black t h derivations of applied mathematics (web draft, 2006)(246s) mcetsinhvienzone com

16 2.3 Dividing power series through successively smaller powers.. 24 2.4 Dividing power series through successively larger powers.. The book, a work yet in progress but a complete, enti

Derivations of Applied Mathematics

Thaddeus H Black

Revised 14 December 2006

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Derivations of Applied Mathematics.

14 December 2006

Copyright c

Published by the Debian Project [7]

This book is free software You can redistribute and/or modify it under theterms of the GNU General Public License [11], version 2

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1.1 Applied mathematics 1

1.2 Rigor 2

1.2.1 Axiom and definition 2

1.2.2 Mathematical extension 4

1.3 Complex numbers and complex variables 5

1.4 On the text 5

2 Classical algebra and geometry 7 2.1 Basic arithmetic relationships 7

2.1.1 Commutivity, associativity, distributivity 7

2.1.2 Negative numbers 9

2.1.3 Inequality 10

2.1.4 The change of variable 11

2.2 Quadratics 11

2.3 Notation for series sums and products 13

2.4 The arithmetic series 15

2.5 Powers and roots 15

2.5.1 Notation and integral powers 15

2.5.2 Roots 17

2.5.3 Powers of products and powers of powers 19

2.5.4 Sums of powers 19

2.5.5 Summary and remarks 20

2.6 Multiplying and dividing power series 20

2.6.1 Multiplying power series 21

2.6.2 Dividing power series 21

2.6.3 Common quotients and the geometric series 26 SinhVienZone.Com

2.6.4 Variations on the geometric series 26

2.7 Constants and variables 27

2.8 Exponentials and logarithms 29

2.8.1 The logarithm 29

2.8.2 Properties of the logarithm 30

2.9 Triangles and other polygons: simple facts 30

2.9.1 Triangle area 31

2.9.2 The triangle inequalities 31

2.9.3 The sum of interior angles 32

2.10 The Pythagorean theorem 33

2.11 Functions 35

2.12 Complex numbers (introduction) 36

2.12.1 Rectangular complex multiplication 38

2.12.2 Complex conjugation 38

2.12.3 Power series and analytic functions (preview) 40

3 Trigonometry 43 3.1 Definitions 43

3.2 Simple properties 45

3.3 Scalars, vectors, and vector notation 45

3.4 Rotation 49

3.5 Trigonometric sums and differences 51

3.5.1 Variations on the sums and differences 52

3.5.2 Trigonometric functions of double and half angles 53

3.6 Trigonometrics of the hour angles 53

3.7 The laws of sines and cosines 57

3.8 Summary of properties 58

3.9 Cylindrical and spherical coordinates 60

3.10 The complex triangle inequalities 62

3.11 De Moivre’s theorem 63

4 The derivative 65 4.1 Infinitesimals and limits 65

4.1.1 The infinitesimal 66

4.1.2 Limits 67

4.2 Combinatorics 68

4.2.1 Combinations and permutations 68

4.2.2 Pascal’s triangle 70

4.3 The binomial theorem 70

4.3.1 Expanding the binomial 70 SinhVienZone.Com

CONTENTS v

4.3.2 Powers of numbers near unity 71

4.3.3 Complex powers of numbers near unity 72

4.4 The derivative 73

4.4.1 The derivative of the power series 73

4.4.2 The Leibnitz notation 74

4.4.3 The derivative of a function of a complex variable 76

4.4.4 The derivative of za 77

4.4.5 The logarithmic derivative 77

4.5 Basic manipulation of the derivative 78

4.5.1 The derivative chain rule 78

4.5.2 The derivative product rule 79

4.6 Extrema and higher derivatives 80

4.7 L’Hˆopital’s rule 82

4.8 The Newton-Raphson iteration 83

5 The complex exponential 87 5.1 The real exponential 87

5.2 The natural logarithm 90

5.3 Fast and slow functions 91

5.4 Euler’s formula 92

5.5 Complex exponentials and de Moivre 96

5.6 Complex trigonometrics 96

5.7 Summary of properties 97

5.8 Derivatives of complex exponentials 97

5.8.1 Derivatives of sine and cosine 97

