Figure 4.2 e: Oscillation time T of the spiral spring in s as function of the deflecting force
D measured in Nm
1
with constant mass M.
41
After we have taken into account the inevitable measurement errors, we can start con-
necting the measured points by a curve or a mathematical calculation instruction, to look
for a function which describes the dependence of the two quantities. If we succeed in
finding such a function, we have achieved real progress: A mathematical formula is usu-
ally short and concise; it can be stapled, processed and conveyed to others much easier
than extensive value tables. With its help we are able to interpolate more closely between
the measurements and to extrapolate beyond the measured area, which suggests further
experiments. Finally it is the first step towards a theory, and with it to the understanding
of the experiment.
Insert: History: T. Brahe measured in his laboratory the position of the planet
Mars at different times. From that value table J. Kepler found the ellipse as an
interpolating function for the orbit curve. This result influenced I. Newton in finding
his gravitation law.
Therefore, for physical reasons, we have to deal with functions, first with real functions
of a real variable.
Mathematically, we can consider a function y = f(x) as an unambiguous mapping x
f(x) of a point x of the area D
f
, (the definition domain of f) of the independent
variable x (also known as abscissa or argument) onto a point f(x) of the area W
f
(the
value domain of f) of the dependent variable y (also known as ordinate or function
value).
While the declaration of the definition domain in addition to the mapping prescription
is absolutely necessary for a function, and often influences the properties of the function,
the exact statement of the value domain W
f
:= {f(x)|x D
f
} is in most cases of less
importance and sometimes takes much effort.
Figure 4.3: Function f as a mapping of the definition domain D
f
into the value domain
W
f
(with two arrows, which lead from two pre-image points to one image point)
42
The pre-image set D
f
is in most cases, just as is the set of images W
f
, a part of the real
number axis R
1
. The unambiguity included in the definition of a real function means that
to each x there is one and only one y = f(x). (It is however possible that two different
pre-image points are mapped into one and the same image point.) To summarize in
mathematical shorthand:
y = f(x) function: x D
f
R
1
!y = f(x) : y W
f
R
1
The arithmetic for real functions of a real variable follows according to the rules of
the field R with both the Commutative and Associative Laws, as well as the connecting
Distributive Law, which we have put together for the numbers in Chapter 2 : for example,
the sum or the difference of two real functions f
1
(x) ± f
2
(x) = (f
1
± f
2
)(x) =: g(x) gives
a new real function, as well as the real multiple r · f (x) = (r · f)(x) =: g(x) with r R
and analogously also the product f
1
(x) · f
2
(x) = (f
1
· f
2
)(x) =: g(x) or, if f
2
(x) 6= 0 all
over the definition domain, the quotient, too.
f
1
(x)
f
2
(x)
=
f
1
f
2
(x) =: g(x).
4.2 Basic Set of Functions
It is surprising that we can manage to go through daily physics with a basic set of very
few functions which moreover you are mostly acquainted with from school. In this section
we will introduce these basic set of functions as examples, then discuss some of their
characteristics, and come back to them again and again.
4.2.1 Rational Functions
We start with the constant function y = c, independent of x. Afterwards we come to
linear functions y = s ·x + c with the graph of a straight line having a gradient s and the
ordinate section c. We proceed to the standard parabola y = x
2
and the higher powers
y = x
n
with n N. Also the standard hyperbola y =
1
x
= x
1
and y =
1
x
2
which you are
surely familiar with.
Straight line and parabola are for example defined over the whole real axis: D
f
= R. For
the hyperbola we must omit the origin: D
f
= R \ {0}. Also in the image domain of the
hyperbola the origin is missing: W
f
= R \ {0}. For the parabola the image domain is
only the positive half-line including zero: y 0. The following figure shows the graphs of
these simple examples:
43
Figure 4.4: Graphs of simple functions
According to the calculation rules of the field of real numbers R we get from the straight
line and the standard parabola y = x
2
all functions of second degree y = ax
2
+ bx + c as
well as all further polynomial functions of higher, e.g. mth degree:
y = P
m
(x) = a
0
+ a
1
x + a
2
x
2
+ . . . + a
m
x
m
=
m
X
k=0
a
k
x
k
.
