מתימטיקה

שתף אותנו בקישורים לספרים וסיכומי הרצאות שמצאת ברשת

2006-11-15T18:55:00.000-08:00

אתר זה מציע קישורים לספרי לימוד, סיכומי הרצאות וטקסטים ללא תשלום בתחומי המתימטיקה הנמצאים ברשת האינטרנט

תוכל לספר לנו על קישורים נוספים שמצאת באמצעות ההערות שבסוף ההודעה הנוכחית, הקלד את כתובת האינטרנט של הספר ואם תרצה תוכל להוסיף פרטים כמו: שם הספר, שם המחבר ואת הערותיך לתוכן הספר

MathWorld

2006-11-14T21:27:00.000-08:00

MathWorld

MathWorldTM is the web's most complete mathematical resource, assembled over more than a decade by internet encyclopedist Eric W. Weisstein with assistance from the mathematics and internet communities.

MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. Like the vibrant and constantly evolving discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries.

Although it is often difficult to find explanations for technical subjects that are both clear and accessible, this website bridges the gap by placing an interlinked framework of mathematical exposition and illustrative examples at the fingertips of every internet user.

LiveGraphics3D Applets

On the Shoulders of Giants: New Approaches to Numeracy (1990)

2004-11-18T05:06:00.000-08:00

On the Shoulders of Giants: New Approaches to Numeracy (1990)

Front Matter i-viii
Pattern 1-10 (skim)
Dimension 11-60 (skim)
Quantity 61-94 (skim)
Uncertainty 95-138 (skim)
Shape 139-182 (skim)
Change 183-218 (skim)
Biographies 219-222 (skim)
Index 223-232 (skim)

The Elements of Non-Euclidean Geometry

2004-11-16T08:00:00.000-08:00

The Elements of Non-Euclidean Geometry

Table of Contents
Prelude: Gauss mappings of plane curves
Gauss mappings of surfaces
Example 1: The shoe surface
Example 2: Menn's surface
Example 3a: The perturbed monkey saddle
Example 3b: The handkerchief surface
Example 4: Surfaces of revolution
Example 5: Canal surfaces
Characterizations of Gaussian cusps
Part 1
Part 2
Part 3
Singularities of families of mappings
Projections to lines
Focal and parallel surfaces
Projections to planes
Singularities and extrinsic geometry
References
Movies

Gauss: Disquisitiones

2004-11-16T07:52:00.000-08:00

Gauss: Disquisitiones

Preface
Table of Contents
Prelude: Gauss mappings of plane curves
Gauss mappings of surfaces
Example 1: The shoe surface
Example 2: Menn's surface
Example 3a: The perturbed monkey saddle
Example 3b: The handkerchief surface
Example 4: Surfaces of revolution
Example 5: Canal surfaces
Characterizations of Gaussian cusps
Part 1
Part 2
Part 3
Singularities of families of mappings
Projections to lines
Focal and parallel surfaces
Projections to planes
Singularities and extrinsic geometry
References
Movies

