數學物理(第1卷)（簡體書）

如果說數學是大自然的語言，那麼，物理學就是大自然的詩歌。數學是物理學的出色工具，數學是物理學唯一能夠表達自己且不失真確性的語言。另外，數學在物理學中的應用也富有戲劇性的發展。
《數學物理(第1卷)》是為學習物理學的讀者編寫的數學基礎教材，不僅如此，《數學物理(第1卷)》還對那些學習數學的人們非常有益，即可以將抽象思維化為活龍活現的應用。現有的數學物理方法著作，通常是像詞典那樣將諸如矩陣對角化、張量分析、Legendre多項式和各種各樣的積分公式等匯集起來，很少強調主題的系統發展，因而缺乏生氣；《數學物理(第1卷)》則不同，作者試圖在形式和應用上、抽象化和具體問題上達到一種平衡，為了使內容編排最優化和自成一體，《數學物理(第1卷)》盡可能多地引入必要的數學形式，這自然包括大量的定量、性質、引理和推論的陳述和證明，以及豐富多樣的練習題。同時，作者希望通過學習《數學物理(第1卷)》，讀者能夠很清楚地發現在物理學中使用數學思想及方法的威力和局限性，這些都是許多物理學和數學教程中很少能明確指出的。《數學物理(第1卷)》的另一個突出特點是，除了用較現代的方法處理經典的數學物理問題外，還引入了很多有較強物理應用意義的較現代的數學方法和思想，從涵蓋的知識面來看，已遠遠超出通常數學物理方法教程的范圍，因此可以供更大范圍的讀者來參考選用。值得一提的是，《數學物理(第1卷)》還將一些近現代的著名科學家的小傳及照片穿插于全書各處，這使得《數學物理(第1卷)》生色不少。
《數學物理(第1卷)》的內容非常豐富，可作為物理系高年級學生、研究生和教師的參考書，也適合專門學習數學及對數學方法感興趣的讀者閱讀。

Preface
Note to the Reader
List of Symbols
0 Mathematical Preliminaries
0.1 Sets
0.2 Maps
0.3 Metric Spaces
0.4 Cardinality
0.5 Mathematical Induction
0.6 Problems

Ⅰ Finite-Dimensional Vector Spaces
1 Vectors and Transformations
1.1 Vector Spaces
1.2 Inner Product
1.3 Linear Transformations
1.4 Algebras
1.5 Problems
2 Operator Algebra
2.1 Algebra of L (V)
2.2 Derivatives of Functions of Operators
2.3Conjugation of Operators
2.4Hermitian and Unitary Operators
2.5Projection Operators
2.6Operators in Numerical Analysis
2.7Problems
3 Matrices: Operator Representations
3.1Matrices
3.2 Operations on Matrices
3.3Orthonormal Bases
3.4 Change of Basis and Similarity Transformation
3.5 The Determinant
3.6The Trace
3.7 Problems
4 Spectral Decomposition
4.1Direct Sums
4.2Invariant Subspaces
4.3Eigenvalues and Eigenvectors
4.4 Spectral Decomposition
4.5Functions of Operators
4.6 Polar Decomposition
4.7Real Vector Spaces
4.8Problems

Ⅱ Infinite-Dimensional Vector Spaces
5 Hilbert Spaces
5.1The Question of Convergence
5.2The Space of Square-Integrable Functions
5.3Problems
6 Generalized Functions
6.1Continuous Index
6.2 Generalized Functions
6.3Problems
7 Classical Orthogonal Polynomials
7.1General Properties
7.2 Classification
7.3Recurrence Relations
7.4 Examples of Classical Orthogonal Polynomials
7.5 Expansion in Terms of Orthogonal Polynomials
7.6 Generating Functions
7.7 Problems
8 F0urier Analysis
8.1 Fourier Series
8.2 The Fourier Transform
8.3 Problems

Ⅲ Complex Analysis
9 Complex Calculus
9.1 Complex Functions
9.2 Analytic Funcfions
9.3 Conformal Maps
9.4 Integration of Complex Functions
9.5 Derivafives as Integrals
9.6 Taylor and Laurent Series
9.7 Problems
10 Calculus of Residues
10.1 Residues
10.2 Class if I cation of I so lated Singularities
10.3 Evaluation ofDeftnite Integrals
10.4 Problems
11 Complex Analysis：A dvanced~opics
11.1 Meromorphic Functions
11.2 Multivalued Functions
11.3 Analytic Confinuation
11.4 The Gamma and Beta Functions
11.5 Method of Steepest Descent
11.6 Problems

Ⅳ Differential Equations
12 Separation of Variables in Spherical Coordinates
12.1 PDEs of Mathematical Physics
12.2 Separation of the Angular Part of the Laplacian
12.3 Construction of Eigenvalues of L2
12.4 Eigenvectors ofL2：Spherical Harmonics
12.5 Problems
13 Second-Order Linear Differential Equations
13.1 General Prope~ies of 0DEs
13.2 Existence and Uniqueness for First.Order DEs
13.3 General Properties Of SOLDEs
13.4 The Wronskian
13.5 Adioint Diflferential Operators
13.6 Power-Series Solutions of SOLDEs
13.7 SOLDEs with Constant Coefficients
13.8 The WKB Method
13.9 Numerical Solutions ofDEs
13.10 Problems
14 Complex Analysis Of SoLDEs
14.1 Analytic Prope~ies of Complex DEs
14.2 Complex SOLDEs
14.3 Fuchsian Difierential Equations
14.4 The HypergeometriC Function
14.5 Confluent Hypergeometric Functions
14.6 Problems
15 Integral Transforms and Differential Equations
15.1 Integral Representation of the Hypergeometric Function
15.2 Integral Representation of the Confluent Hypergeometrie Function
1 5.3 Integral Representation of Bessel Functions
15.4 Asymptotic Behavior Of Bessel Functions
15.5 Problems

Ⅴ Operators on Hilbert Spaces
16 An Introduction to Operator Theory
1 6.1 From Abstract to Integral and Differential 0perators
16.2 Bounded Opera torsin Hilbert Spaces
16.3 Spectra of Linear Operators
16.4 Compact Sets
16.5 Compact Operators
16.6 Spectrum of Compact Operators
16.7 Spectral Theorem for Compact Operators
16.8 Resolvents
16.9 Problems
17 Integral Equations
17.1 Classification

Ⅵ GreenS Functions
Ⅶ Groups and Manifolds
Ⅷ Lie Croups and Their Applications
Bibvliography
Index