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E-raamat: Asymptotic Analysis and Perturbation Theory

(Arkansas State University, Jonesboro, USA)
  • Formaat: 550 pages
  • Ilmumisaeg: 18-Jul-2013
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781040220009
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Asymptotic Analysis and Perturbation TheorySuurem pilt
  • Formaat: 550 pages
  • Ilmumisaeg: 18-Jul-2013
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781040220009
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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 subject of asymptotics easy to comprehend. Along the way, it explores the properties of some important functions in applied mathematics. Although the book emphasizes problem solving, some proofs are scattered throughout to give readers a justification for the methods used.

List of Figures
ix
List of Tables
xiii
Preface xv
Acknowledgments xvii
About the Author xix
Symbol Description xxi
1 Introduction to Asymptotics
1(36)
1.1 Basic Definitions
1(7)
1.1.1 Definition of ~ and <<
1(3)
1.1.2 Hierarchy of Functions
4(2)
1.1.3 Big O and Little o Notation
6(2)
1.2 Limits via Asymptotics
8(5)
1.3 Asymptotic Series
13(9)
1.4 Inverse Functions
22(8)
1.4.1 Reversion of Series
26(4)
1.5 Dominant Balance
30(7)
2 Asymptotics of Integrals
37(74)
2.1 Integrating Taylor Series
37(7)
2.2 Repeated Integration by Parts
44(9)
2.2.1 Optimal asymptotic approximation
48(5)
2.3 Laplace's Method
53(16)
2.3.1 Properties of IΓ(x)
59(2)
2.3.2 Watson's Lemma
61(8)
2.4 Review of Complex Numbers
69(21)
2.4.1 Analytic Functions
73(4)
2.4.2 Contour Integration
77(3)
2.4.3 Gevrey Asymptotics
80(4)
2.4.4 Asymptotics for Oscillatory Functions
84(6)
2.5 Method of Stationary Phase
90(7)
2.6 Method of Steepest Descents
97(14)
2.6.1 Saddle Points
101(10)
3 Speeding Up Convergence
111(52)
3.1 Shanks Transformation
111(6)
3.1.1 Generalized Shanks Transformation
114(3)
3.2 Richardson Extrapolation
117(7)
3.2.1 Generalized Richardson Extrapolation
120(4)
3.3 Euler Summation
124(6)
3.4 Borel Summation
130(14)
3.4.1 Generalized Borel Summation
132(5)
3.4.2 Stieltjes Series
137(7)
3.5 Continued Fractions
144(10)
3.6 Pade Approximants
154(9)
3.6.1 Two-point Pade
158(5)
4 Differential Equations
163(44)
4.1 Classification of Differential Equations
163(18)
4.1.1 Linear vs. Non-Linear
166(2)
4.1.2 Homogeneous vs. Inhomogeneous
168(5)
4.1.3 Initial Conditions vs. Boundary Conditions
173(2)
4.1.4 Regular Singular Points vs. Irregular Singular Points
175(6)
4.2 First Order Equations
181(6)
4.2.1 Separable Equations
181(3)
4.2.2 First Order Lincar Equations
184(3)
4.3 Taylor Series Solutions
187(10)
4.4 Frobenius Method
197(10)
5 Asymptotic Series Solutions for Differential Equations
207(46)
5.1 Behavior for Irregular Singular Points
207(10)
5.2 Full Asymptotic Expansion
217(11)
5.3 Local Analysis of Inhomogeneous Equations
228(15)
5.3.1 Variation of Parameters
234(9)
5.4 Local Analysis for Non-linear Equations
243(10)
6 Difference Equations
253(64)
6.1 Classification of Difference Equations
253(10)
6.1.1 Anti-differences
256(3)
6.1.2 Regular and Irregular Singular Points
259(4)
6.2 First Order Linear Equations
263(11)
6.2.1 Solving General First Order Linear Equations
265(4)
6.2.2 The Digamma Function
269(5)
6.3 Analysis of Linear Differcncc Equations
274(12)
6.3.1 Full Stirling Series
278(3)
6.3.2 Taylor Series Solution
281(5)
6.4 The Euler-Maclaurin Formula
286(15)
6.4.1 The Bernoulli Numbers
289(5)
6.4.2 Applications of the Euler-Maclaurin Formula
294(7)
6.5 Taylor-like and Frobenius-like Series Expansions
301(16)
7 Perturbation Theory
317(72)
7.1 Introduction to Perturbation Theory
317(9)
7.2 Regular Perturbation for Differential Equations
326(11)
7.3 Singular Perturbation for Differential Equations
337(15)
7.4 Asymptotic Matching
352(37)
7.4.1 Van Dyke Method
362(12)
7.4.2 Dealing with Logarithmic Terms
374(6)
7.4.3 Multiple Boundary Layers
380(9)
8 WKBJ Theory
389(54)
8.1 The Exponential Approximation
391(12)
8.2 Region of Validity
403(14)
8.3 Turning Points
417(26)
8.3.1 One Simple Root Turning Point Problem
426(2)
8.3.2 Parabolic Turning Point Problems
428(8)
8.3.3 The Two-turning Point Schrodinger Equation
436(7)
9 Multiple-Scale Analysis
443(36)
9.1 Strained Coordinates Method (Poincare-Lindstedt)
443(14)
9.2 The Multiple-Scale Procedure
457(8)
9.3 Two-Variable Expansion Method
465(14)
Appendix-Guide to the Special Functions 479(16)
Answers to Odd-Numbered Problems 495(24)
Bibliography 519(2)
Index 521
William Paulsen is a professor of mathematics at Arkansas State University, where he teaches asymptotics to undergraduate and graduate students. He is the author of Abstract Algebra: An Interactive Approach (CRC Press, 2009) and has published over 15 papers in applied mathematics, one of which proves that Penrose tiles can be three-colored, thus resolving a 30-year-old open problem posed by John H. Conway. Dr. Paulsen has also programmed several new games and puzzles in Javascript and C++, including Duelling Dimensions, which was syndicated through Knight Features. He received a Ph.D. in mathematics from Washington University in St. Louis.
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