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Table of Contents

Preface xi

Chapter 1 Numbers and Elementary Mathematics 1

1.1 Proofs without words 3

1.2 Constants 5

1.3 Special numbers 13

1.4 Interval analysis 24

1.5 Fractal Arithmetic 25

1.6 Max-Plus Algebra 26

1.7 Coupled-analogues of Functions 27

1.8 Number theory 28

1.9 Series and products 47

Chapter 2 Algebra 67

2.1 Elementary algebra 69

2.2 Polynomials 73

2.3 Vector algebra 78

2.4 Linear and matrix algebra 83

2.5 Abstract algebra 106

Chapter 3 Discrete Mathematics 133

3.1 Sets 135

3.2 Combinatorics 140

3.3 Graphs 151

3.4 Combinatorial design theory 172

3.5 Difference equations 184

Chapter 4 Geometry 191

4.1 Euclidean geometry 193

4.2 Grades and Degrees 193

4.3 Coordinate systems in the plane 194

4.4 Plane symmetries or isometries 200

4.5 Other transformations of the plane 207

4.6 Lines 209

4.7 Polygons 211

vi Table of Contents

4.8 Surfaces of revolution: the torus 219

4.9 Quadrics 219

4.10 Spherical geometry and trigonometry 224

4.11 Conics 229

4.12 Special plane curves 240

4.13 Coordinate systems in space 249

4.14 Space symmetries or isometries 252

4.15 Other transformations of space 255

4.16 Direction angles and direction cosines 257

4.17 Planes 257

4.18 Lines in space 259

4.19 Polyhedra 261

4.20 Cylinders 265

4.21 Cones 265

4.22 Differential geometry 267

Chapter 5 Analysis 275

5.1 Differential calculus 277

5.2 Differential forms 288

5.3 Integration 291

5.4 Table of indefinite integrals 305

5.5 Table of definite integrals 343

5.6 Ordinary differential equations 350

5.7 Partial differential equations 362

5.8 Integral equations 375

5.9 Tensor analysis 378

5.10 Orthogonal coordinate systems 388

5.11 Real analysis 393

5.12 Generalized functions 403

5.13 Complex analysis 405

5.14 Significant Mathematical Equations 417

Chapter 6 Special Functions 419

6.1 Ceiling and floor functions 421

6.2 Exponentiation 421

6.3 Exponential function 422

6.4 Logarithmic functions 422

6.5 Trigonometric functions 424

6.6 Circular functions and planar triangles 433

6.7 Tables of trigonometric functions 437

6.8 Angle conversion 440

6.9 Inverse circular functions 441

6.10 Hyperbolic functions 443

6.11 Inverse hyperbolic functions 447

6.12 Gudermannian function 449

6.13 Orthogonal polynomials 451

6.14 Clebsch–Gordan coefficients 458

6.15 Bessel functions 460

6.16 Beta function 469

6.17 Elliptic integrals 470

6.18 Jacobian elliptic functions 473

6.19 Error functions 475

6.20 Fresnel integrals 476

6.21 Gamma function 478

6.22 Hypergeometric functions 481

6.23 Lambert Function 483

6.24 Legendre functions 484

6.25 Polylogarithms 488

6.26 Prolate Spheroidal Wave Functions 489

6.27 Sine, cosine, and exponential integrals 490

6.28 Weierstrass Elliptic Function 492

6.29 Integral transforms: List 493

6.30 Integral transforms: Preliminaries 494

6.31 Fourier integral transform 494

6.32 Discrete Fourier transform (DFT) 500

6.33 Fast Fourier transform (FFT) 502

6.34 Multidimensional Fourier transforms 502

6.35 Hankel transform 503

6.36 Hartley transform 504

6.37 Hilbert transform 505

6.38 Laplace transform 508

6.39 Mellin transform 512

6.40 Z-Transform 512

6.41 Tables of transforms 517

Chapter 7 Probability and Statistics 533

7.1 Probability theory 535

7.2 Classical probability problems 545

7.3 Probability distributions 553

7.4 Queuing theory 562

7.5 Markov chains 565

7.6 Random number generation 568

7.7 Random matrices 574

7.8 Control charts and reliability 575

7.9 Statistics 580

viii Table of Contents