5.8.2 Derivatives of the trigonometrics 100

5.8.3 Derivatives of the inverse trigonometrics 100

5.9 The actuality of complex quantities 102

6 Primes, roots and averages 105 6.1 Prime numbers 105

6.1.1 The infinite supply of primes 105

6.1.2 Compositional uniqueness 106

6.1.3 Rational and irrational numbers 109

6.2 The existence and number of roots 110

6.2.1 Polynomial roots 110

6.2.2 The fundamental theorem of algebra 111

6.3 Addition and averages 112

6.3.1 Serial and parallel addition 112

6.3.2 Averages 115 SinhVienZone.Com

7 The integral 119

7.1 The concept of the integral 119

7.1.1 An introductory example 120

7.1.2 Generalizing the introductory example 123

7.1.3 The balanced definition and the trapezoid rule 123

7.2 The antiderivative 124

7.3 Operators, linearity and multiple integrals 126

7.3.1 Operators 126

7.3.2 A formalism 127

7.3.3 Linearity 128

7.3.4 Summational and integrodifferential transitivity 129

7.3.5 Multiple integrals 130

7.4 Areas and volumes 131

7.4.1 The area of a circle 131

7.4.2 The volume of a cone 132

7.4.3 The surface area and volume of a sphere 133

7.5 Checking integrations 136

7.6 Contour integration 137

7.7 Discontinuities 138

7.8 Remarks (and exercises) 141

8 The Taylor series 143 8.1 The power series expansion of 1/(1 − z)n+1 143

8.1.1 The formula 144

8.1.2 The proof by induction 145

8.1.3 Convergence 146

8.1.4 General remarks on mathematical induction 148

8.2 Shifting a power series’ expansion point 149

8.3 Expanding functions in Taylor series 151

8.4 Analytic continuation 152

8.5 Branch points 154

8.6 Cauchy’s integral formula 155

8.6.1 The meaning of the symbol dz 156

8.6.2 Integrating along the contour 156

8.6.3 The formula 160

8.7 Taylor series for specific functions 161

8.8 Bounds 164

8.9 Calculating 2π 165

8.10 The multidimensional Taylor series 166 SinhVienZone.Com

CONTENTS vii

9.1 Integration by antiderivative 169

9.2 Integration by substitution 170

9.3 Integration by parts 171

9.4 Integration by unknown coefficients 173

9.5 Integration by closed contour 176

9.6 Integration by partial-fraction expansion 178

9.6.1 Partial-fraction expansion 178

9.6.2 Multiple poles 180

9.6.3 Integrating rational functions 182

9.7 Integration by Taylor series 184

10 Cubics and quartics 185 10.1 Vieta’s transform 186

10.2 Cubics 186

10.3 Superfluous roots 189

10.4 Edge cases 191

10.5 Quartics 193

10.6 Guessing the roots 195

11 The matrix (to be written) 199 A Hex and other notational matters 203 A.1 Hexadecimal numerals 204

A.2 Avoiding notational clutter 205

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List of Figures

1.1 Two triangles 4

2.1 Multiplicative commutivity 8

2.2 The sum of a triangle’s inner angles: turning at the corner 32

2.3 A right triangle 34

2.4 The Pythagorean theorem 34

2.5 The complex (or Argand) plane 37

3.1 The sine and the cosine 44

3.2 The sine function 45

3.3 A two-dimensional vector u = ˆxx + ˆyy 47

3.4 A three-dimensional vector v = ˆxx + ˆyy + ˆzz 47

3.5 Vector basis rotation 50

3.6 The 0x18 hours in a circle 55

3.7 Calculating the hour trigonometrics 55

3.8 The laws of sines and cosines 57

3.9 A point on a sphere 61

4.1 The plan for Pascal’s triangle 70

4.2 Pascal’s triangle 71

4.3 A local extremum 80

4.4 A level inflection 81

4.5 The Newton-Raphson iteration 84

5.1 The natural exponential 90

5.2 The natural logarithm 91

5.3 The complex exponential and Euler’s formula 94

5.4 The derivatives of the sine and cosine functions 99

7.1 Areas representing discrete sums 120

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7.2 An area representing an infinite sum of infinitesimals 122

