| Description |
1 online resource |
| Contents |
1. Introduction to Asymptotics -- 1.1. Basic Definitions -- 1.1.1. Definition of ̃ and <<- 1.1.2. Hierarchy of Functions -- 1.1.3. Big O and Little o Notation -- 1.2. Limits via Asymptotics -- 1.3. Asymptotic Series -- 1.4. Inverse Functions -- 1.4.1. Reversion of Series 1.5. Dominant Balance -- 2. Asymptotics of Integrals -- 2.1. Integrating Taylor Series -- 2.2. Repeated Integration by Parts 2.2.1. Optimal asymptotic approximation -- 2.3. Laplace's Method -- 2.3.1. Properties of IK (x) -- 2.3.2. Watson's Lemma -- 2.4. Review of Complex Numbers -- 2.4.1. Analytic Functions -- 2.4.2. Contour Integration -- 2.4.3. Gevrey Asymptotics -- 2.4.4. Asymptotics for Oscillatory Functions -- 2.5. Method of Stationary Phase -- 2.6. Method of Steepest Descents -- 2.6.1. Saddle Points -- 3. Speeding Up Convergence -- 3.1. Shanks Transformation -- 3.1.1. Generalized Shanks Transformation 3.2. Richardson Extrapolation -- 3.2.1. Generalized Richardson Extrapolation -- 3.3. Euler Summation -- 3.4. Borel Summation -- 3.4.1. Generalized Borel Summation -- 3.4.2. Stieltjes Series -- 3.5. Continued Fractions -- 3.6. Pade Approximants -- 3.6.1. Two-point Pade -- 4. Differential Equations -- 4.1. Classification of Differential Equations -- 4.1.1. Linear vs. Non-Linear -- 4.1.2. Homogeneous vs. Inhomogeneous -- 4.1.3. Initial Conditions vs. Boundary Conditions -- 4.1.4. Regular Singular Points vs. Irregular Singular Points -- 4.2. First Order Equations -- 4.2.1. Separable Equations -- 4.2.2. First Line Lincar Equations -- 4.3. Taylor Series Solutions -- 4.4. Frobenius Method -- 5. Asymptotic Series Solutions for Differential Equations -- 5.1. Behavior for Irregular Singular Points 5.2. Full Asymptotic Expansion -- 5.3. Local Analysis of Inhomogeneous Equations -- 5.3.1. Variation of Parameters -- 5.4. Local Analysis for Non-linear Equations -- 6. Difference Equations -- 6.1. Classification of Difference Equations -- 6.1.1. Anti-differences -- 6.1.2. Regular and Irregular Singular Points 6.2. First Order Linear Equations -- 6.2.1. Solving General First Order Linear Equations -- 6.2.2. The Digamma Function -- 6.3. Analysis of Linear Differential Equations -- 6.3.1. Full Stirling Series -- 6.3.2. Taylor Series Solution -- 6.4. The Euler-Maclaurin Formula -- 6.4.1. The Bernoulli Numbers -- 6.4.2. Applications of the Euler-Maclaurin Formula -- 6.5. Taylor-like and Frobenius-like Series Expansions -- 7. Perturbation Theory -- 7.1. Introduction to Perturbation Theory -- 7.2. Regular Perturbation for Differential Equations -- 7.3. Singular Perturbation for Differential Equations -- 7.4. Asymptotic Matching -- 7.4.1. Van Dyke Method -- 7.4.2. Dealing with Logarithmic Terms -- 7.4.3. Multiple Boundary Layers -- 8. WKBJ Theory -- 8.1. The Exponential Approximation -- 8.2. Region of Validity -- 8.3. Turning Points -- 8.3.1. One Simple Root Turning Point Problem -- 8.3.2. Parabolic Turning Point Problems -- 8.3.3. The Two-Turn Point Schrodinger Equation -- 9. Multiple-Scale Analysis -- 9.1. Strained Coordinates Method (Poincare-Lindstedt) -- 9.2. The Multiple-Scale Procedure -- 9.3. Two-Variable Expansion Method |
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Front Cover; Contents; List of Figures; List of Tables; Preface; Acknowledgments; About the Author; Symbol Description; Chapter 1 Introduction to Asymptotics; Chapter 2 Asymptotics of Integrals; Chapter 3 Speeding Up Convergence; Chapter 4 Differential Equations; Chapter 5 Asymptotic Series Solutions for Differential Equation; Chapter 6 Difference Equations; Chapter 7 Perturbation Theory; Chapter 8 WKBJ Theory; Chapter 9 Multiple-Scale Analysis; Guide to the Special Functions; Answers to Odd-NumberedProblems; Bibliography; Back Cover |
| Summary |
Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. Suitable for those who have completed the standard calculus sequence, the book assumes no prior knowledge of differential equations. It explains the exact solution of only the simplest differential equations, such as first-order linear and separable equations. With varying levels of problems in each section, this self-contained text makes the difficult su |
| Notes |
"A CRC title." |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Perturbation (Mathematics) -- Textbooks
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Differential equations -- Asymptotic theory -- Textbooks
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MATHEMATICS -- Differential Equations -- General.
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Differential equations -- Asymptotic theory
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Perturbation (Mathematics)
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| Genre/Form |
Textbooks
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| Form |
Electronic book
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| ISBN |
1466515120 |
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9781466515123 |
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