7.10 Confidence intervals 588

7.11 Tests of hypotheses 595

7.12 Linear regression 609

7.13 Analysis of variance (ANOVA) 613

7.14 Sample size 620

7.15 Contingency tables 623

7.16 Acceptance sampling 626

7.17 Probability tables 628

Chapter 8 Scientific Computing 645

8.1 Basic numerical analysis 646

8.2 Numerical linear algebra 659

8.3 Numerical integration and differentiation 668

8.4 Programming techniques 688

Chapter 9 Mathematical Formulas from the Sciences 689

9.1 Acoustics 691

9.2 Astrophysics 692

9.3 Atmospheric physics 694

9.4 Atomic Physics 695

9.5 Basic mechanics 696

9.6 Beam dynamics 698

9.7 Biological Models 699

9.8 Chemistry 700

9.9 Classical mechanics 701

9.10 Coordinate systems – Astronomical 702

9.11 Coordinate systems – Terrestrial 703

9.12 Earthquake engineering 704

9.13 Economics (Macro) 705

9.14 Electromagnetic Transmission 707

9.15 Electrostatics and magnetism 708

9.16 Electromagnetic Field Equations 709

9.17 Electronic circuits 710

9.18 Epidemiology 711

9.19 Fluid mechanics 712

9.20 Human body 713

9.21 Modeling physical systems 714

9.22 Optics 715

9.23 Population genetics 716

9.24 Quantum mechanics 717

9.25 Quaternions 719

9.26 Radar 720

9.27 Relativistic mechanics 721

9.28 Solid mechanics 722

9.29 Statistical mechanics 723

9.30 Thermodynamics 724

Chapter 10 Miscellaneous 725

10.1 Calendar computations 727

10.2 Cellular automata 728

10.3 Communication theory 729

10.4 Control theory 734

10.5 Computer languages 736

10.6 Compressive Sensing 737

10.7 Constrained Least Squares 738

10.8 Cryptography 739

10.9 Discrete dynamical systems and chaos 740

10.10 Elliptic curves 743

10.11 Financial formulas 746

10.12 Game theory 754

10.13 Knot theory 757

10.14 Lattices 759

10.15 Logic 761

10.16 Moments of inertia 766

10.17 Music 767

10.18 Operations research 769

10.19 Proof Methods 781

10.20 Recreational mathematics 782

10.21 Risk analysis and decision rules 783

10.22 Signal processing 785

10.23 Units 794

10.24 Voting power 801

10.25 Greek alphabet 803

10.26 Braille code 803

10.27 Morse code 803

10.28 Bar Codes 804

List of References 805

List of Figures 809

List of Notations 811

Index 819

It has long been the established policy of CRC Press to publish, in handbook form,the most up-to-date, authoritative, logically arranged, and readily usable referencematerial available

Just as pocket calculators have replaced tables of square roots and trig functions;the internet has made printed tabulation of many tables and formulas unnecessary

As the content and capabilities of the internet continue to grow, the content of thisbook also evolves For this edition of Standard Mathematical Tables and Formulaethe content was reconsidered and reviewed The criteria for inclusion in this editionincludes:

• information that is immediately useful as a reference (e.g., interpretation ofpowers of 10);

• information that is useful and not commonly known (e.g., proof methods);

• information that is more complete or concise than that which can be easilyfound on the internet (e.g., table of conformal mappings);

• information difficult to find on the internet due to the challenges of entering anappropriate query (e.g., integral tables)

Applying these criteria, practitioners from mathematics, engineering, and the ences have made changes in several sections and have added new material

sci-• The “Mathematical Formulas from the Sciences” chapter now includes topicsfrom biology, chemistry, and radar

• Material has been augmented in many areas, including: acceptance sampling,card games, lattices, and set operations