Even the general rational function
y(x) = R(x) =
P
m
(x)
Q
n
(x)
with a polynomial of mth degree P
m
(x) in the numerator and a polynomial of the nth
degree Q
n
(x) in the denominator you are surely familiar with, for example y =
1
x
2
+1
,
the Lorentz distribution, which among other things describes the natural line width of a
spectral line with D = R and 0 < y 1 or y =
x
2
+1
x1
. These rational functions are defined
for all x except for those values x
m
, where the denominator vanishes: Q
n
(x
m
) = 0.
44
Exercise 4.1 Graphs, definition domains and image domains
State the graphs and maximal definition domains of following functions and if possible
also the image domains:
a) f(x) = 2x 2; b) f(x) = 2 2x
2
; c) f(x) = x
2
2x 3; d) f(x) =
1
3
x
3
3;
e) f(x) = x
4
4; f) f(x) =
1
1x
; g) f(x) =
2x3
x1
; h) f(x) =
1
x
2
1
;
i) f(x) =
1
(x1)
2
; j) f(x) =
x+2
x
2
4
; k) f(x) =
x
2
+5
x2
.
4.2.2 Trigonometric Functions
A further group of fundamental functions for all natural sciences which you already know
from school are the trigonometric functions. They play a central role in all periodic
processes, whether it is in space or in time, for example during the oscillation of a pendu-
lum, for the description of light or sound waves, and even for the vibration of a string. In
the following figure a unit circle is pivoted rotatably around the centre carrying a virtual
ink cartridge on its circumference at the end of the red radius. Please click with your
mouse on the circular disc, pull the underlying sheet of paper out to the right under the
uniformly rotating disc and look at the curve which the cartridge has drawn on the paper.
ONLINE ONLY
Figure 4.5 shows a virtually rotatable circle disc carrying an ink car-
tridge on its circumference, under which per mouse click a picture
of the graph of y = sin x can be extracted.
With the help of the projection of the revolving pointer the cartridge has drawn for us
onto the 2-axis the “length of the opposite leg” in the right-angled triangle built by the
circulating radius of length one as hypotenuse, i.e. the graph of the function y = sin x,
the “sine” as function of the angle x.
Clearly, this construction rule gives a periodic function, meaning that in intervals of 2π
of the independent variable the dependent variable takes on the same values: sin(x+2π) =
sin x, generally:
y = f(x) periodic with 2π: f(x + 2π) = f(x)
Out of the sine function by simple operations we can build other trigonometric functions,
which have received their own names due to their importance:
We get the cosine-function” y = cos x analogously just like the sine function as the
45
“length of the adjacent leg of the angle x in the right-angled triangle composed by the
rotating radius and the sine, or as the projection of the circulating radius, that is now on
the 1-axis. The fundamental connection:
cos
2
x + sin
2
x = 1
follows with the Pythagoras Theorem directly from the triangle marked in the figure. The
ink cartridge would have obviously drawn the cosine immediately, if we had started with
the angle
π
2
instead of 0:
cos x = sin(x +
π
2
).
So the cosine function is in fact a sine function shifted to the left by the “Phase”
π
2
.
Also the cosine is periodic with the period 2π : cos(x + 2π) = cos x.
Figure 4.6: Graph of the cosine
From sine and cosine, through division we get two further important trigonometric func-
tions: the
tangent: y = tan x =
sin x
cos x
and the
46
cotangent: y = cot x =
cos x
sin x
=
1
tan x
.
Insert: Notations: In the German literature you may often find also
tg x instead of tan x and ctg x instead of cot x.
Figure 4.7: Tangent and cotangent
Tangent and cotangent are periodic with the period π : tan(x + π) = tan x.
In Chapter 6 we will learn how to calculate the functional value of even the trigonometric
functions, e.g. of y = sin x for every value of the variable x through elementary calculations
such as addition and multiplication.
