Mathematical physics: a modern introduction to its foundations
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
New York [u.a.]
Springer
2002
|
| Edition: | corr. 3. print. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 1025 S. Ill., graph. Darst. |
| ISBN: | 0387985794 |
Staff View
MARC
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| 264 | 1 | |a New York [u.a.] |b Springer |c 2002 | |
| 300 | |a XXII, 1025 S. |b Ill., graph. Darst. | ||
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| 650 | 4 | |a Mathematische Physik - Lehrbuch | |
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Record in the Search Index
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| adam_text |
Contents
Preface v
Note to the Reader xi
List of Symbols xxi
0 Mathematical Preliminaries 1
0.1 Sets 1
0.2 Maps 4
0.3 Metric Spaces 7
0.4 Cardinality 10
0.5 Mathematical Induction 12
0.6 Problems 14
1 Finite Dimensional Vector Spaces 17
1 Vectors and Transformations 19
1.1 Vector Spaces 19
1.2 Inner Product 23
1.3 Linear Transformations 32
1.4 Algebras 41
1.5 Problems 44
2 Operator Algebra 49
2.1 Algebra of £(V) 49
xiv CONTENTS
2.2 Derivatives of Functions of Operators 56
2.3 Conjugation of Operators 61
2.4 Hermitian and Unitary Operators 63
2.5 Projection Operators 67
2.6 Operators in Numerical Analysis 70
2.7 Problems 76
3 Matrices: Operator Representations 82
3.1 Matrices 82
3.2 Operations on Matrices 87
3.3 Orthonormal Bases 89
3.4 Change of Basis and Similarity Transformation 91
3.5 The Determinant 93
3.6 The Trace 101
3.7 Problems 103
4 Spectral Decomposition 109
4.1 Direct Sums 109
4.2 Invariant Subspaces 112
4.3 Eigenvalues and Eigenvectors 114
4.4 Spectral Decomposition 117
4.5 Functions of Operators 125
4.6 Polar Decomposition 129
4.7 Real Vector Spaces 130
4.8 Problems 138
II Infinite Dimensional Vector Spaces 143
5 Hilbert Spaces 145
5.1 The Question of Convergence 145
5.2 The Space of Square Integrable Functions 150
5.3 Problems 157
6 Generalized Functions 159
6.1 Continuous Index 159
6.2 Generalized Functions 165
6.3 Problems 169
7 Classical Orthogonal Polynomials 172
7.1 General Properties 172
7.2 Classification 175
7.3 Recurrence Relations 176
7.4 Examples of Classical Orthogonal Polynomials 179
CONTENTS XV
7.5 Expansion in Terms of Orthogonal Polynomials 186
7.6 Generating Functions 190
7.7 Problems 190
8 Fourier Analysis 196
8.1 Fourier Series 196
8.2 The Fourier Transform 208
8.3 Problems 220
III Complex Analysis 225
9 Complex Calculus 227
9.1 Complex Functions 227
9.2 Analytic Functions 228
9.3 Conformal Maps 236
9.4 Integration of Complex Functions 241
9.5 Derivatives as Integrals 248
9.6 Taylor and Laurent Series 252
9.7 Problems 263
10 Calculus of Residues 270
10.1 Residues 270
10.2 Classification of Isolated Singularities 273
10.3 Evaluation of Definite Integrals 275
10.4 Problems 290
11 Complex Analysis: Advanced Topics 293
11.1 Meromorphic Functions 293
11.2 Multivalued Functions 295
11.3 Analytic Continuation 302
11.4 The Gamma and Beta Functions 309
11.5 Method of Steepest Descent 312
11.6 Problems 319
IV Differential Equations 325
12 Separation of Variables in Spherical Coordinates 327
12.1 PDEs of Mathematical Physics 327
12.2 Separation of the Angular Part of the Laplacian 331
12.3 Construction of Eigenvalues of L2 334
12.4 Eigenvectors of L2: Spherical Harmonics 338
12.5 Problems 346
xvi CONTENTS
13 Second Order Linear Differential Equations 348
13.1 General Properties of ODEs 349
13.2 Existence and Uniqueness for First Order DEs 350
13.3 General Properties of SOLDEs 352
13.4 TheWronskian 355
13.5 Adjoint Differential Operators 364
13.6 Power Series Solutions of SOLDEs 367
13.7 SOLDEs with Constant Coefficients 376
13.8 The WKB Method 380
13.9 Numerical Solutions of DEs 383
13.10 Problems 394
14 Complex Analysis of SOLDEs 400
14.1 Analytic Properties of Complex DEs 401
14.2 Complex SOLDEs 404
14.3 Fuchsian Differential Equations 410
14.4 The Hypergeometric Function 413
14.5 Confluent Hypergeometric Functions 419
14.6 Problems 426
15 Integral Transforms and Differential Equations 433
15.1 Integral Representation of the Hypergeometric Function . . . 434
15.2 Integral Representation of the Confluent Hypergeometric
Function 437
15.3 Integral Representation of Bessel Functions 438
15.4 Asymptotic Behavior of Bessel Functions 443
15.5 Problems 445
V Operators on Hilbert Spaces 449
16 An Introduction to Operator Theory 451
16.1 From Abstract to Integral and Differential Operators 451
16.2 Bounded Operators in Hilbert Spaces 453
16.3 Spectra of Linear Operators 457
16.4 Compact Sets 458
16.5 Compact Operators 464
16.6 Spectrum of Compact Operators 467
16.7 Spectral Theorem for Compact Operators 473
16.8 Resolvents 480
16.9 Problems 485
17 Integral Equations 488
17.1 Classification 488
CONTENTS XVii
17.2 Fredholm Integral Equations 494
17.3 Problems 505
18 Sturm Liouville Systems: Formalism 507
18.1 Unbounded Operators with Compact Resolvent 507
18.2 Sturm Liouville Systems and SOLDEs 513
18.3 Other Properties of Sturm Liouville Systems 517
18.4 Problems 522
19 Sturm Liouville Systems: Examples 524
19.1 Expansions in Terms of Eigenfunctions 524