Elements of Geometry and Trigonometry

2004-11-16T07:36:00.000-08:00

The Cornell Library Historical Mathematics Monographs

An Elementary Treatise on Pure Geometry

2004-11-16T07:00:00.000-08:00

An Elementary Treatise on Pure Geometry with Numerous Examples

The Cornell Library Historical Mathematics Monographs

2004-11-16T06:58:00.000-08:00

The Cornell Library Historical Mathematics Monographs

2004-11-16T06:49:00.000-08:00

The Cornell Library Historical Mathematics Monographs

Pravin Varaiya's Home Page

2004-11-16T06:41:00.000-08:00

Pravin Varaiya's Home Page

Functions of Complex Variable

2004-11-16T06:34:00.000-08:00

Functions of Complex Variable

The Cornell Library Historical Mathematics Monographs

2004-11-16T06:29:00.000-08:00

The Cornell Library Historical Mathematics MonographsBR>

The Cornell Library Historical Mathematics Monographs

2004-11-16T06:22:00.000-08:00

The Cornell Library Historical Mathematics Monographs

2004-11-16T06:15:00.000-08:00

The Cornell Library Historical Mathematics Monographs

2004-11-16T05:49:00.000-08:00

The Cornell Library Historical Mathematics Monographs

2004-11-16T05:46:00.000-08:00

The Cornell Library Historical Mathematics Monographs

2004-11-16T05:38:00.000-08:00

The Cornell Library Historical Mathematics Monographs

2004-11-16T05:27:00.000-08:00

The Cornell Library Historical Mathematics Monographs

2004-11-16T05:23:00.000-08:00

The Cornell Library Historical Mathematics Monographs

Toeplitz and Circulant Matrices

2004-11-16T05:04:00.000-08:00

Toeplitz and Circulant Matrices
Toeplitz and Circulant Matices: A Review, by R. M. Gray. A very old (1971, revised 1977, 1993, 1997, 1998, 2000, 2001, 2002.) but still occasionally useful tutorial on Toeplitz and circulant matrices. The file is in Adobe portable document format (pdf). Free readers can be downloaded from Adobe.

Wolfram Research, Inc.

2004-11-16T04:57:00.000-08:00

Wolfram Research, Inc.

Table of Contents
What Is Mathematica?About This Book Acknowledgments Tour
Part 1. A Practical Introduction to Mathematica1.0 Running Mathematica 1.1 Numerical Calculations 1.2 Building Up Calculations 1.3 Using the Mathematica System 1.4 Algebraic Calculations 1.5 Symbolic Mathematics 1.6 Numerical Mathematics 1.7 Functions and Programs 1.8 Lists 1.9 Graphics and Sound 1.10 Files and External Operations
Part 2. Principles of Mathematica2.1 Expressions 2.2 Functional Operations 2.3 Patterns 2.4 Transformation Rules and Definitions 2.5 Evaluation of Expressions 2.6 Modularity and the Naming of Things 2.7 Textual Output 2.8 Strings, Names and Messages 2.9 The Structure of Graphics and Sound 2.10 Input and Output 2.11 Global Aspects of Mathematica Sessions
Part 3. Advanced Mathematics in Mathematica3.1 Numbers 3.2 Mathematical Functions 3.3 Algebraic Manipulation 3.4 Manipulating Equations 3.5 Calculus 3.6 Power Series, Limits and Residues 3.7 Linear Algebra 3.8 Numerical Operations on Data 3.9 Numerical Operations on Functions
Appendix. Mathematica Reference GuideA.1 Basic Objects A.2 Input Syntax A.3 Some General Notations and Conventions A.4 Evaluation A.5 Patterns and Transformation Rules A.6 Input and Output A.7 Mathematica Sessions and Global Objects A.8 Listing of Built-in Mathematica Objects

Differential and Integral Calculus

2004-11-15T18:51:00.000-08:00

The Cornell Library Historical Mathematics Monographs Differential and Integral Calculus

The Cornell Library Historical Mathematics Monographs

2004-11-15T16:16:00.000-08:00

The Cornell Library Historical Mathematics Monographs

Preface
Acknowledgements
1 XML for Data
2 XML Protocols
3 Writing XML with Java
4 Converting Flat Files to XML
5 Reading XML
6 SAX
7 The XMLReader Interface
8 SAX Filters
9 The Document Object Model
10 Creating New XML Documents with DOM
11 The Document Object Model Core
12 The DOM Traversal Module
13 Output from DOM
14 JDOM
15 The JDOM Model
16 XPath
17 XSLT
A XML APIs Quick Reference
B SOAP Schemas
Recommended Reading
Examples
I've extracted out all the

