Fourier分析及其应用
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作者(美)杰拉尔德·弗兰德
出版社高等教育出版社
ISBN9787040535969
出版时间2020-04
装帧精装
开本16开
定价169元
货号1202044332
上书时间2024-04-15
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目录
1 Overture
1.1 Some equations of mathematical physics
1.2 Linear differential operators
1.3 Separation of variables
2 Fourier Series
2.1 The Fourier series of a periodic function
2.2 A convergence theorem
2.3 Derivatives, integrals, and uniform convergence
2.4 Fourier series on intervals
2.5 Some applications
2.6 Further remarks on Fourier series
3 Orthogonal Sets of Functions
3.1 Vectors and inner products
3.2 Functions and inner products
3.3 Convergence and completeness
3.4 More about L2 spaces; the dominated convergence theorem
3.5 Regular Sturm-LiouviUe problems
3.6 Singular Sturm-Liouville problems
4 Some Boundary Value Problems
4.1 Some useful techniques
4.2 One-dimensional heat flow
4.3 One-dimensional wave motion
4.4 The Dirichlct problem
4.5 Multiple Fourier series and applications
5 Bessel Functions
5.1 Solutions of Bessels equation
5.2 Bessel function identities
5.3 Asymptotics and zeros of Bessel functions
5.4 Orthogonal sets of Bessel functions
5.5 Applications of Bessel functions
5.6 Variants of Bessel functions
6 Orthogonal Polynomials
6.1 Introduction
6.2 Legendrepolynomials
6.3 Spherical coordinates and Legendre functions
6.4 Hermite polynomials
6.5 Laguerre polynomials
6.6 Other orthogonal bases
7 The Fourier Transform
7.1 Convolutions
7.2 The Fourier transform
7.3 Some applications
7.4 Fourier transforms and Sturm-LiouviUe problems
7.5 Multivariable convolutions and Fourier transforms
7.6 Transforms related to the Fourier transform
8 The Laplace Transform
8.1 The Laplace transform
8.2 The inversion formula
8.3 Applications: Ordinary differential equations
8.4 Applications: Partial differential equations
8.5 Applications: Integral equations
8.6 Asymptotics of Laplace transforms
9 Generalized Functions
9.1 Distributions
9.2 Convergence, convolution, and approximation
9.3 More examples: Periodic distributions and finite parts
9.4 Tempered distributions and Fourier transforms
9.5 Weak solutions of differential equations
10 Greens Functions
10.1 Greens functions for ordinary differential operators
10.2 Greens functions for partial differential operators
10.3 Greens functions and regular Sturm-Liouville problems
10.4 Greens functions and singular Sturm-Liouville problems
Appendices
1 Some physical derivations
2 Summary of complex variable theory
3 The gamma function
4 Calculations in polar coordinates
5 The fundamental theorem of ordinary differential equations
Answers to the Exercises
References
Index of Symbols
Index
内容摘要
本书在适合于高年级本科生学习的水平上讲述Fourier级数和Fourier积分、特征函数展开以及相关论题的理论和应用，内容涵盖Bessel函数、正交多项式和Laplace变换，还包括了对于常微分方程和偏微分方程的广义函数和Green函数的一些章节。本书几乎专门处理这些主题的可用于物理和工程中的那些方面，同时也包含了广泛的各类应用。在理论方面，它使用现代分析中的思想来展开技术背后的概念和推理，而叉没有陷入严格证明的技术细节的陷阱中。
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