Asymptotic analysis and perturbation theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2014
|
| Series: | A Chapman & Hall book
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XXIII, 526 S. graph. Darst. |
| ISBN: | 9781466515116 |
Staff View
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Record in the Search Index
| _version_ | 1804150650091601920 |
|---|---|
| adam_text | Contents
List of Figures
ix
List of Tables
xiii
Preface
xv
Acknowledgments
xvii
About the Author
xix
Symbol Description
xxi
1
Introduction to Asymptotics
1
1.1
Ваьіс
Definitions
......................... 1
1.1.1
Definition of ~ and
<S
.................. 1
1.1.2
Hierarchy of Functions
.................. 4
1.1.3
Big
О
and Little
о
Notation
............... 6
1.2
Limits via
Asymptot
ics .....................
8
1.3
Asymptotic Series
........................ 13
1.4
Inverse Functions
........................ 22
1.4.1
Reversion of Series
.................... 20
1.5
Dominant Balance
........................ 30
2
Asymptotics of Integrals
37
2.1
Integrating Taylor Series
.................... 37
2.2
Repeated
Integrar
ion by Parts
................. 44
2.2.1
Optimal asymptotic approximation
........... 48
2.3
Laplace« Method
........................ 53
2.3.1
Properties of
Г(х)
.................... 59
2.3.2
Vatsoii s Lemma
.....................
til
2.4
Review of Complex Numbers
.................. 69
2.4.1
Analytic Functions
.................... 73
2.4.2
Contour Integration
................... 77
2.4.3
Gevrey Asymptotics
................... 80
2.4.4
Asymptotics for Oscillatory Functions
......... 84
2.5
Method (if Stationary Phase
.................. 90
2.(j Method of Steepest Descents
.................. 97
2.0.1
Saddle Points
....................... 101
vi
Contents
3
Speeding Up Convergence
3.1
Shanks Transformation
.....................
Ш
3.1.1
Generalized Shanks Transformation
.......... 114
3.2
R u
harcison
Extrapolation
.................... 117
3.2.1
G
ene
raliz
et
і
Richardson Extrapolation
.........
12U
3.3
Euler
Summation
........................ 1-4
3.4
Borei
Summation
........................ 130
3.4.1
Generalized
Borei
Summation
.............. 132
3.4.2
Stieltjes
Seiies
...................... 13
3.5
Continued Fractions
....................... 1-14
3.0
Padé
Approximatifs
.......................
154
3.6.1
Two-point
Padé .....................
158
4
Differential Equations
163
4.1
Classification of Differential Equations
............. 103
4.1.1
Linear vs. Non-Linear
..................
ICG
4.1.2
Homogeneous vs. Inlioinogeneous
............ 168
4.1.3
Initial Conditions vs. Boundary Conditions
...... 173
4.1.4
Regular
Singular Points vs. Irregular Singular Points
. 175
4.2
First Order Equations
...................... 181
4.2.1
Separable Equations
................... 181
4.2.2
First Order Linear Equations
.............. 184
4.3
Taylor Series Solutions
..................... 187
4.4
Frobenius Method
........................ 197
5
Asymptotic Series Solutions for Differential Equations
207
5.1
Behavior for Irregular Singular Points
.............
2U7
5.2
Full Asymptotic Expansion
................... 217
5.3
Local Analysis of Inhomogeneous Equations
......... 228
5.3.1
Variation of Parameters
................. 234
5.4
Local Analysis for Non-linear Equations
............ 243
6
Difference Equations
253
6.1
Classification of Difference Equations
............. 253
6.1.1
Anti-differences
...................... 256
6.1.2
Regular and Irregular Singular Points
......... 259
6.2
First Order Linear Equations
.................. 263
6.2.1
Solving General First Order Linear Equations
..... 265
6.2.2
The
Digamma
Function
................. 269
6.3
Analysis of Linear
Difiere
ace Equations
............ 274
6.3.1
Full Stirling Series
.................... 278
6.3.2
Taylor Series Solution
.................. 281
6.4
The
Euler-Maelaurin
Formula
................. 286
6.4.1
The Bernoulli Numbers
................. 289
6.4.2
Applications of the Euler-Maclaurin Formula
.....
