Lectures on applications-oriented mathematics
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| Format: | Buch |
| Sprache: | English |
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San Francisco, Calif.
Holden-Day
1969
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Datensatz im Suchindex
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| adam_text | TABLE OF CONTENTS
INTRODUCTION by Victor Twersky
CHAPTER 1. DISTRIBUTIONS 1
1.1 DELTA FUNCTIONS AND DISTRIBUTIONS 1
Introduction 1
Symbolic Functions 2
Change of Variables 4
Derivatives of Distributions 6
1.2 APPLICATIONS OF DISTRIBUTIONS 8
Integration of Distributions 8
Limits of Symbolic Functions 11
Symbolic Functions in More than One Dimension 13
Illustration from Electrostatics 15
Equation of Continuity for Electric Charge 16
1.3 DISTRIBUTIONS AND TRANSFORMS 17
An Integral Representation for 6(x) 17
The Fourier Integral 20
Finite Parts of Integrals 23
Operational Methods and Laplace Transform 29
Solution of the Heat Equation 32
Fourier Series 35
CHAPTER 2. SPECTRAL THEORY OF OPERATORS 41
2.1 LINEAR VECTOR SPACES 41
Introduction 41
Examples 42
Function Spaces 43
Event Spaces 44
Inner Product 45
Schwarz Inequality; Projection 48
Applications to Probability 50
2.2 LINEAR OPERATORS 53
Adjoint Operator 54
Quantum Mechanics 56
Uncertainty Principle 57
Spectral Representation 58
Functions of an Operator 62
Continuous Spectrum 63
Partial Differential Equations 65
CHAPTER 3. ASYMPTOTIC METHODS 68
INTRODUCTION 68
3.1 ASYMPTOTIC SERIES 68
Definition of Asymptotic Expansion 70
ix
x Table of Contents
Operations with Asymptotic Series 73
Integration by Parts 74
Improving an Expansion 76
3.2 WATSON S LEMMA 77
Stirling s Formula 81
Laplace s Method 84
3.3 METHODS OF STEEPEST DESCENT 86
3.4 METHOD OF STATIONARY PHASE 91
Two Dimensions 97
Finite Limits 98
3.5 AIRY INTEGRAL 99
Contour Integration Procedures 101
Pole Near Stationary Point 108
CHAPTER 4. DIFFERENCE EQUATIONS 110
4.1 RECURRENCE RELATIONS AND DIFFERENCE EQUATIONS ... 110
Introduction 110
Finite Difference Operators Ill
Connection between Differences and Derivatives 113
Inverse of a Difference Operator 115
Simple Recurrence Relations 115
4.2 DIFFERENCE EQUATIONS OF ORDER HIGHER THAN ONE .... 119
Nonhomogeneous Difference Equations 122
4.3 THE METHOD OF THE GENERATING FUNCTION 124
Linear Difference Equations with Nonconstant Coefficients 126
Method of Integral Transform 127
Linear First Order Equations 128
4.4 SUMMATION BY PARTS 129
Generalization of Summation by Parts 131
Euler Maclaurin Formula 132
4.5 SPECIAL METHODS 133
Poisson Process 136
CHAPTER 5. COMPLEX INTEGRATION 138
INTRODUCTION 138
5.1 ANALYTIC FUNCTIONS 138
Singularities 139
Residues 143
Full Period Integrals of Trigonometric Functions 144
Cauchy Principal Value 148
5.2 INTEGRALS OF RATIONAL FUNCTIONS 149
Integrals of Exponentials Multiplied by Rational Functions 151
Evaluation of Infinite Series 152
5.3 INTEGRALS INVOLVING BRANCH POINTS 155
5.4 CONTOUR INTEGRAL REPRESENTATIONS OF
SOLUTIONS TO THE WAVE EQUATION 164
Table of Contents xi
CHAPTER 6. SYMBOLIC METHODS 170
INTRODUCTION 170
6.1 DIFFERENTIAL OPERATORS 170
6.2 DIFFERENCE OPERATORS 176
6.3 LAPLACE TRANSFORMS 178
6.4 SYMBOLIC METHODS AND GENERATING FUNCTIONS 188
6.5 NONCOMMUTATIVE OPERATORS 196
CHAPTER 7. PROBABILITY 206
7.1 INTRODUCTION 206
The Algebra of Sets 207
Finite Sample Space 208
7.2 ELEMENTARY PROPERTIES OF PROBABILITY 209
Method of Assigning Probability 210
Expected Values : . 210
Indicator Functions 212
Dependent Events 214
Independent Events 214
Product Spaces 216
7.3 INFINITE SAMPLE SPACES 218
Counting 219
Measure 222
Integration 224
CHAPTER 8. PERTURBATION THEORY 231
8.1 REGULAR PERTURBATION THEORY 231
8.2 THE FREDHOLM EXPANSION 236
Perturbation of Eigenvalues 241
8.3 EXAMPLE OF SINGULAR PERTURBATION PROBLEM 243
Boundary Layer Effect 246
Correction to Eigenvalues 247
8.4 SINGULAR PERTURBATION PROCEDURE 249
BIBLIOGRAPHY 257
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| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-12-23T11:42:10Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003554609 |
| oclc_num | 441420092 |
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| physical | XI,257 S.m.Abb. |
| publishDate | 1969 |
| publishDateSearch | 1969 |
| publishDateSort | 1969 |
| publisher | Holden-Day |
| record_format | marc |
| spellingShingle | Friedman, Bernard Lectures on applications-oriented mathematics Angewandte Mathematik (DE-588)4142443-8 gnd Mathematik (DE-588)4037944-9 gnd Naturwissenschaftler (DE-588)4041423-1 gnd |
| subject_GND | (DE-588)4142443-8 (DE-588)4037944-9 (DE-588)4041423-1 |
| title | Lectures on applications-oriented mathematics |
| title_auth | Lectures on applications-oriented mathematics |
| title_exact_search | Lectures on applications-oriented mathematics |
| title_full | Lectures on applications-oriented mathematics |
| title_fullStr | Lectures on applications-oriented mathematics |
| title_full_unstemmed | Lectures on applications-oriented mathematics |
| title_short | Lectures on applications-oriented mathematics |
| title_sort | lectures on applications oriented mathematics |
| topic | Angewandte Mathematik (DE-588)4142443-8 gnd Mathematik (DE-588)4037944-9 gnd Naturwissenschaftler (DE-588)4041423-1 gnd |
| topic_facet | Angewandte Mathematik Mathematik Naturwissenschaftler |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003554609&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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