Graph Theory - Reinhard Diestel

2004-11-14T05:37:00.000-08:00

Graph Theory

1. The Basics
1.1 Graphs ....................................... 2
1.2 The degree of a vertex ....................... 4
1.3 Paths and cycles ............................. 6
1.4 Connectivity ................................. 9
1.5 Trees and forests ............................ 12
1.6 Bipartite graphs ............................. 14
1.7 Contraction and minors ....................... 16
1.8 Euler tours .................................. 18
1.9 Some linear algebra .......................... 20
1.10 Other notions of graphs ...................... 25
Exercises .................................... 26
Notes ........................................ 28
2. Matching
2.1 Matching in bipartite graphs ................. 29
2.2 Matching in general graphs ................... 34
2.3 Path covers .................................. 39
Exercises .................................... 40
Notes ........................................ 42
3. Connectivity
3.1 2-Connected graphs and subgraphs ............. 43
3.2 The structure of 3-connected graphs........... 45
3.3 Menger's theorem ............................. 50
3.4 Mader's theorem .............................. 56
3.5 Edge-disjoint spanning trees ................. 58
3.6 Paths between given pairs of vertices ........ 61
Exercises .................................... 63
Notes ........................................ 65
4. Planar Graphs
4.1 Topological prerequisites .................... 68
4.2 Plane graphs ................................. 70
4.3 Drawings ..................................... 76
4.4 Planar graphs: Kuratowski's theorem .......... 80
4.5 Algebraic planarity criteria ................. 85
4.6 Plane duality ................................ 87
Exercises .................................... 89
Notes ........................................ 92
5. Colouring
5.1 Colouring maps and planar graphs ............. 96
5.2 Colouring vertices ........................... 98
5.3 Colouring edges .............................. 103
5.4 List colouring ............................... 105
5.5 Perfect graphs ............................... 110
Exercises .................................... 117
Notes ........................................ 120
6. Flows
6.1 Circulations ................................. 124
6.2 Flows in networks ............................ 125
6.3 Group-valued flows ........................... 128
6.4 k-Flows for small k .......................... 133
6.5 Flow-colouring duality ....................... 136
6.6 Tutte's flow conjectures ..................... 140
Exercises .................................... 144
Notes ........................................ 145
7. Substructures in Dense Graphs
7.1 Subgraphs .................................... 148
7.2 Szemerédi's regularity lemma ................. 153
7.3 Applying the regularity lemma ................ 160
Exercises .................................... 165
Notes ........................................ 166
8. Substructures in Sparse Graphs
8.1 Topological minors ........................... 170
8.2 Minors ....................................... 179
8.3 Hadwiger's conjecture ........................ 181
Exercises .................................... 184
Notes ........................................ 186
9. Ramsey Theory for Graphs
9.1 Ramsey's original theorems ................... 190
9.2 Ramsey numbers ............................... 193
9.3 Induced Ramsey theorems ...................... 197
9.4 Ramsey properties and connectivity ........... 207
Exercises .................................... 208
Notes ........................................ 210
10. Hamilton Cycles
10.1 Simple sufficient conditions ................. 213
10.2 Hamilton cycles and degree sequence .......... 216
10.3 Hamilton cycles in the square of a graph ..... 218
Exercises .................................... 226
Notes ........................................ 227
11. Random Graphs
11.1 The notion of a random graph ................. 230
11.2 The probabilistic method ..................... 235
11.3 Properties of almost all graphs .............. 238
11.4 Threshold functions and second moments ....... 242
Exercises .................................... 247
Notes ........................................ 249
12. Minors, Trees, and WQO
12.1 Well-quasi-ordering .......................... 251
12.2 The minor theorem for trees .................. 253
12.3 Tree-decompositions .......................... 255
12.4 Tree-width and forbidden minors .............. 263
12.5 The graph minor theorem ......................

3F3 - Random Processes, Optimal Filtering and Model-based Signal

2004-11-10T22:57:00.000-08:00

[PDF] 3F3 - Random Processes, Optimal Filtering and Model-based Signal
PDF/Adobe Acrobat - View as HTMLPage 1. 3F3 - Random Processes, Optimal Filtering and Model-based SignalProcessing SJ Godsill February 25, 2004 Page 2. 3F3 - Random ... www-sigproc.eng.cam.ac.uk/~sjg/teaching/3f3/lecture.pdf

Short course on Ttrigonometry and Complex Numbers

2004-11-10T22:32:00.000-08:00

David Joyce's Home Page

Euclid's Elements
Compass geometry
a short course on trigonometry
a short course on complex numbers
History of Mathematics, Hilbert's address of 1900 and his 23 mathematical problems

