Hassani-Mathematical Physics-a modern intro

书籍推荐 · 发表于 2018-7-13 21:48:22

Preface
Note to the Reader
List of Symbols
o 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 VectorSpaces
1 Vectors and Transformations
1.1 VectorSpaces . . . . .
1.2 Inner Product .
1.3 Linear Transformations .
1.4 Algebras.
1.5 Problems..........
2 Operator Algebra
2.1 Algebra «sV)
2.2 Derivatives of Functions of Operators .
2.3 Conjugationof Operators . . . .
2.4 Hermitian and Unitary Operators
2.5 Projection Operators . . . . . . .
2.6 Operatorsin Numerical Analysis
2.7 Problems.............
3 Matrices: Operator Representations
3.1 Matrices..........
3.2 Operations on Matrices .. . . . .
3.3 OrthonormalBases .
3.4 Change of Basis and Similarity Transformation .
3.5 The Determinant . . ..
3.6 The Trace
3.7 Problems......
4 Spectral Decomposition
4.1 Direct Sums . . . . . . . . . .
4.2 InvariantSubspaces . . . . . .
4.3 Eigeuvaluesand Eigenvectors .
4.4 Spectral Decomposition
4.5 Functions of Operators
4.6 Polar Decomposition
4.7 Real VectorSpaces
4.8 Problems.......
IT Infinite-Dimensional Vector Spaces
5 Hilbert Spaces
5.1 The Question of Convergence .
5.2 The Space of Square-IntegrableFunctions
5.3 Problems...................
6 Generalized Functions
6.1 ContinuousIndex
6.2 Generalized Functions .
6.3 Problems........
7 Classical Orthogonal Polynomials
7.1 General Properties . .
7.2 Classification.........
7.3 RecurrenceRelations ....
7.4 Examples of Classical OrthogonalPolynomials.
Expansion in Terms of Orthogonal Polynomials ..
Generating Functions . . . . . . . . . . . .
Problems .
8 Fourier Analysis
8.1 Fourier Series .....
8.2 The Fourier Transform .
8.3 Problems........
III Complex Analysis
9 Complex Calculus 227
9.1 Complex Functions " 227
9.2 Analytic Functions. . . . . . . " 228
9.3 ConformalMaps .. . . . . . . . . . . . 236
9A Integration of Complex Functions . . . . . . . . . . . . . " 241
9.5 Derivativesas Integrals 248
9.6 Taylor and Laurent Series . 252
9.7 Problems............................ 263
10 Calculus of Residues
10.1 Residues . . . . . . . . . . . . . . . .
10.2 Classificationof Isolated Singularities
10.3 Evaluation of Definite Integrals . . ..
lOA Problems .
11 Complex Analysis: Advanced Topics
ILl Meromorphic Functions . . .
11.2 MultivaluedFunctions . . . . . . . .
11.3 Analytic Continuation . . . . . . . . .
1104 The Gamma and Beta Functions. . . .
11.5 Method of Steepest Descent . . . . . .
11.6 Problems. . . . . . . . . . . . . . . .
IV DifferentialEquations 325
12 Separation of Variables in Spherical Coordinates 327
12.1 PDEs of Mathematical Physics .. . . . . . . . . . . . . . . 327
12.2 Separation of theAngular Part of the Laplacian. . . . . . .. 331
12.3 Construction of Eigenvalues of L2. . . . . . . . . . . 334
1204 Eigenvectors of L2: Spherical Harmonics . . 338
12.5 Problems.. . . . . . . . . . . . . . ., . . . . . . . 346……