科学家与工程师的数学方法/Mathematical methods for scientists and engineers
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作 者：Peter B. Kahn 著
分类号：
ISBN：9780486435169
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简介
Appropriate for advanced undergraduate and graduate students in a variety of scientific and engineering fields, this text introduces linear and nonlinear problems and their associated models. The first part covers linear systems, emphasizing perturbation or approximation techniques and asymptotic methods. The second part comprises nonlinear problems, including weakly nonlinear oscillatory systems and nonlinear difference equations. The two parts, both of which include exercises, merge smoothly, and many of the nonlinear techniques arise from the study of the linear systems. After covering preliminaries, the author examines some properties of matrices and introduces the gamma function. A discussion of related functions and the concept of asymptotic expansions and approximations follows, leading to an exposition on the Euler-MacLaurin sum expansion. The Laplace method of evaluating integrals dependent upon a large parameter is profiled in detail, and the first section of the text concludes with an exploration of the asymptotic behavior of second-order linear differential equations and an introduction to the related perturbation theory. The second part of the book opens with an extensive discussion of weakly nonlinear oscillating systems originating in the simple harmonic oscillator. Topics include aspects of harmonic motion and the concept of secular terms, equilibrium points and the phase plane, conservative and nonconservative systems, methods of averaging and of multiple times scales, higher-order calculations, error analysis, and one-dimensional iterative maps and the onset of chaos.
目录
PARTⅠ LINEAR SYSTEMS
0 Miscellaneous Resources
0.1 Abbreviations,
0.2 Numerical Results,
0.3 Trigonometric Identities and Differential Equations,
0.4 Taylor-Series Expansions,
0.5 Inequalities,
0.6 Graphs and Analytical Expressions,
Bibliography,
1 Matrix Theory
1.1 Notation and Preliminary Remarks,
1.2 Eigenvalues and Eigenvectors,
1.3 The Gram-Schmidt Procedure,
1.4 Diagonalization of Matrices,
1.5 The Cayley-Hamilton (CH) Theorem,
1.6 Perturbation Theory,
1.7 Concluding Remarks,
2 The Gamma and Related Functions
2.1 The Gamma Function,
2.2 Dirichlet Integrals,
2.3 Beta Functions,
2.4 Applications and Exercises,
2.5 The Riemann Zeta Function,
2.6 The Dirac Delta Function,
Bibliography,
3 Elements of Asymptotics
3.1 Introduction,
3.2 The Asymptotic Symbols,
3.3 The Error Function,
3.4 Sums,
3.5 Asymptotic Series,
Bibliography,
4 Evaluation of Sums: The Euler-MacLaurin Sum Expansion
……
PAPRT NONLINEAR SYSTEMS
Appendix A Disussion of Euler's Constant
Index
0 Miscellaneous Resources
0.1 Abbreviations,
0.2 Numerical Results,
0.3 Trigonometric Identities and Differential Equations,
0.4 Taylor-Series Expansions,
0.5 Inequalities,
0.6 Graphs and Analytical Expressions,
Bibliography,
1 Matrix Theory
1.1 Notation and Preliminary Remarks,
1.2 Eigenvalues and Eigenvectors,
1.3 The Gram-Schmidt Procedure,
1.4 Diagonalization of Matrices,
1.5 The Cayley-Hamilton (CH) Theorem,
1.6 Perturbation Theory,
1.7 Concluding Remarks,
2 The Gamma and Related Functions
2.1 The Gamma Function,
2.2 Dirichlet Integrals,
2.3 Beta Functions,
2.4 Applications and Exercises,
2.5 The Riemann Zeta Function,
2.6 The Dirac Delta Function,
Bibliography,
3 Elements of Asymptotics
3.1 Introduction,
3.2 The Asymptotic Symbols,
3.3 The Error Function,
3.4 Sums,
3.5 Asymptotic Series,
Bibliography,
4 Evaluation of Sums: The Euler-MacLaurin Sum Expansion
……
PAPRT NONLINEAR SYSTEMS
Appendix A Disussion of Euler's Constant
Index
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