Introduction to Methods of Applied Mathematics

2004-11-09T20:21:00.000-08:00

Introduction to Methods of Applied Mathematics
Contents
Anti-Copyright xiii
Preface xv
0.1 Advice to Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
0.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
0.3 Warnings and Disclaimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
0.4 Suggested Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
0.5 About the Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
I Algebra 1
1 Sets and Functions 3
1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Single Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Inverses and Multi-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Transforming Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Vectors 13
2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 The Kronecker Delta and Einstein Summation Convention . . . . . . . . . . . 15
2.1.3 The Dot and Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Sets of Vectors in n Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
II Calculus 27
3 Differential Calculus 29
3.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6.1 Application: Using Taylor’s Theorem to Approximate Functions. . . . . . . . 41
3.6.2 Application: Finite Difference Schemes . . . . . . . . . . . . . . . . . . . . . . 43
3.7 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
i
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.8.4 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.8.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.8.6 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.8.7 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.11 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.12 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Integral Calculus 71
4.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 The Fundamental Theorem of Integral Calculus . . . . . . . . . . . . . . . . . . . . . 76
4.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.1 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6.3 The Fundamental Theorem of Integration . . . . . . . . . . . . . . . . . . . . 82
4.6.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.9 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.10 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Vector Calculus 95
5.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Gradient, Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.6 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.7 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
III Functions of a Complex Variable 113
6 Complex Numbers 115
6.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4 Arithmetic and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.5 Integer Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.6 Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
ii
7 Functions of a Complex Variable 127
7.1 Curves and Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 The Point at Infinity and the Stereographic Projection . . . . . . . . . . . . . . . . . 129
7.3 A Gentle Introduction to Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.4 Cartesian and Modulus-Argument Form . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.5 Graphing Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . 133
7.6 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.8 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.9 Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8 Analytic Functions 155
8.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.2 Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.3 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.4 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.4.1 Categorization of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.4.2 Isolated and Non-Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . 167
8.5 Application: Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9 Analytic Continuation 173
9.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.2 Analytic Continuation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.3 Analytic Functions Defined in Terms of Real Variables . . . . . . . . . . . . . . . . . 175
9.3.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
9.3.2 Analytic Functions Defined in Terms of Their Real or Imaginary Parts . . . . 180
10 Contour Integration and the Cauchy-Goursat Theorem 183
10.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
10.2.1 Maximum Modulus Integral Bound . . . . . . . . . . . . . . . . . . . . . . . . 185
10.3 The Cauchy-Goursat Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.4 Contour Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
10.5 Morera’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
10.6 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
10.7 Fundamental Theorem of Calculus via Primitives . . . . . . . . . . . . . . . . . . . . 190
10.7.1 Line Integrals and Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.7.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.8 Fundamental Theorem of Calculus via Complex Calculus . . . . . . . . . . . . . . . 190
11 Cauchy’s Integral Formula 193
11.1 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
11.2 The Argument Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
11.3 Rouche’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
12 Series and Convergence 201
12.1 Series of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
12.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
12.1.2 Special Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
12.1.3 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
12.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
12.2.1 Tests for Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 208
12.2.2 Uniform Convergence and Continuous Functions. . . . . . . . . . . . . . . . . 209
12.3 Uniformly Convergent Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
12.4 Integration and Differentiation of Power Series . . . . . . . . . . . . . . . . . . . . . 213
iii
12.5 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
12.5.1 Newton’s Binomial Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
12.6 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
13 The Residue Theorem 221
13.1 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
13.2 Cauchy Principal Value for Real Integrals . . . . . . . . . . . . . . . . . . . . . . . . 225
13.2.1 The Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
13.3 Cauchy Principal Value for Contour Integrals . . . . . . . . . . . . . . . . . . . . . . 228
13.4 Integrals on the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
13.5 Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
13.6 Fourier Cosine and Sine Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
13.7 Contour Integration and Branch Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . 236
13.8 Exploiting Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
13.8.1 Wedge Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
13.8.2 Box Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
13.9 Definite Integrals Involving Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . 241