19.2 Separation in Cartesian Coordinates 526
19.3 Separation in Cylindrical Coordinates 535
19.4 Separation in Spherical Coordinates 540
19.5 Problems 545
VI Green's Functions 551
20 Green's Functions in One Dimension 553
20.1 Calculation of Some Green's Functions 554
20.2 Formal Considerations 557
20.3 Green's Functions for SOLDOs 565
20.4 Eigenfunction Expansion of Green's Functions 577
20.5 Problems 580
21 Multidimensional Green's Functions: Formalism 583
21.1 Properties of Partial Differential Equations 584
21.2 Multidimensional GFs and Delta Functions 592
21.3 Formal Development 596
21.4 Integral Equations and GFs 600
21.5 Perturbation Theory 603
21.6 Problems 610
22 Multidimensional Green's Functions: Applications 613
22.1 Elliptic Equations 613
22.2 Parabolic Equations 621
22.3 Hyperbolic Equations 626
22.4 The Fourier Transform Technique 628
22.5 The Eigenfunction Expansion Technique 636
22.6 Problems 641
xviii CONTENTS
VII Groups and Manifolds 649
23 Group Theory 651
23.1 Groups 652
23.2 Subgroups 656
23.3 Group Action 663
23.4 The Symmetric Group Sn 664
23.5 Problems 669
24 Group Representation Theory 673
24.1 Definitions and Examples 673
24.2 Orthogonality Properties 680
24.3 Analysis of Representations 685
24.4 Group Algebra 687
24.5 Relationship of Characters to Those of a Subgroup 692
24.6 Irreducible Basis Functions 695
24.7 Tensor Product of Representations 699
24.8 Representations of the Symmetric Group 707
24.9 Problems 723
25 Algebra of Tensors 728
25.1 Multilinear Mappings 729
25.2 Symmetries of Tensors 736
25.3 Exterior Algebra 739
25.4 Inner Product Revisited 749
25.5 The Hodge Star Operator 756
25.6 Problems 758
26 Analysis of Tensors 763
26.1 Differentiable Manifolds 763
26.2 Curves and Tangent Vectors 770
26.3 Differential of a Map 776
26.4 Tensor Fields on Manifolds 780
26.5 Exterior Calculus 791
26.6 Symplectic Geometry 801
26.7 Problems 808
VIII Lie Groups and Their Applications 813
27 Lie Groups and Lie Algebras 815
27.1 Lie Groups and Their Algebras 815
27.2 An Outline of Lie Algebra Theory 833
27.3 Representation of Compact Lie Groups 845
CONTENTS xix
27.4 Representation of the General Linear Group 856
27.5 Representation of Lie Algebras 859
27.6 Problems 876
28 Differential Geometry 882
28.1 Vector Fields and Curvature 883
28.2 Riemannian Manifolds 887
28.3 Covariant Derivative and Geodesies 897
28.4 Isometries and Killing Vector Fields 908
28.5 Geodesic Deviation and Curvature 913
28.6 General Theory of Relativity 918
28.7 Problems 932
29 Lie Groups and Differential Equations 936
29.1 Symmetries of Algebraic Equations 936
29.2 Symmetry Groups of Differential Equations 941
29.3 The Central Theorems 951
29.4 Application to Some Known PDEs 956
29.5 Application to ODEs 964
29.6 Problems 970
30 Calculus of Variations, Symmetries, and Conservation Laws 973
30.1 The Calculus of Variations 973
30.2 Symmetry Groups of Variational Problems 988
30.3 Conservation Laws and Noether's Theorem 992
30.4 Application to Classical Field Theory 997
30.5 Problems 1000
Bibliography 1003
Index 1007 |
| any_adam_object | 1 |
| author | Hassani, Sadri |
| author_GND | (DE-588)114976838X |
| author_facet | Hassani, Sadri |
| author_role | aut |
| author_sort | Hassani, Sadri |
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| building | Verbundindex |
| bvnumber | BV014303802 |
| classification_rvk | SK 950 |
| ctrlnum | (OCoLC)248771951 (DE-599)BVBBV014303802 |
| dewey-full | 530.15 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.15 |
| dewey-search | 530.15 |
| dewey-sort | 3530.15 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| edition | corr. 3. print. |
| format | Book |
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| spelling | Hassani, Sadri Verfasser (DE-588)114976838X aut Mathematical physics a modern introduction to its foundations Sadri Hassani corr. 3. print. New York [u.a.] Springer 2002 XXII, 1025 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematische Physik - Lehrbuch Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009811442&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hassani, Sadri Mathematical physics a modern introduction to its foundations Mathematische Physik - Lehrbuch Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4037952-8 |
| title | Mathematical physics a modern introduction to its foundations |
| title_auth | Mathematical physics a modern introduction to its foundations |
| title_exact_search | Mathematical physics a modern introduction to its foundations |
| title_full | Mathematical physics a modern introduction to its foundations Sadri Hassani |
| title_fullStr | Mathematical physics a modern introduction to its foundations Sadri Hassani |
| title_full_unstemmed | Mathematical physics a modern introduction to its foundations Sadri Hassani |
| title_short | Mathematical physics |
| title_sort | mathematical physics a modern introduction to its foundations |
| title_sub | a modern introduction to its foundations |
| topic | Mathematische Physik - Lehrbuch Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Mathematische Physik - Lehrbuch Mathematische Physik |
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