• New material has been added on the following topics: continuous wavelet form, contour integration, coupled analogues, financial options, fractal arith-metic, generating functions, linear temporal logic, matrix pseudospectra, maxplus algebra, proof methods, and two dimensional integrals

trans-• Descriptions of new functions have been added: Lambert, prolate spheroidal,and Weierstrass

Of course, the same successful format which has characterized earlier editions of theHandbookhas been retained Material is presented in a multi-sectional format, witheach section containing a valuable collection of fundamental reference material—tabular and expository

xii Preface

In line with the established policy of CRC Press, the Handbook will be updated

in as current and timely manner as is possible Suggestions for the inclusion of newmaterial in subsequent editions and comments regarding the present edition are wel-comed The home page for this book, which will include errata, will be maintained

MATLABR is a registered trademark of The MathWorks, Inc

For product information please contact:

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ZwillingerBooks@gmail.com

Daniel Zwillinger

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Troy, New York

Contributors

George E Andrews

Evan Pugh University Professor in

Mathematics

The Pennsylvania State University

University Park, Pennsylvania

Professor of Scientific Computing

Florida State University

Portland, Oregon

Dr Joseph J RushananMITRE CorporationBedford, Massachusetts

Dr Les ServiMITRE CorporationBedford, Massachusetts

Dr Michael T StraussPresident HMENewburyport, Massachusetts

Dr Nico M TemmeCentrum Wiskunde & InformaticaAmsterdam, The NetherlandsAhmed I Zayed

Professor in Department ofMathematical SciencesDePaul UniversityChicago, Illinois

Chapter 1

Numbers and Elementary Mathematics

1.1 PROOFS WITHOUT WORDS 3

1.2 CONSTANTS 5

1.2.1 Divisibility tests 5

1.2.2 Decimal multiples and prefixes 6

1.2.3 Binary prefixes 6

1.2.4 Interpretations of powers of 10 7

1.2.5 Numerals in different languages 8

1.2.6 Roman numerals 8

1.2.7 Types of numbers 9

1.2.8 Representation of numbers 10

1.2.9 Representation of complex numbers – DeMoivre’s theorem 10

1.2.10 Arrow notation 11

1.2.11 Ones and Twos Complement 11

1.2.12 Symmetric base three representation 11

1.2.13 Hexadecimal addition and subtraction table 12

1.2.14 Hexadecimal multiplication table 12

1.3 SPECIAL NUMBERS 13

1.3.1 Powers of 2 13

1.3.2 Powers of 10 in hexadecimal 13

1.3.3 Special constants 14

1.3.4 Factorials 16

1.3.5 Bernoulli polynomials and numbers 17

1.3.6 Euler polynomials and numbers 18

1.3.7 Fibonacci numbers 18

1.3.8 Sums of powers of integers 19

1.3.9 Negative integer powers 20

1.3.10 Integer sequences 21

1.3.11 p-adic Numbers 23

1.3.12 de Bruijn sequences 23

1.4 INTERVAL ANALYSIS 24

1.5 FRACTAL ARITHMETIC 25

1.6 MAX-PLUS ALGEBRA 26

1.7 COUPLED-ANALOGUES OF FUNCTIONS 27

1.7.1 Coupled-operations 27

1.8 NUMBER THEORY 28

1.8.1 Congruences 28

1.8.2 Chinese remainder theorem 29

1.8.3 Continued fractions 30

1.8.4 Diophantine equations 32

1.8.5 Greatest common divisor 35

1.8.6 Least common multiple 35

1.8.7 Möbius function 36

1.8.8 Prime numbers 37

1.8.9 Prime numbers of special forms 39

1.8.10 Prime numbers less than 7,000 42

1.8.11 Factorization table 44

1.8.12 Euler totient function 46

1.9 SERIES AND PRODUCTS 47

1.9.1 Definitions 47

1.9.2 General properties 47

1.9.3 Convergence tests 49

1.9.4 Types of series 50

1.9.5 Fourier series 54

1.9.6 Series expansions of special functions 59

1.9.7 Summation formulas 63

1.9.8 Faster convergence: Shanks transformation 63

1.9.9 Summability methods 64

1.9.10 Operations with power series 64

1.9.11 Miscellaneous sums 64

1.9.12 Infinite products 65

1.9.13 Infinite products and infinite series 65

1.1 PROOFS WITHOUT WORDS 3

1.1 PROOFS WITHOUT WORDS

—the Chou pei suan ching

(author unknown, circa B.C. 200?)