13.10Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
IV Ordinary Differential Equations 245
14 First Order Differential Equations 247
14.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
14.2 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
14.2.1 Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
14.3 One Parameter Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
14.4 Integrable Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
14.4.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
14.4.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
14.4.3 Homogeneous Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . 254
14.5 The First Order, Linear Differential Equation . . . . . . . . . . . . . . . . . . . . . . 257
14.5.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
14.5.2 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
14.5.3 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
14.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
14.6.1 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . 260
14.7 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
14.8 Equations in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
14.8.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
14.8.2 Regular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
14.8.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
14.8.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
14.9 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
14.10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
14.11Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
14.12Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
15 First Order Linear Systems of Differential Equations 281
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
15.2 Using Eigenvalues and Eigenvectors to find Homogeneous Solutions . . . . . . . . . . 281
15.3 Matrices and Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
15.4 Using the Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
15.5 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
iv
15.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
16 Theory of Linear Ordinary Differential Equations 297
16.1 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
16.2 Nature of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
16.3 Transformation to a First Order System . . . . . . . . . . . . . . . . . . . . . . . . . 300
16.4 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
16.4.1 Derivative of a Determinant. . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
16.4.2 The Wronskian of a Set of Functions. . . . . . . . . . . . . . . . . . . . . . . 301
16.4.3 The Wronskian of the Solutions to a Differential Equation . . . . . . . . . . . 302
16.5 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
16.6 The Fundamental Set of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
16.7 Adjoint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
16.8 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
16.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
16.10Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
16.11Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
17 Techniques for Linear Differential Equations 315
17.1 Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
17.1.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
17.1.2 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
17.1.3 Higher Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
17.2 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
17.2.1 Real-Valued Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
17.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
17.4 Equations Without Explicit Dependence on y . . . . . . . . . . . . . . . . . . . . . . 325
17.5 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
17.6 *Reduction of Order and the Adjoint Equation . . . . . . . . . . . . . . . . . . . . . 326
17.7 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
17.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
18 Techniques for Nonlinear Differential Equations 331
18.1 Bernoulli Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
18.2 Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
18.3 Exchanging the Dependent and Independent Variables . . . . . . . . . . . . . . . . . 334
18.4 Autonomous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
18.5 *Equidimensional-in-x Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
18.6 *Equidimensional-in-y Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
18.7 *Scale-Invariant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
19 Transformations and Canonical Forms 341
19.1 The Constant Coefficient Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
19.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
19.2.1 Second Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
19.2.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 344
19.3 Transformations of the Independent Variable . . . . . . . . . . . . . . . . . . . . . . 344
19.3.1 Transformation to the form u” + a(x) u = 0 . . . . . . . . . . . . . . . . . . 344
19.3.2 Transformation to a Constant Coefficient Equation . . . . . . . . . . . . . . . 345
19.4 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
19.4.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
19.4.2 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
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20 The Dirac Delta Function 351
20.1 Derivative of the Heaviside Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
20.2 The Delta Function as a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
20.3 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
20.4 Non-Rectangular Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 353
21 Inhomogeneous Differential Equations 355
21.1 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
21.2 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
21.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
21.3.1 Second Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 358
21.3.2 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 360
21.4 Piecewise Continuous Coefficients and Inhomogeneities . . . . . . . . . . . . . . . . . 362
21.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
21.5.1 Eliminating Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . 364
21.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions365
21.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions 365
21.6 Green Functions for First Order Equations . . . . . . . . . . . . . . . . . . . . . . . 367
21.7 Green Functions for Second Order Equations . . . . . . . . . . . . . . . . . . . . . . 368
21.7.1 Green Functions for Sturm-Liouville Problems . . . . . . . . . . . . . . . . . 374
21.7.2 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
21.7.3 Problems with Unmixed Boundary Conditions . . . . . . . . . . . . . . . . . 377
21.7.4 Problems with Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . 378
21.8 Green Functions for Higher Order Problems . . . . . . . . . . . . . . . . . . . . . . . 380
21.9 Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