7.3 Integration by the trapezoid rule 124

7.4 The area of a circle 132

7.5 The volume of a cone 133

7.6 A sphere 134

7.7 An element of a sphere’s surface 134

7.8 A contour of integration 138

7.9 The Heaviside unit step u(t) 139

7.10 The Dirac delta δ(t) 139

8.1 A complex contour of integration in two parts 157

8.2 A Cauchy contour integral 161

9.1 Integration by closed contour 177

10.1 Vieta’s transform, plotted logarithmically 187

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List of Tables

2.1 Basic properties of arithmetic 8

2.2 Power properties and definitions 16

2.3 Dividing power series through successively smaller powers 24

2.4 Dividing power series through successively larger powers 25

2.5 General properties of the logarithm 31

3.1 Simple properties of the trigonometric functions 46

3.2 Trigonometric functions of the hour angles 56

3.3 Further properties of the trigonometric functions 59

3.4 Rectangular, cylindrical and spherical coordinate relations 61

5.1 Complex exponential properties 98

5.2 Derivatives of the trigonometrics 101

5.3 Derivatives of the inverse trigonometrics 103

6.1 Parallel and serial addition identities 114

7.1 Basic derivatives for the antiderivative 126

8.1 Taylor series 163

10.1 The method to extract the three roots of the general cubic 189

10.2 The method to extract the four roots of the general quartic 196

B.1 The Roman and Greek alphabets 208

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I never meant to write this book It emerged unheralded, unexpectedly.The book began in 1983 when a high-school classmate challenged me toprove the Pythagorean theorem on the spot I lost the dare, but looking theproof up later I recorded it on loose leaves, adding to it the derivations of

a few other theorems of interest to me From such a kernel the notes grewover time, until family and friends suggested that the notes might make thematerial for a book

The book, a work yet in progress but a complete, entire book as itstands, first frames coherently the simplest, most basic derivations of ap-plied mathematics, treating quadratics, trigonometrics, exponentials, deriva-tives, integrals, series, complex variables and, of course, the aforementionedPythagorean theorem These and others establish the book’s foundation inChs 2 through 9 Later chapters build upon the foundation, deriving resultsless general or more advanced Such is the book’s plan

A book can follow convention or depart from it; yet, though occasionaldeparture might render a book original, frequent departure seldom renders

a book good Whether this particular book is original or good, neither orboth, is for the reader to tell, but in any case the book does both follow anddepart Convention is a peculiar thing: at its best, it evolves or accumulatesonly gradually, patiently storing up the long, hidden wisdom of generationspast; yet herein arises the ancient dilemma Convention, in all its richness,

in all its profundity, can, sometimes, stagnate at a local maximum, a hillockwhence higher ground is achievable not by gradual ascent but only by descentfirst—or by a leap Descent risks a bog A leap risks a fall One ought notrun such risks without cause, even in such an inherently unconservativediscipline as mathematics

Well, the book does risk It risks one leap at least: it employs mal numerals

hexadeci-Decimal numerals are fine in history and anthropology (man has tenfingers), finance and accounting (dollars, cents, pounds, shillings, pence: the

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base hardly matters), law and engineering (the physical units are arbitraryanyway); but they are merely serviceable in mathematical theory, neveraesthetic There unfortunately really is no gradual way to bridge the gap

to hexadecimal (shifting to base eleven, thence to twelve, etc., is no use)

If one wishes to reach hexadecimal ground, one must leap Twenty years

of keeping my own private notes in hex have persuaded me that the leapjustifies the risk In other matters, by contrast, the book leaps seldom Thebook in general walks a tolerably conventional applied mathematical line.The book belongs to the emerging tradition of open-source software,where at the time of this writing it fills a void Nevertheless it is a book, not aprogram Lore among open-source developers holds that open developmentinherently leads to superior work Well, maybe Often it does in fact.Personally with regard to my own work, I should rather not make too manyclaims It would be vain to deny that professional editing and formal peerreview, neither of which the book enjoys, had substantial value On the otherhand, it does not do to despise the amateur (literally, one who does for thelove of it: not such a bad motive, after all1) on principle, either—unlessone would on the same principle despise a Washington or an Einstein, or aDebian Developer [7] Open source has a spirit to it which leads readers to

be far more generous with their feedback than ever could be the case with

a traditional, proprietary book Such readers, among whom a surprisingconcentration of talent and expertise are found, enrich the work freely Thishas value, too

The book’s peculiar mission and program lend it an unusual quantity

of discursive footnotes These footnotes offer nonessential material which,while edifying, coheres insufficiently well to join the main narrative Thefootnote is an imperfect messenger, of course Catching the reader’s eye,

it can break the flow of otherwise good prose Modern publishing offersvarious alternatives to the footnote—numbered examples, sidebars, specialfonts, colored inks, etc Some of these are merely trendy Others, likenumbered examples, really do help the right kind of book; but for thisbook the humble footnote, long sanctioned by an earlier era of publishing,extensively employed by such sages as Gibbon [12] and Shirer [23], seemsthe most able messenger In this book it shall have many messages to bear.The book provides a bibliography listing other books I have referred towhile writing Mathematics by its very nature promotes queer bibliogra-phies, however, for its methods and truths are established by derivationrather than authority Much of the book consists of common mathematical