Besides the Pythagoras Relation cos
2
x + sin
2
x = 1 the
trigonometric addition theorems:
cos(a ±b) = cos a cos b sin a sin b
sin(a ±b) = sin a cos b ± cos a sin b
are of major importance, and experience shows that we have to remind you of them and
to recommend that they be learned by heart. In Chapter 8 we will learn to derive them
in a much more elegant way than you did in school.
47
Exercise 4.2 Trigonometric Functions:
Sketch the graphs and the definition domains of the following functions, and also the value
domains except for the last example:
a) y = 1 + sin x, b) y = sin x + cos x, c) y = sin x cos x, d) y = x + sin x,
e) y = x sin x, f) y =
1
sin x
, g) y =
1
tan x
und h) y =
sin x
x
.
4.2.3 Exponential Functions
While raising to powers b
n
, we have until now introduced only natural numbers n N as
exponents, which indicate how often a real base b occurs as a factor:
b
n
:= b · b · b · . . . · b with n factors b
and we have got the calculation rules:
b
n
b
m
= b
n+m
and (b
n
)
m
= b
n·m
for n, m N.
We then have added negative exponents by the definition b
n
:=
1
b
n
and through the
convention b
0
:= 1 extended the set of exponents to integers n Z.
In order to get to the exponential functions we have to allow real numbers x as expo-
nents instead of taking only integers n (like with the bases b): y = b
x
with x, b R and
to restrict ourselves to positive bases b, without changing the calculation rules for the
powers, i.e. with the following
multiplication theorems for exponential functions:
b
x+y
= b
x
b
y
, (b
x
)
y
= b
x·y
with x, y, b R, b > 0
Of central importance for all natural sciences is the natural exponential function with
the irrational number e defined in Section 3.5 as base:
y = e
x
=: exp x,
Its graph with its characteristically fast growth can be directly measured in the following
figure:
48
ONLINE ONLY
Figure 4.8 illustrates the building of the exponential function e.g.
during the increase of the number of biological cells with a fixed
division rate.
For physicists the inverse function y =
1
e
x
= e
x
is also of great importance, especially for
all damping and decay processes. This function, too, is accessible to measurements, e.g.
during a radioactive decay, in which the still available amount of matter determines the
decay: N(t) = N(0)e
t
T
, where N(t) is the number of nuclei at a time t and T the decay
time:
Figure 4.9: Inverse exponential function, e.g. during a radioactive decay
Even for the exponential functions we will get to know a method in Chapter 6 which
will enable us to calculate the functional value y = e
x
for every value of the variable x
by elementary calculation operations like addition and multiplication with every desired
accuracy.
The following combinations of both the natural exponential functions have received special
names due to their importance, which we will not understand until later: The
hyperbolic cosine: y = cosh x :=
e
x
+ e
x
2
also known as catenary, because in the gravitational field of the earth a chain sags between
two suspension points according to this functional curve, and the
49
hyperbolic sine: y = sinh x :=
e
x
e
x
2
both connected by the easily verifiable relation:
cosh
2
x sinh
2
x = 1.
In addition analogously to the trigonometric functions, we get the quotient of both, the
hyperbolic tangent: y = tanh x :=
sinh x
cosh x
=
e
x
e
x
e
x
+ e
x
and the
hyperbolic cotangent: y = coth x :=
1
tanh x
=
e
x
+ e
x
e
x
e
x
.
The following figure shows the graphs of these functions, which are summarized under the
term hyperbolic functions.
Figure 4.10: Hyperbolic functions
50
Insert: Notations: The notation of the hyperbolic functions in the literature
is not unique: also the following short hand notations are commonly used: ch x =
cosh x, sh x = sinh x and th x = tanh x.
Insert: Hyperbolic: The name “hyperbolic” comes from the equation cosh
2
z
sinh
2
z = 1: With x = cosh z and y = sinh z in a Cartesian coordinate system
this is the parameter representation x
2
y
2
= 1 of a standard hyperbola which has
the bisectors of the first and fourth quadrant as asymptotes and cuts the abscissa
at x = ±1: Analogously with the unit circle we can draw the right branch of the
hyperbola: cosh x is the projection of the moving point on the 1-axis and sinh x the
projection on the 2-axis, as can be seen from the following figure.