21.10Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
21.11Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
22 Difference Equations 391
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
22.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
22.3 Homogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
22.4 Inhomogeneous First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
22.5 Homogeneous Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . 395
22.6 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
23 Series Solutions of Differential Equations 399
23.1 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
23.1.1 Taylor Series Expansion for a Second Order Differential Equation . . . . . . . 402
23.2 Regular Singular Points of Second Order Equations . . . . . . . . . . . . . . . . . . . 407
23.2.1 Indicial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
23.2.2 The Case: Double Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
23.2.3 The Case: Roots Differ by an Integer . . . . . . . . . . . . . . . . . . . . . . 412
23.3 Irregular Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
23.4 The Point at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
23.5 Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
23.6 Quiz Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
24 Asymptotic Expansions 421
24.1 Asymptotic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
24.2 Leading Order Behavior of Differential Equations . . . . . . . . . . . . . . . . . . . . 423
24.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
24.4 Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
24.5 Asymptotic Expansions of Differential Equations . . . . . . . . . . . . . . . . . . . . 433
24.5.1 The Parabolic Cylinder Equation. . . . . . . . . . . . . . . . . . . . . . . . . 433
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25 Hilbert Spaces 437
25.1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
25.2 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
25.3 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
25.4 Linear Independence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
25.5 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
25.6 Gramm-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
25.7 Orthonormal Function Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
25.8 Sets Of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
25.9 Least Squares Fit to a Function and Completeness . . . . . . . . . . . . . . . . . . . 446
25.10Closure Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
25.11Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
26 Self Adjoint Linear Operators 453
26.1 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
26.2 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
27 Self-Adjoint Boundary Value Problems 455
27.1 Summary of Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
27.2 Formally Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
27.3 Self-Adjoint Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
27.4 Self-Adjoint Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
27.5 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
28 Fourier Series 463
28.1 An Eigenvalue Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
28.2 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
28.3 Least Squares Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
28.4 Fourier Series for Functions Defined on Arbitrary Ranges . . . . . . . . . . . . . . . 470
28.5 Fourier Cosine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
28.6 Fourier Sine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
28.7 Complex Fourier Series and Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . 474
28.8 Behavior of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
28.9 Gibb’s Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
28.10Integrating and Differentiating Fourier Series . . . . . . . . . . . . . . . . . . . . . . 481
29 Regular Sturm-Liouville Problems 485
29.1 Derivation of the Sturm-Liouville Form . . . . . . . . . . . . . . . . . . . . . . . . . 485
29.2 Properties of Regular Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . 486
29.3 Solving Differential Equations With Eigenfunction Expansions . . . . . . . . . . . . 493
30 Integrals and Convergence 497
30.1 Uniform Convergence of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
30.2 The Riemann-Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
30.3 Cauchy Principal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
30.3.1 Integrals on an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 498
30.3.2 Singular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
31 The Laplace Transform 501
31.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
31.2 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
31.2.1 ˆ f(s) with Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
31.2.2 ˆ f(s) with Branch Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
31.2.3 Asymptotic Behavior of ˆ f(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
31.3 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
vii
31.4 Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 511
31.5 Systems of Constant Coefficient Differential Equations . . . . . . . . . . . . . . . . . 512
32 The Fourier Transform 515
32.1 Derivation from a Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
32.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
32.2.1 A Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
32.3 Evaluating Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
32.3.1 Integrals that Converge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
32.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent. . 520
32.3.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
32.4 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
32.4.1 Closure Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
32.4.2 Fourier Transform of a Derivative. . . . . . . . . . . . . . . . . . . . . . . . . 523
32.4.3 Fourier Convolution Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 523
32.4.4 Parseval’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
32.4.5 Shift Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
32.4.6 Fourier Transform of x f(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
32.5 Solving Differential Equations with the Fourier Transform . . . . . . . . . . . . . . . 527
32.6 The Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . 528
32.6.1 The Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 528
32.6.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
32.7 Properties of the Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . 530
32.7.1 Transforms of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
32.7.2 Convolution Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
32.7.3 Cosine and Sine Transform in Terms of the Fourier Transform . . . . . . . . 532
32.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms . . . . . 533
33 The Gamma Function 535
33.1 Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
33.2 Hankel’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536
33.3 Gauss’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