2У4
Contents
vii
6.
Γ)
Taylor-like and Frobenius-like Series Expansions
....... 301
7
Perturbation Theory
317
7.1
Introduction to Perturbation Theory
............. 317
7.2
Regular Perturbation for Differential Equations
....... 326
7.3
Singular Perturbation for Differential Equations
....... 337
7.4
Asymptotic Matching
...................... 352
7.4.1
Van Dyke Method
....................
3G2
7.4.2
Dealing with Logarithmic Terms
............ 374
7.4.3
Multiple Boundary Layers
................ 38Ü
8
WKBJ Theory
389
8.1
The Exponential Approximation
................ 391
8.2
Region of Validity
........................ 403
8.3
Turning Points
.......................... 417
5.3.1 One Simple Root Turning Point Problem
....... 426
8.3.2
Parabolic Turning Point Problems
........... 428
8.3.3
The Two-turning Point
Schrödinger
Equation
..... 436
9
Multiple-Scale Analysis
443
9.1
Strained Coordinates Method
(Poincaré-Lindstedt)
..... 443
9.2
The Multiple-Scale Procedure
................. 457
9.3
Two-Variable Expansion Method
................ 465
Appendix—Guide to the Special Functions
479
Answers to Odd-Numbered Problems
495
Bibliography
519
Index
521
Mathematics
Asymptotic Analysis and
Perturbation Theory
Beneficial to both beginning students and researchers. Asymptotic
Analysis and Perturbation Theory immediateiy 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.
Features
►
Requires no prior knowledge of differential equations
•
Gives the necessary background on complex variables
►
Contains an abundance of graphs and tables that illustrate how
well the asymptotic approximations come near to the actual
solutions and how the approximation methods can be applied
to many types of problems
•
Presents informal proofs to enrich readers understanding of
the material
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|
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| institution | BVB |
| isbn | 9781466515116 |
| language | English |
| lccn | 2013019702 |
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| owner_facet | DE-703 DE-83 |
| physical | XXIII, 526 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | CRC Press |
| record_format | marc |
| series2 | A Chapman & Hall book |
| spelling | Paulsen, William Verfasser (DE-588)141936401 aut Asymptotic analysis and perturbation theory William Paulsen Boca Raton [u.a.] CRC Press 2014 XXIII, 526 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier A Chapman & Hall book Perturbation (Mathematics) Textbooks Differential equations Asymptotic theory Textbooks Asymptotik (DE-588)4126634-1 gnd rswk-swf Störungstheorie (DE-588)4128420-3 gnd rswk-swf Störungstheorie (DE-588)4128420-3 s Asymptotik (DE-588)4126634-1 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026190360&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026190360&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Paulsen, William Asymptotic analysis and perturbation theory Perturbation (Mathematics) Textbooks Differential equations Asymptotic theory Textbooks Asymptotik (DE-588)4126634-1 gnd Störungstheorie (DE-588)4128420-3 gnd |
| subject_GND | (DE-588)4126634-1 (DE-588)4128420-3 |
| title | Asymptotic analysis and perturbation theory |
| title_auth | Asymptotic analysis and perturbation theory |
| title_exact_search | Asymptotic analysis and perturbation theory |
| title_full | Asymptotic analysis and perturbation theory William Paulsen |
| title_fullStr | Asymptotic analysis and perturbation theory William Paulsen |
| title_full_unstemmed | Asymptotic analysis and perturbation theory William Paulsen |
| title_short | Asymptotic analysis and perturbation theory |
| title_sort | asymptotic analysis and perturbation theory |
| topic | Perturbation (Mathematics) Textbooks Differential equations Asymptotic theory Textbooks Asymptotik (DE-588)4126634-1 gnd Störungstheorie (DE-588)4128420-3 gnd |
| topic_facet | Perturbation (Mathematics) Textbooks Differential equations Asymptotic theory Textbooks Asymptotik Störungstheorie |
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