Figure 4.11: Right branch of the standard hyperbola with cosh x and sinh x to be
compared with cos x and sin x in the unit circle
Exercise 4.3 Exponential functions:
Sketch the graphs for the following functions for x 0: a) y = 1 e
x
, which describes
e.g. the voltage during the charging of a capacitor
b) y = x + e
x
,
c) the simple Poisson distribution y = xe
x
for totally independent statistic events,
d) the quadratic Poisson distribution y = x
2
e
x
,
e) y = sin x + e
x
,
f) a damped oscillation y = e
x
sin x,
51
g) the reciprocal chain line y =
1
cosh x
h) the Bose-Einstein distribution function of quantum statistics y =
1
e
x
1
or
i) the corresponding Fermi-Dirac distribution for particles with half-integer spin, e.g. con-
ducting electrons y =
1
e
x
+1
,
j) the Planck formula for the spectral intensity distribution of the frequencies of a radiating
cavity y =
x
3
e
x
1
.
You may most easily check your sketches online with our function plotter or e.g. graph.tk
or www.wolframalpha.com.
4.2.4 Functions with Kinks and Cracks
In addition to these sample functions, physicists use a few functions whose graphs show
kinks (or corners) and cracks (or jumps). Among these, the following two are of special
importance for us:
The first is the
absolute value function: y = |x| :=
x for x 0
x for x < 0
This function is defined over the whole number axis, but as for the standard parabola the
value domain covers only the non-negative half-line: y 0. The following figure shows
its graph with the “kink” at x = 0.
Exercise 4.4 Absolute value functions:
Sketch the graphs and the value domains of the following functions:
a) y = 1 |
x
a
|, b) y = x + |x|, c) y =
1
|x|
and d) y = |x|cos x.
The second function is one you most likely have not encountered yet: the Heaviside step
function y = θ(x), defined through:
Heaviside step function:
θ(x) := 1 for x > 0,
θ(x) := 0 for x < 0 and
θ(0) :=
1
2
.
52
Figure 4.12: Graph of the absolute value function
The figure shows its graph with the characteristic two part step at x = 0.
We can easily imagine that the Heaviside function in physics is used among other things
for start and stop situations and to describe steps and barriers.
Insert: Distributions: From the viewpoint of mathematics the step function
is a sample of a discontinuous function. Thus it offers an access to the generalized
functions, called distributions, of which the most important example in physics is
the so-called Dirac δ-distribution.
The calculation with the θ-function requires a little practice which we will gain further
on: First we establish that
θ(ax) = θ(x),
if the argument is multiplied with a positive real number a > 0. Then we consider
θ(x) = 1 θ(x).
In order to get an idea of θ(x+a), we realize that the function vanishes where the argument
is x + a < 0, thus x < a, i.e. that the graph is “upstairs at a. Analogously θ(x a)
means “upstairs at +a and θ(a x) “downstairs at +a.
Of further interest are the products of two step functions: for example θ(x)θ(x+a) = θ(x).
With the same sign of the variables, the smaller argument gets its way. With different
signs of variables in the argument, we receive either identically 0, as with θ(x)θ(x a)
or a barrier as for θ(x)θ(x + a) = θ(x) θ(x a) with the following graph: “upstairs at
0 and downstairs at +a:
53
Figure 4.13: Heaviside function θ(x) : “upstairs at 0”.
Figure 4.14: Graph of θ(x): “downstairs at 0”.
Exercise 4.5 Heaviside function: with a>0
a) Sketch θ(x a),
b) Sketch θ(x)θ(x a), θ(x)θ(x + a) and θ(x)θ(x a),
c) Visualize θ(x)θ(x + a) = θ(x + a) θ(x), θ(x)θ(x a)
and θ(x + a)θ(a x) = θ(x + a) θ(x a),
d) Draw the graph of θ(x)e
x
,
e) Sketch the triangle function (1 |
x
a
|)θ(x + a)θ(a x).