33.4 Weierstrass’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
33.5 Stirling’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
34 Bessel Functions 543
34.1 Bessel’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
34.2 Frobeneius Series Solution about z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 543
34.2.1 Behavior at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
34.3 Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
34.3.1 The Bessel Function Satisfies Bessel’s Equation . . . . . . . . . . . . . . . . . 547
34.3.2 Series Expansion of the Bessel Function . . . . . . . . . . . . . . . . . . . . . 548
34.3.3 Bessel Functions of Non-Integer Order . . . . . . . . . . . . . . . . . . . . . . 549
34.3.4 Recursion Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
34.3.5 Bessel Functions of Half-Integer Order . . . . . . . . . . . . . . . . . . . . . . 553
34.4 Neumann Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
34.5 Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
34.6 Hankel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
34.7 The Modified Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
V Partial Differential Equations 561
35 Transforming Equations 563
viii
36 Classification of Partial Differential Equations 565
36.1 Classification of Second Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . 565
36.1.1 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
36.1.2 Parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
36.1.3 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
36.2 Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
37 Separation of Variables 573
37.1 Eigensolutions of Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . 573
37.2 Homogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . . 573
37.3 Time-Independent Sources and Boundary Conditions . . . . . . . . . . . . . . . . . . 574
37.4 Inhomogeneous Equations with Homogeneous Boundary Conditions . . . . . . . . . 576
37.5 Inhomogeneous Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
37.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
37.7 General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
38 Finite Transforms 581
39 The Diffusion Equation 583
40 Laplace’s Equation 585
40.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
40.2 Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
40.2.1 Two Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
41 Waves 587
42 Similarity Methods 589
43 Method of Characteristics 593
43.1 First Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
43.2 First Order Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
43.3 The Method of Characteristics and the Wave Equation . . . . . . . . . . . . . . . . . 594
43.4 The Wave Equation for an Infinite Domain . . . . . . . . . . . . . . . . . . . . . . . 595
43.5 The Wave Equation for a Semi-Infinite Domain . . . . . . . . . . . . . . . . . . . . . 596
43.6 The Wave Equation for a Finite Domain . . . . . . . . . . . . . . . . . . . . . . . . . 597
43.7 Envelopes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
44 Transform Methods 601
44.1 Fourier Transform for Partial Differential Equations . . . . . . . . . . . . . . . . . . 601
44.2 The Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
44.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
45 Green Functions 605
45.1 Inhomogeneous Equations and Homogeneous Boundary Conditions . . . . . . . . . . 605
45.2 Homogeneous Equations and Inhomogeneous Boundary Conditions . . . . . . . . . . 605
45.3 Eigenfunction Expansions for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . 607
45.4 The Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
46 Conformal Mapping 611
47 Non-Cartesian Coordinates 613
47.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
47.2 Laplace’s Equation in a Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
47.3 Laplace’s Equation in an Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
ix
VI Calculus of Variations 619
48 Calculus of Variations 621
48.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
48.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
48.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634
VII Nonlinear Differential Equations 685
49 Nonlinear Ordinary Differential Equations 687
49.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688
49.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
49.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692
50 Nonlinear Partial Differential Equations 705
50.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
50.2 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
50.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709
VIII Appendices 721
A Greek Letters 723
B Notation 725
C Formulas from Complex Variables 727
D Table of Derivatives 729
E Table of Integrals 731
F Definite Integrals 733
G Table of Sums 735
H Table of Taylor Series 737
I Continuous Transforms 739
I.1 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
I.2 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741
I.3 Table of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743
I.4 Table of Fourier Transforms in n Dimensions . . . . . . . . . . . . . . . . . . . . . . 745
I.5 Table of Fourier Cosine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746
I.6 Table of Fourier Sine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747
J Table of Wronskians 749
K Sturm-Liouville Eigenvalue Problems 751
L Green Functions for Ordinary Differential Equations 753
M Trigonometric Identities 755
M.1 Circular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755
M.2 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756
x
N Bessel Functions 759
N.1 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
O Formulas from Linear Algebra 761
P Vector Analysis 763
Q Partial Fractions 765
R Finite Math 767
S Physics 769
T Probability 771
T.1 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
T.2 Playing the Odds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
U Economics 773
V Glossary 775
W whoami 777
xi

Differential Geometry. Honours 1996

2004-11-08T06:22:00.000-08:00

Differential Geometry. Honours 1996

Contents
Co-ordinate independent calculus.
Introduction
Smooth functions
Derivatives as linear operators.
The chain rule.
Diffeomorphisms and the inverse function theorem.
Differentiable manifolds
Co-ordinate charts
Linear manifolds.
Topology of a manifold
Smooth functions on a manifold.
The tangent space.
The derivative of a function.
Co-ordinate tangent vectors and one-forms.
How to calculate.
Submanifolds
Tangent space to a submanifold
Smooth functions between manifolds
The tangent to a smooth map.
Submanifolds again.
Vector fields.
The Lie bracket.
Differential forms.
The exterior algebra of a vector space.
Differential forms and the exterior derivative.
Pulling back differential forms
Integration of differential forms
Orientation.
Integration again
Stokes theorem.
Manifolds with boundary.
Stokes theorem.
Partitions of unity.
Vector fields and the tangent bundle.
Vector fields and derivations.
Tensor products
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