The Pythagorean Theorem

A Property of the Sequence of Odd Integers (Galileo, 1615)

13

1+35+7

1+3+57+9+11

=

1+3+ +(2n–1) (2n+1)+(2n+3)+ +(4n–1)

13

sin(x + y) = sinxcosy + cosxsiny

cos(x + y) = cosxcosy – sinxsiny

d (a,b) (a,ma + c)

x y

c d

a

b

c

d a d

Reprinted from “Proofs Without Words: Exercises in Visual Thinking,” byRoger B Nelsen, 1997, MAA, pages: 3, 40, 49, 60, 70, 72, 115, 120 CopyrightThe Mathematical Association of America All rights reserved

Reprinted from “Proofs Without Words II: More Exercises in Visual Thinking,”

by Roger B Nelsen, 2001, MAA, pages 46, 111 Copyright The Mathematical sociation of America All rights reserved

As-1.2 CONSTANTS

1.2.1 DIVISIBILITY TESTS

1 Divisibility by 2: the last digit is divisible by 2

2 Divisibility by 3: the sum of the digits is divisible by 3

3 Divisibility by 4: the number formed from the last 2 digits is divisible by 4

4 Divisibility by 5: the last digit is either 0 or 5

5 Divisibility by 6: is divisible by both 2 and 3

6 Divisibility by 9: the sum of the digits is divisible by 9

7 Divisibility by 10: the last digit is 0

8 Divisibility by 11: the difference between the sum of the odd digits and thesum of the even digits is divisible by 11

EXAMPLE Consider the number N = 1036728

• The last digit is 8, so N is divisible by 2

• The last two digits are 28 which is divisible by 4, so N is divisible by 4

• The sum of the digits is 27 = 1 + 0 + 3 + 6 + 7 + 2 + 8 This is divisible by 3, so N

is divisible by 3 This is also divisible by 9, so N is divisible by 9

• The sum of the odd digits is 19 = 1 + 3 + 7 + 8 and the sum of the even digits is

8 = 6 + 2; the difference is 19 − 8 = 11 This is divisible by 11, so N is divisible

by 11

1.2.2 DECIMAL MULTIPLES AND PREFIXES

The prefix names and symbols below are taken from Conference Générale des Poids

et Mesures, 1991 The common names are for the United States

Factor Prefix Symbol Common name

0.000 000 000 001 = 10−12 pico p trillionth

0.000 000 000 000 001 = 10−15 femto f quadrillionth0.000 000 000 000 000 001 = 10−18 atto a quintillionth

10−21 zepto z hexillionth

10−24 yocto y heptillionth

1.2.3 BINARY PREFIXES

A byte is 8 bits A kibibyte is 210= 1024 bytes Other prefixes for power of 2 are:

Factor Prefix Symbol

1.2 CONSTANTS 7

1.2.4 INTERPRETATIONS OF POWERS OF 10

10−43 Planck time in seconds

10−35 Planck length in meters

10−30 mass of an electron in kilograms

10−27 mass of a proton in kilograms

10−15 the radius of the hydrogen nucleus (a proton) in meters

10−11 the likelihood of being dealt 13 top honors in bridge

10−10 the (Bohr) radius of a hydrogen atom in meters

10−9 the number of seconds it takes light to travel one foot

10−6 the likelihood of being dealt a royal flush in poker

100 the density of water is 1 gram per milliliter

101 the number of fingers that people have

102 the number of stable elements in the periodic table

104 the speed of the Earth around the sun in meters/second

105 the number of hairs on a human scalp

106 the number of words in the English language

107 the number of seconds in a year

108 the speed of light in meters per second

109 the number of heartbeats in a lifetime for most mammals

1010 the number of people on the earth

1011 the distance from the Earth to the sun in meters

1013 diameter of the solar system in meters

1014 number of cells in the human body

1015 the surface area of the earth in square meters

1016 the number of meters light travels in one year

1017 the age of the universe in seconds

1018 the volume of water in the earth’s oceans in cubic meters

1019 the number of possible positions of Rubik’s cube

1021 the volume of the earth in cubic meters

1024 the number of grains of sand in the Sahara desert

1025 the mass of the earth in kilograms

1030 the mass of the sun in kilograms

1050 the number of atoms in the earth

1052 the mass of the observable universe in kilograms

1054 the number of elements in the monster group

1078 the volume of the universe in cubic meters

(Note: these numbers have been rounded to the nearest power of ten.)

1.2.5 NUMERALS IN DIFFERENT LANGUAGES

1.2.6 ROMAN NUMERALS

The major symbols in Roman numerals are I = 1, V = 5, X = 10, L = 50, C = 100,

D = 500, and M = 1,000 The rules for constructing Roman numerals are:

1 A symbol following one of equal or greater value adds its value (For example,

II = 2, XI = 11, and DV = 505.)

2 A symbol following one of lesser value has the lesser value subtracted fromthe larger value An I is only allowed to precede a V or an X, an X is onlyallowed to precede an L or a C, and a C is only allowed to precede a D or

an M (For example IV = 4, IX = 9, and XL = 40.)

3 When a symbol stands between two of greater value, its value is subtractedfrom the second and the result is added to the first (For example, XIV=10+(5−1) = 14, CIX= 100+(10−1) = 109, DXL= 500+(50−10) = 540.)

4 When two ways exist for representing a number, the one in which the symbol

of larger value occurs earlier in the string is preferred (For example, 14 isrepresented as XIV, not as VIX.)

1.2 CONSTANTS 9

1.2.7 TYPES OF NUMBERS

1 Natural numbers The set of natural numbers, {0, 1, 2, }, is customarilydenoted by N Many authors do not consider 0 to be a natural number

2 Integers The set of integers, {0, ±1, ±2, }, is customarily denoted by Z

3 Rational numbers The set of rational numbers, {p

q | p, q ∈ Z, q 6= 0}, iscustomarily denoted by Q

(a) Two fractions p

sare equal if and only if ps = qr

(b) Addition of fractions is defined by p

Real numbers can be divided into two subsets One subset, the algebraic bers, are real numbers which solve a polynomial equation in one variable withinteger coefficients For example;√

num-2 is an algebraic number because it solvesthe polynomial equation x2

− 2 = 0; and all rational numbers are algebraic.Real numbers that are not algebraic numbers are called transcendental num-bers Examples of transcendental numbers include π and e

5 Definition of infinity The real numbers are extended to R by the inclusion

of +∞ and −∞ with the following definitions

(h) −∞ − ∞ = −∞

(i) ∞ · ∞ = ∞(j) −∞ · (−∞) = ∞

6 Complex numbers The set of complex numbers is customarily denoted

by C They are numbers of the form a + bi, where i2 =−1, and a and b arereal numbers

Operation computation result

addition (a + bi) + (c + di) (a + c) + i(b + d)

multiplication (a + bi)(c + di) (ac− bd) + (ad + bc)i

Properties include: z + w = z + w and zw = z w

1.2.8 REPRESENTATION OF NUMBERS

Numerals as usually written have radix or base 10, so the numeral anan−1 a1a0

represents the number an10n+ an−110n−1+· · · + a2102+ a110 + a0 However,other bases can be used, particularly bases 2, 8, and 16 When a number is written inbase 2, the number is said to be in binary notation The names of other bases are:

When writing a number in base b, the digits used range from 0 to b − 1 If

b > 10, then the digit A stands for 10, B for 11, etc When a base other than 10 isused, it is indicated by a subscript:

as the previous digit Continue this process until a quotient of 0 is obtained

EXAMPLE To convert 573 to base 12, divide 573 by 12, yielding a quotient of 47 and aremainder of 9; hence, “9” is the last digit Divide 47 by 12, yielding a quotient of 3 and

a remainder of 11 (which we represent with a “B”) Divide 3 by 12 yielding a quotient

of 0 and a remainder of 3 Therefore, 57310= 3B912

Converting from base b to base r can be done by converting to and from base

10 However, it is simple to convert from base b to base bn For example, to vert 1101111012to base 16, group the digits in fours (because 16 is 24), yielding

con-1 con-10con-1con-1 con-1con-10con-12, and then convert each group of 4 to base 16 directly, yielding 1BD16