Insert: δ-Function”: The family of functions θ
a
(x) =
θ(x+a)θ(ax)
2a
with the
family parameter a, the “symmetrical box” of width 2a and height
1
2a
(this means area
1), is one of the large number of function sets, whose limits (here the limit a 0)
54
Figure 4.15: Graph of the product θ(x)θ(x + a)
lead to the famous Dirac δ-distribution (casually also called Dirac’s δ-function). We
do not want to deal with them here any further since they are no more functions.
4.3 Nested Functions
Besides the possibilities which the field of real numbers offers to build new functions out
of our basic set of functions with addition, subtraction, multiplication and division, there
exists an important new operation to achieve that goal, namely the means of nested
functions, sometimes also called encapsulated functions. It consists in “inserting one
function into an other one”: If for instance the value domain W
g
of an (“inner”) function
y = g(x) is lying in the definition domain D
f
of an other (“outer”) function y = f(x):
we get y = f(g(x)) with x D
g
, i.e. a new functional dependency which is sometimes
also written as y = (f g)(x). Since we are free in the notation of the independent
and dependent variables, the nesting operation will become particularly clear if we write:
y = f(z) with z = g(x) yields y = f(g(x)):
Figure 4.16: Diagram to visualize the nested function: y = f(g(x))
55
Simple examples are e.g.: z = g(x) = 1 + x
2
with W
g
: z 1 as inner function and
y = f(z) =
1
z
with D
f
= R
1
\ {0} as outer one, which yields the Lorentz distribution
function as nested function y =
1
1+x
2
, or z = sin x with W
g
: 1 z 1 inserted into
y = |x| with D
f
= R yields y = |sin x| to describe a rectified alternating current, or
z = −|2x| with W
g
= R inserted into y = e
z
yields y = exp(−|2x|), an exponential top.
Also the bell-shaped Gaussian function y = exp(x
2
) built out of z = x
2
with W
g
: z 0
and y = e
z
is an interesting nested function which is widely used in all sciences.
Exercise 4.6 Nested Functions: Sketch the graphs of the above mentioned examples
and examine and sketch the following nested functions:
a) y = sin 2x,
b) y = sin x + sin 2x + sin 4x,
c) y = cos
2
x sin
2
x,
d) y = sin(x
2
),
e) y = sin
1
x
,
f) y = (
sin x
x
)
2
, describing e.g. the intensity of light after diffraction,
g) y = tan 2x,
h) the classic Maxwell-Boltzmann velocity distribution of the colliding molecules of an
ideal gas y = x
2
e
x
2
,
i) the Bose-Einstein distribution of the velocities of a gas according to quantum statistics
y =
x
2
e
x
2
1
,
j) the Fermi-Dirac distribution of the velocities in an electron gas y =
x
e
xa
+1
with the
constant a depending on the temperature,
k) Planck’s formula for the spectral intensity of the wavelengths of the radiation of a cavity
y =
1
x
5
[e
1
x
1]
,
l) y = e
sin x
,
m) y = 1 |2x| and
n) y =
1
|2x|
.
You may easily check your sketches with our online function plotter or
e.g. graph.tk or www.wolframalpha.com.
56
ONLINE ONLY
Figure 4.17 is a function plotter: It shows you in a Cartesian
coordinate system the graphs of all the functions which you can
build out of our basic set of functions as linear combinations,
products or nested functions:
You may type in the interesting function into the box above on the
right using x as symbol for the independent variable and writing
the function in computer manner (with a real number r R):
The plotter knows the number pi :=π, but it does not know the
Euler number e.
Addition, subtraction and division as usual: x + r, x r, x/r
Multiplication with the star instead of the point symbol: r
?
x := r·x,
raising to a power with the hat: x
r := x
r
and r
x := r
x
,
square roots with sqrt(x) :=
x, other roots must be written as
broken exponents,
trigonometric functions with brackets: sin(x) := sin x, cos(x) :=
cos x, tan(x) := tan x,
exponential functions with exp(x) := e
x
, because the plotter does
not know the number e,
hyperbolic functions also with brackets: sinh(x) :=
sinh x, cosh(x) := cosh x, tanh(x) := tanh x.