1 The expression is derived from an observation I seem to recall George F Will making.SinhVienZone.Com

knowledge or of proofs I have worked out with my own pencil from variousideas gleaned who knows where over the years The latter proofs are perhapsoriginal or semi-original from my personal point of view, but it is unlikelythat many if any of them are truly new To the initiated, the mathematicsitself often tends to suggest the form of the proof: if to me, then surely also

to others who came before; and even where a proof is new the idea provenprobably is not

Some of the books in the bibliography are indeed quite good, but youshould not necessarily interpret inclusion as more than a source acknowl-edgment by me They happen to be books I have on my own bookshelf forwhatever reason (had bought it for a college class years ago, had found itonce at a yard sale for 25 cents, etc.), or have borrowed, in which I lookedsomething up while writing

As to a grand goal, underlying purpose or hidden plan, the book hasnone, other than to derive as many useful mathematical results as possibleand to record the derivations together in an orderly manner in a single vol-ume What constitutes “useful” or “orderly” is a matter of perspective andjudgment, of course My own peculiar heterogeneous background in mil-itary service, building construction, electrical engineering, electromagneticanalysis and Debian development, my nativity, residence and citizenship inthe United States, undoubtedly bias the selection and presentation to somedegree How other authors go about writing their books, I do not know,but I suppose that what is true for me is true for many of them also: webegin by organizing notes for our own use, then observe that the same notesmay prove useful to others, and then undertake to revise the notes and tobring them into a form which actually is useful to others Whether this booksucceeds in the last point is for the reader to judge

THB

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Chapter 1

Introduction

This is a book of applied mathematical proofs If you have seen a matical result, if you want to know why the result is so, you can look forthe proof here

mathe-The book’s purpose is to convey the essential ideas underlying the tions of a large number of mathematical results useful in the modeling ofphysical systems To this end, the book emphasizes main threads of math-ematical argument and the motivation underlying the main threads, deem-phasizing formal mathematical rigor It derives mathematical results fromthe purely applied perspective of the scientist and the engineer

deriva-The book’s chapters are topical This first chapter treats a few ductory matters of general interest

intro-1.1 Applied mathematics

What is applied mathematics?

Applied mathematics is a branch of mathematics that concernsitself with the application of mathematical knowledge to otherdomains The question of what is applied mathematics doesnot answer to logical classification so much as to the sociology

of professionals who use mathematics [1]

That is about right, on both counts In this book we shall define plied mathematics to be correct mathematics useful to scientists, engineersand the like; proceeding not from reduced, well defined sets of axioms butrather directly from a nebulous mass of natural arithmetical, geometricaland classical-algebraic idealizations of physical systems; demonstrable butgenerally lacking the detailed rigor of the professional mathematician

ap-SinhVienZone.Com

1.2 Rigor

It is impossible to write such a book as this without some discussion of ematical rigor Applied and professional mathematics differ principally andessentially in the layer of abstract definitions the latter subimposes beneaththe physical ideas the former seeks to model Notions of mathematical rigorfit far more comfortably in the abstract realm of the professional mathe-matician; they do not always translate so gracefully to the applied realm

math-Of this difference, the applied mathematical reader and practitioner needs

to be aware

1.2.1 Axiom and definition

Ideally, a professional mathematician knows or precisely specifies in advancethe set of fundamental axioms he means to use to derive a result A primeaesthetic here is irreducibility: no axiom in the set should overlap the others

or be specifiable in terms of the others Geometrical argument—proof bysketch—is distrusted The professional mathematical literature discouragesundue pedantry indeed, but its readers do implicitly demand a convincingassurance that its writers could derive results in pedantic detail if calledupon to do so Precise definition here is critically important, which is whythe professional mathematician tends not to accept blithe statements such

as that

1

0 = ∞,without first inquiring as to exactly what is meant by symbols like 0 and ∞.The applied mathematician begins from a different base His ideal liesnot in precise definition or irreducible axiom, but rather in the elegant mod-eling of the essential features of some physical system Here, mathematicaldefinitions tend to be made up ad hoc along the way, based on previousexperience solving similar problems, adapted implicitly to suit the model athand If you ask the applied mathematician exactly what his axioms are,which symbolic algebra he is using, he usually doesn’t know; what he knows

is that the bridge has its footings in certain soils with specified tolerances,suffers such-and-such a wind load, etc To avoid error, the applied mathe-matician relies not on abstract formalism but rather on a thorough mentalgrasp of the essential physical features of the phenomenon he is trying tomodel An equation like