1.2.9 REPRESENTATION OF COMPLEX NUMBERS –

m↑↑ n = m ↑ m ↑ · · · ↑ m| {z }

n

m↑↑↑ n = m ↑↑ m ↑↑ · · · ↑↑ m| {z }

n

For example, m ↑ n = mn, m ↑↑ 2 = mm, and m ↑↑ 3 = m(m m )

1.2.11 ONES AND TWOS COMPLEMENT

One’s and two’s complement are ways to represent numbers in a computer Forpositive values the binary representation, the ones’ complement representation, andthe twos’ complement representation are the same

• Ones’ complement represents integers from − 2N −1− 1
to 2N −1− 1 Fornegative values, the binary representation of the absolute value is obtained, andthen all of the bits are inverted (i.e., swapping 0’s for 1’s and vice versa)

• Twos’ complement represents integers from −2N −1to 2N −1− 1 For negativevales, the two’s complement representation is the same as the value one added

to the ones’ complement representation

Number Ones’ complement Twos’ complement

1.2.12 SYMMETRIC BASE THREE REPRESENTATION

In the symmetric base three representation, powers of 3 are added and subtracted

to represent numbers; the symbols {↓, 0, ↑} represent {−1, 0, 1} For example, onewrites ↑↓↓ for 5 since 5 = 9−3−1 To negate a number in symmetric base three, turnits symbol upside down, e.g., −5 =↓↑↑ Basic arithmetic operations are especiallysimple in this base

1.2.13 HEXADECIMAL ADDITION AND SUBTRACTION TABLE

A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

Example: 6 + 2 = 8; hence 8 − 6 = 2 and 8 − 2 = 6

Example: 4 + E = 12; hence 12 − 4 = E and 12 − E = 4

1.3.3 SPECIAL CONSTANTS

1.3.3.1 The constant π

The transcendental number π is defined as the ratio of the circumference of a circle

to the diameter It is also the ratio of the area of a circle to the square of the radius(r) and appears in several formulas in geometry and trigonometry

circumference of a circle = 2πr, volume of a sphere = 4

3πr

area of a circle = πr2, surface area of a sphere = 4πr2.One method of computing π is to use the infinite series for the function tan−1x andone of the identities



π = lim

vuut2 −

vuut2 +

vu

t2 +

vu

The factorial of n, denoted n!, is the product of all positive integers less than or equal

to n; n! = n·(n−1)·(n−2) · · · 2·1 By definition, 0! = 1 If n is a negative integer(n = −1, −2, ) then n! = ±∞ The generalization of the factorial function tonon-integer arguments is the gamma function (seepage 478); when n is an integer,Γ(n) = (n− 1)!

The double factorial of n is denoted by n!! and is defined as n!! = n(n − 2)(n −4)· · · , where the last term in the product is 2 or 1, depending on whether n is even

or odd The shifted factorial (also called Pochhammer’s symbol) is denoted by (a)n

n!≈√2πe n

e

n+ 1

1 + 112n+

1288n2+



(1.3.5)and Burnside’s formula

n!≈√2π

n +1 2

1.3.5 BERNOULLI POLYNOMIALS AND NUMBERS

The Bernoulli polynomials Bn(x) are defined by the generating function

0 Bn(x) dx = 0 for n≥ 1 The identity Bk+1(x + 1)− Bk+1(x) = (k + 1)xk

means that sums of powers can be computed via Bernoulli polynomials

1k+ 2k+· · · + nk =Bk+1(n + 1)− Bk+1(0)

The Bernoulli numbers are the Bernoulli polynomials evaluated at 0: Bn = Bn(0)