The plotter knows only the three usual logarithms: ln(x) := log
e
x,
ld(x) := log
2
x and lg(x) := log
10
x. The absolute value function
and the Heaviside function must be synthesized by interval division.
In any case only round brackets are allowed.
You may change the scale in both directions independently within
a wide range through a click at the magnifying glass symbol. If
you are ready with the preparations, you should start the plotting
by the return button. Of course this simple function plotter
programmed by Thomas Fuhrmann computes the desired functions
only at a few points and reproduces the graph only roughly.
Especially in the neighbourhood of singularities the graphs must
be taken with some caution.
Now, please play around with the plotter. I hope you will enjoy yourself!
If you are at the end of your wishes and fantasy, I would propose to study the building of
interesting series: for instance
a) in the interval [-0.99,0.99]: first 1, then 1 + x, then 1 + x + x
2, and +x
3, +x
4, etc.,
and always compared with
1
1x
,
b) in the interval [-0.1,0.1]: 1 x
2/2 + x
4/2
?
3
?
4 x
6/2
?
3
?
4
?
5
?
6
?
+ ... etc.,
compared with cos(x),
57
c) in the interval [-pi,3
?
pi]: sin(x) sin(2
?
x)/2 + sin(3
?
x)/3 sin(4
?
x)/4 + . . .
What does this series yield?
d) in the interval [-pi,3
?
pi]: sin(x) + sin(3
?
x)/3 + sin(5
?
x)/5 + sin(7
?
x)/7 + . . .
What do physicists need this series for?
4.4 Mirror Symmetry
Several properties of functions deserve to be considered in more detail:
Symmetry properties play an important role in all sciences: think for instance of
crystals. A symmetric problem has mostly also a symmetric solution. Frequently this
fact saves work. There are many kinds of symmetries. We want to select one of these,
the mirror symmetry. Therefore we examine in this chapter the behavior of the functions
y = f(x) , resp. of their graphs against reflections first in the 2-axis, i.e. in the straight
line x = 0, if x is turned into x.
In this case y = f(x) is turned into f(x). In general there is no simple connection
between f(x) and f(x) for a given x. Take for example f (x) = x+1 for x = 3 : f (3) = 4,
while f(3) = 2. There exist however functions with a simple connection between the
function values before and after the reflection. These functions are of special interest for
physicists and mathematicians and have a special name:
A function which is symmetric against reflections in the 2-axis is called even:
y = f(x) even f(x) = f(x).
For instance y = x
2
, y = cos x and y = |x| are even functions, their graphs turn into
themselves through a reflection in the y-axis. The name “even” comes from the fact that
all powers with even numbers as exponents are even functions.
On the other hand, a function is called odd, if it is antisymmetric against a reflection
in the 2-axis, i.e. it is turned into its negative or equivalently if the graph is unchanged
through a point reflection in the origin:
y = f(x) odd f(x) = f(x),
58
for instance y =
1
x
, y = x
3
or y = sin x.
The straight line function y = s · x + c is for c 6= 0 neither even nor odd. Every function
can however be split into an even and an odd part:
f(x) =
f(x)+f(x)
2
+
f(x)f(x)
2
= f
+
(x) + f
(x) with the
even part: f
+
(x) =
f(x) + f(x)
2
= f
+
(x)
and the
odd part: f
(x) =
f(x) f(x)
2
= f
(x).
For instance c is the even part of the straight line function y = s · x + c and s · x is the
odd part.
Exercise 4.7 Symmetry properties of functions:
1) Examine the following functions for mirror symmetry:
a) y = x
4
, b) y = x
5
, c) y =
sin x
x
, d) y = tan x, e) y = cot x,
f) y = sinh x, g) y = cosh x and h) y = −|x|.
2) Determine the even and odd part of e.g.:
a) f(x) = x(x + 1), b) f(x) = x sin x + cos x, c) y = e
x
and d) y = θ(x).
4.5 Boundedness
Our next step is to transfer the boundedness, known to us from sequences, onto functions.
A function is said to be bounded above in an interval [a, b], if there is an upper bound for
the functional values in this interval:
y = f(x) bounded above in [a, b] B : B f(x) x [a, b]
59