1

0 = ∞SinhVienZone.Com

The irascible Oliver Heaviside, responsible for the important appliedmathematical technique of phasor analysis, once said,

It is shocking that young people should be addling their brainsover mere logical subtleties, trying to understand the proof ofone obvious fact in terms of something equally obvious [2]Exaggeration, perhaps, but from the applied mathematical perspectiveHeaviside nevertheless had a point The professional mathematicians

R Courant and D Hilbert put it more soberly in 1924 when they wrote,Since the seventeenth century, physical intuition has served as

a vital source for mathematical problems and methods Recenttrends and fashions have, however, weakened the connection be-tween mathematics and physics; mathematicians, turning awayfrom the roots of mathematics in intuition, have concentrated onrefinement and emphasized the postulational side of mathemat-ics, and at times have overlooked the unity of their science withphysics and other fields In many cases, physicists have ceased

to appreciate the attitudes of mathematicians [6, Preface]

Although the present book treats “the attitudes of mathematicians” withgreater deference than some of the unnamed 1924 physicists might havedone, still, Courant and Hilbert could have been speaking for the engineersand other applied mathematicians of our own day as well as for the physicists

of theirs To the applied mathematician, the mathematics is not principallymeant to be developed and appreciated for its own sake; it is meant to beused This book adopts the Courant-Hilbert perspective

The introduction you are now reading is not the right venue for an essay

on why both kinds of mathematics—applied and professional (or pure)—are needed Each kind has its place; and although it is a stylistic error

to mix the two indiscriminately, clearly the two have much to do with oneanother However this may be, this book is a book of derivations of appliedmathematics The derivations here proceed by a purely applied approach

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Figure 1.1: Two triangles.

ex-More often, however, the results achieved by extension are unsurprisingand not very interesting in themselves Such extended results are the faithfulservants of mathematical rigor Consider for example the triangle on the left

of Fig 1.1 This triangle is evidently composed of two right triangles of areas

A1 = b1h

2 ,

A2 = b2h

2(each right triangle is exactly half a rectangle) Hence the main triangle’sarea is

A = A1+ A2= (b1+ b2)h

bh

2 .Very well What about the triangle on the right? Its b1 is not shown on thefigure, and what is that −b2, anyway? Answer: the triangle is composed ofthe difference of two right triangles, with b1 the base of the larger, overallone: b1 = b + (−b2) The b2 is negative because the sense of the small righttriangle’s area in the proof is negative: the small area is subtracted fromSinhVienZone.Com

1.3 COMPLEX NUMBERS AND COMPLEX VARIABLES 5

the large rather than added By extension on this basis, the main triangle’sarea is again seen to be A = bh/2 The proof is exactly the same In fact,once the central idea of adding two right triangles is grasped, the extension

is really rather obvious—too obvious to be allowed to burden such a book

as this

Excepting the uncommon cases where extension reveals something teresting or new, this book generally leaves the mere extension of proofs—including the validation of edge cases and over-the-edge cases—as an exercise

in-to the interested reader

1.3 Complex numbers and complex variables

More than a mastery of mere logical details, it is an holistic view of themathematics and of its use in the modeling of physical systems which is themark of the applied mathematician A feel for the math is the great thing.Formal definitions, axioms, symbolic algebras and the like, though oftenuseful, are felt to be secondary The book’s rapidly staged development ofcomplex numbers and complex variables is planned on this sensibility.Sections 2.12, 3.11, 4.3.3, 4.4, 6.2, 8.4, 8.5, 8.6 and 9.5, plus all of Chap-ter 5, constitute the book’s principal stages of complex development Inthese sections and throughout the book, the reader comes to appreciate thatmost mathematical properties which apply for real numbers apply equallyfor complex, that few properties concern real numbers alone

1.4 On the text

The book gives numerals in hexadecimal It denotes variables in Greekletters as well as Roman Readers unfamiliar with the hexadecimal notationwill find a brief orientation thereto in Appendix A Readers unfamiliar withthe Greek alphabet will find it in Appendix B