A generating function for the Bernoulli numbers is

Nn/Dn Note that B2m+1= 0 for m≥ 1

1.3.6 EULER POLYNOMIALS AND NUMBERS

The Euler polynomials En(x) are defined by the generating function

∞

X

n=0

Entnn! =

!n

− 1−

√52

!n#

(1.3.13)Note that Fnis the integer nearest to φn/√

5 as n→ ∞, where φ is the golden ratio

1.3 SPECIAL NUMBERS 19

1.3.8 SUMS OF POWERS OF INTEGERS

1 Define the sum of the first n kth-powers



a3nk

+· · · +

k + 12

1.3.9 NEGATIVE INTEGER POWERS

Riemann’s zeta function is defined to be ζ(n) =P∞

k=1 k1n(it is defined for Re n > 1and extended to C) Related functions are

is an integer and p does not divide a or b Define the p-adic norm of x as |x|p= p−n

and also define |0|p= 0 The p-adic norm has the properties:

1 |x|p≥ 0 for all non-negative rational numbers x

2 |x|p= 0 if and only if x = 0

3 For all non-negative rational numbers x and y

(a) |xy|p=|x|p|y|p

(b) |x + y|p≤ max (|x|p,|y|p)≤ |x|p+|y|p

Note the product formula: |x|Qp∈{2,3,5,7,11, }|x|p= 1

Let Qpbe the topological completion of Q with respect to | · |p Then Qpis thefield of p-adic numbers The elements of Qp can be viewed as infinite series: theseries P∞

n=0anconverges to a point in Qpif and only if limn→∞|an|p= 0

EXAMPLE The number140

297 = 22· 3−3· 5 · 7 · 11−1has the different p-adic norms:

/q!distinct sequences.) The table below contains some small examples

1.4 INTERVAL ANALYSIS

1 Definitions

(a) An interval x is a subset of the real line:

x = [x, x] ={z ∈ R | x ≤ z ≤ x}.(b) A thin interval is a real number: x is thin if x = x

(c) mid(x) = x + x

2(d) rad(x) = x− x

2

(e) |x| = mag(x) = maxz∈x|z|

(f) hxi = mig(x) = minz∈x|z|

The MATLABR package INTLAB performs interval computations

2 Interval arithmetic rules

3 Interval arithmetic properties

commutative x + y = y + x xy = yx

associative x + (y + z) = (x + y) + z x(yz) = (xy)z

identity elements 0 + x = x + 0 = x 1∗ y = y ∗ 1 = ysub-distributivity x(y ± z) ⊆ xy ± xz (equality holds if x is thin)sub-cancellation x− y ⊆ (x + z) − (y + z) x

1 The expffunction satisfies expf(x⊕ y) = expfx⊙ expfy and is the uniquesolution to the differential equationd f A(x)

d f x = A(x) with A(0) = 1

2 The lnffunction satisfies lnf(x⊙ y) = lnfx⊕ lnfy

1.6 MAX-PLUS ALGEBRA

The max-plus algebra is an algebraic structure with real numbers and two tions (called “plus” and “times”); “plus” is the operation of taking a maximum, and

opera-“times” is the “standard addition” operation Mathematically:

The max-plus semiring Rmax is the set R ∪ {−∞} with two operations called

“plus” (⊕) and “times” (⊗) defined as follows:

• If a = b, then x can be any value with x ≤ b

• If a > b, then there is no solution

3 In Rmaxmatrix multiplication is associative

5 The Kleene star of the matrix A is the matrix A∗= I + A + A2+· · ·

6 The completed max-plus semiring Rmaxis the same as Rmaxwith the additionalelement +∞ and the convention (−∞) + (+∞) = (+∞) + (−∞) = −∞

opera-“times” is the ? ?standard addition” operation Mathematically:

The... expffunction satisfies expf(x⊕ y) = expfx⊙ expfy and is the uniquesolution to the differential equationd f A(x)

d f x... max-plus semiring Rmax is the set R ∪ {−∞} with two operations called

“plus” (⊕) and “times” (⊗) defined as follows:

• If a = b, then x can be any value with x ≤ b

•