Licensed to the public under the GNU General Public Licence [11], sion 2, this book meets the Debian Free Software Guidelines [8]

ver-If you cite an equation, section, chapter, figure or other item from thisbook, it is recommended that you include in your citation the book’s precisedraft date as given on the title page The reason is that equation numbers,chapter numbers and the like are numbered automatically by the LATEXtypesetting software: such numbers can change arbitrarily from draft todraft If an example citation helps, see [5] in the bibliography

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2.1 Basic arithmetic relationships

This section states some arithmetical rules

2.1.1 Commutivity, associativity, distributivity, identity and

inversion

Refer to Table 2.1, whose rules apply equally to real and complex bers (§ 2.12) Most of the rules are appreciated at once if the meaning ofthe symbols is understood In the case of multiplicative commutivity, oneimagines a rectangle with sides of lengths a and b, then the same rectan-gle turned on its side, as in Fig 2.1: since the area of the rectangle is thesame in either case, and since the area is the length times the width in ei-ther case (the area is more or less a matter of counting the little squares),evidently multiplicative commutivity holds A similar argument validatesmultiplicative associativity, except that here we compute the volume of athree-dimensional rectangular box, which box we turn various ways

num-Multiplicative inversion lacks an obvious interpretation when a = 0

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Table 2.1: Basic properties of arithmetic.

(a)(1/a) = 1 Multiplicative inversion(a)(b + c) = ab + ac Distributivity

Figure 2.1: Multiplicative commutivity

a b

b aSinhVienZone.Com

2.1 BASIC ARITHMETIC RELATIONSHIPS 9

Looking ahead in the book, we note that the multiplicative properties

do not always hold for more general linear transformations For example,matrix multiplication is not commutative and vector cross-multiplication isnot associative Where associativity does not hold and parentheses do nototherwise group, right-to-left association is notationally implicit:1

The first three of the four equations are unsurprising, but the last is esting Why would a negative count −a of a negative quantity −b come to

inter-1 The fine C and C++ programming languages are unfortunately stuck with the reverse order of association, along with division inharmoniously on the same level of syntactic precedence as multiplication Standard mathematical notation is more elegant:

abc/uvw = (a)(bc)

(u)(vw).SinhVienZone.Com

a positive product +ab? To see why, consider the progression

.(+3)(−b) = −3b,(+2)(−b) = −2b,(+1)(−b) = −1b,(0)(−b) = 0b,(−1)(−b) = +1b,(−2)(−b) = +2b,(−3)(−b) = +3b,

The logic of arithmetic demands that the product of two negative numbers

be positive for this reason

2.1.3 Inequality

If2

a < b,this necessarily implies that

2.2 QUADRATICS 112.1.4 The change of variable

The applied mathematician very often finds it convenient to change ables, introducing new symbols to stand in place of old For this we havethe change of variable or assignment notation3

vari-Q ← P

This means, “in place of P , put Q;” or, “let Q now equal P ” For example,

if a2 + b2 = c2, then the change of variable 2µ ← a yields the new form(2µ)2+ b2 = c2

Similar to the change of variable notation is the definition notation

Q ≡ P

This means, “let the new symbol Q represent P ”4

The two notations logically mean about the same thing Subjectively,

Q ≡ P identifies a quantity P sufficiently interesting to be given a permanentname Q, whereas Q ← P implies nothing especially interesting about P or Q;

it just introduces a (perhaps temporary) new symbol Q to ease the algebra.The concepts grow clearer as examples of the usage arise in the book

k = k + 1 mean? It looks like a claim that k and k + 1 are the same, which is impossible The notation k ← k + 1 by contrast is unambiguous; it means to increment k by one However, the latter notation admittedly has seen only scattered use in the literature The C and C++ programming languages use == for equality and = for assignment (change of variable), as the reader may be aware.

4

One would never write k ≡ k + 1 Even k ← k + 1 can confuse readers inasmuch as

it appears to imply two different values for the same symbol k, but the latter notation is sometimes used anyway when new symbols are unwanted or because more precise alter- natives (like k n = k n−1 + 1) seem overwrought Still, usually it is better to introduce a new symbol, as in j ← k + 1.

In some books, ≡ is printed as4=.

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(where i is the imaginary unit, a number defined such that i2 = −1, troduced in more detail in § 2.12 below) Useful as these four forms are,however, none of them can directly factor a more general quadratic5 expres-sion like

It follows that the two solutions of the quadratic equation

6 A root of f (z) is a value of z for which f (z) = 0.

7 It suggests it because the expressions on the left and right sides of (2.3) are both quadratic (the highest power is z2) and have the same roots Substituting into the equation the values of z 1 and z 2 and simplifying proves the suggestion correct.

8 The form of the quadratic formula which usually appears in print is

x = −b ±√b 2 − 4ac

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2.3 NOTATION FOR SERIES SUMS AND PRODUCTS 13

2.3 Notation for series sums and products

Sums and products of series arise so frequently in mathematical work thatone finds it convenient to define terse notations to express them The sum-mation notation

sym-The product shorthand

z =3

2 ±

s

„ 3 2

« 2

− 2 = 1 or 2.

9 The hexadecimal numeral 0x56 represents the same number the decimal numeral 86 represents The book’s preface explains why the book represents such numbers in hex- adecimal Appendix A tells how to read the numerals.

10 Section 7.3 speaks further of the dummy variable.

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is very frequently used The notation n! is pronounced “n factorial.” garding the notation n!/m!, this can of course be regarded correctly as n!divided by m! , but it usually proves more amenable to regard the notation

Re-as a single unit.11

Because multiplication in its more general sense as linear transformation

is not always commutative, we specify that

b

Y

k=a

f (k) = [f (b)][f (b − 1)][f(b − 2)] · · · [f(a + 2)][f(a + 1)][f(a)]

rather than the reverse order of multiplication.12 Multiplication proceedsfrom right to left In the event that the reverse order of multiplication isneeded, we shall use the notation

b

a

k=a

f (k) = [f (a)][f (a + 1)][f (a + 2)] · · · [f(b − 2)][f(b − 1)][f(b)].Note that for the sake of definitional consistency,

k=1f (k).” However, experience shows the latter notation to beextremely useful in expressing more sophisticated mathematical ideas Weshall use such notation extensively in this book

11 One reason among others for this is that factorials rapidly multiply to extremely large sizes, overflowing computer registers during numerical computation If you can avoid unnecessary multiplication by regarding n!/m! as a single unit, this is a win.

12 The extant mathematical literature lacks an established standard on the order of multiplication implied by the “Q” symbol, but this is the order we shall use in this book.SinhVienZone.Com

2.4 THE ARITHMETIC SERIES 15

2.4 The arithmetic series

A simple yet useful application of the series sum of § 2.3 is the arithmeticseries

b

X

k=a

k = a + (a + 1) + (a + 2) + · · · + b

Pairing a with b, then a+1 with b−1, then a+2 with b−2, etc., the average

of each pair is [a+b]/2; thus the average of the entire series is [a+b]/2 (Thepairing may or may not leave an unpaired element at the series midpoint

k = [a + b]/2, but this changes nothing.) The series has b − a + 1 terms.Hence,

b

X

k=a

k = (b − a + 1)a + b2 (2.6)Success with this arithmetic series leads one to wonder about the geo-metric series P∞

k=0zk Section 2.6.3 addresses that point

2.5 Powers and roots

This necessarily tedious section discusses powers and roots It offers nosurprises Table 2.2 summarizes its definitions and results Readers seekingmore rewarding reading may prefer just to glance at the table then to skipdirectly to the start of the next section

In this section, the exponents k, m, n, p, q, r and s are integers,13 butthe exponents a and b are arbitrary real numbers

2.5.1 Notation and integral powers

The power notation

zn

13 In case the word is unfamiliar to the reader who has learned arithmetic in another language than English, the integers are the negative, zero and positive counting numbers , −3, −2, −1, 0, 1, 2, 3, The corresponding adjective is integral (although the word

“integral” is also used as a noun and adjective indicating an infinite sum of infinitesimals; see Ch 7) Traditionally, the letters i, j, k, m, n, M and N are used to represent integers (i is sometimes avoided because the same letter represents the imaginary unit), but this section needs more integer symbols so it uses p, q, r and s, too.

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Table 2.2: Power properties and definitions.

lim

→0 + = lim

E→∞

„ 1 E

2.5 POWERS AND ROOTS 17Notice that in general,

zn−1= z

n

z .This leads us to extend the definition to negative integral powers with

Taking the u and v formulas together, then,

(z1/n)n= z = (zn)1/n (2.12)

for any z and integral n

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The number z1/n is called the nth root of z—or in the very common case

n = 2, the square root of z, often written

√z

When z is real and nonnegative, the last notation is usually implicitly taken

to mean the real, nonnegative square root In any case, the power and rootoperations mutually invert one another

What about powers expressible neither as n nor as 1/n, such as the 3/2power? If z and w are numbers related by

wq= z,then

wpq = zp.Taking the qth root,

wp = (zp)1/q.But w = z1/q, so this is

(z1/q)p = (zp)1/q,which says that it does not matter whether one applies the power or theroot first; the result is the same Extending (2.10) therefore, we define zp/qsuch that

(z1/q)p = zp/q= (zp)1/q (2.13)Since any real number can be approximated arbitrarily closely by a ratio ofintegers, (2.13) implies a power definition for all real exponents

Equation (2.13) is this subsection’s main result However, § 2.5.3 willfind it useful if we can also show here that

(z1/q)1/s= z1/qs= (z1/s)1/q (2.14)The proof is straightforward If

w ≡ z1/qs,then raising to the qs power yields

2.5 POWERS AND ROOTS 19

Per (2.11),

(uv)p = upvp.Raising this equation to the 1/q power, we have that

for any real a

On the other hand, per (2.10),

With (2.9), (2.15) and (2.16), one can reason that

z(p/q)+(r/s)= (zps+rq)1/qs= (zpszrq)1/qs= zp/qzr/s,

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or in other words that

Table 2.2 on page 16 summarizes this section’s definitions and results.Looking ahead to § 2.12, § 3.11 and Ch 5, we observe that nothing inthe foregoing analysis requires the base variables z, w, u and v to be realnumbers; if complex (§ 2.12), the formulas remain valid Still, the analysisdoes imply that the various exponents m, n, p/q, a, b and so on are realnumbers This restriction, we shall remove later, purposely defining theaction of a complex exponent to comport with the results found here Withsuch a definition the results apply not only for all bases but also for allexponents, real or complex

2.6 Multiplying and dividing power series

A power series16 is a weighted sum of integral powers:

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2.6 MULTIPLYING AND DIVIDING POWER SERIES 212.6.1 Multiplying power series

Given two power series

2.6.2 Dividing power series

The quotient Q(z) = B(z)/A(z) of two power series is a little harder tocalculate The calculation is by long division For example,

If you feel that you understand the example, then that is really all there

is to it, and you can skip the rest of the subsection if you like One sometimeswants to express the long division of power series more formally, however.That is what the rest of the subsection is about

Formally, we prepare the long division B(z)/A(z) by writing

B(z) = A(z)Qn(z) + Rn(z), (2.23)

where Rn(z) is a remainder (being the part of B(z) remaining to be divided);

17 If Q(z) is a quotient and R(z) a remainder, then B(z) is a dividend (or numerator ) and A(z) a divisor (or denominator ) Such are the Latin-derived names of the parts of a long division.

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QN(z) = 0 while RN(z) = B(z), but the thrust of the long division process

is to build Qn(z) up by wearing Rn(z) down The goal is to grind Rn(z)away to nothing, to make it disappear as n → −∞

As in the example, we pursue the goal by choosing from Rn(z) an easilydivisible piece containing the whole high-order term of Rn(z) The piece

we choose is (rnn/aK)zn−KA(z), which we add and subtract from (2.23) toobtain

B(z) = A(z)Qn−1(z) + Rn−1(z)and observing from the definition of Qn(z) that Qn−1(z) = Qn(z) +

2.6 MULTIPLYING AND DIVIDING POWER SERIES 23

where no term remains in Rn−1(z) higher than a zn−1 term

To begin the actual long division, we initialize

RN(z) = B(z),

for which (2.23) is trivially true Then we iterate per (2.25) as many times

as desired If an infinite number of times, then so long as Rn(z) tends tovanish as n → −∞, it follows from (2.23) that

B(z)

Iterating only a finite number of times leaves a remainder,

B(z)A(z) = Qn(z) +

Rn(z)

except that it may happen that Rn(z) = 0 for sufficiently small n

Table 2.3 summarizes the long-division procedure

It should be observed in light of Table 2.3 that if

18 If a more formal demonstration of (2.28) is wanted, then consider per (2.25) that

Table 2.3: Dividing power series through successively smaller powers.

A(z) = Q−∞(z)SinhVienZone.Com