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E-grāmata: Guide To Mathematical Methods For Physicists, A: With Problems And Solutions

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(Sorbonne Univ, Paris, France), (Univ Of Rome Tor Vergata, Italy), (Univ Of Milano-bicocca, Italy)
  • Formāts: 340 pages
  • Sērija : Essential Textbooks in Physics
  • Izdošanas datums: 07-Jul-2017
  • Izdevniecība: World Scientific Europe Ltd
  • Valoda: eng
  • ISBN-13: 9781786343468
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Guide To Mathematical Methods For Physicists, A: With Problems And SolutionsPalielināt
  • Formāts: 340 pages
  • Sērija : Essential Textbooks in Physics
  • Izdošanas datums: 07-Jul-2017
  • Izdevniecība: World Scientific Europe Ltd
  • Valoda: eng
  • ISBN-13: 9781786343468
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Mathematics plays a fundamental role in the formulation of physical theories. This textbook provides a self-contained and rigorous presentation of the main mathematical tools needed in many fields of Physics, both classical and quantum. It covers topics treated in mathematics courses for final-year undergraduate and graduate physics programmes, including complex function: distributions, Fourier analysis, linear operators, Hilbert spaces and eigenvalue problems. The different topics are organised into two main parts complex analysis and vector spaces in order to stress how seemingly different mathematical tools, for instance the Fourier transform, eigenvalue problems or special functions, are all deeply interconnected. Also contained within each chapter are fully worked examples, problems and detailed solutions. A companion volume covering more advanced topics that enlarge and deepen those treated here is also available.
Preface v
PART I Complex Analysis
1(110)
1 Holomorphic Functions
3(30)
1.1 Complex Functions
3(2)
1.2 Holomorphic Functions
5(6)
1.3 Singularities of Holomorphic Functions
11(10)
1.4 The Riemann Sphere and the Point at Infinity
21(4)
1.5 Elementary Functions
25(5)
1.6 Exercises
30(3)
2 Integration
33(28)
2.1 Curves in the Complex Plane
33(4)
2.2 Line Integral of a Function Along a Curve
37(4)
2.3 Cauchy's Theorem
41(4)
2.4 Primitive of a Holomorphic Function
45(3)
2.5 Holomorphic Functions, Differential Forms and Vector Fields
48(2)
2.6 Cauchy's Integral Formula
50(2)
2.7 Morera's Theorem for a Simply Connected Domain
52(1)
2.8 Other Properties of Holomorphic Functions
53(3)
2.9 Harmonic Functions
56(2)
2.10 Exercises
58(3)
3 Taylor and Laurent Series
61(26)
3.1 Power Series
61(3)
3.2 Taylor Series
64(5)
3.3 Laurent Series
69(6)
3.4 Analytic Continuation
75(10)
3.5 Exercises
85(2)
4 Residues
87(24)
4.1 Residue of a Function at an Isolated Singularity
87(3)
4.2 Residue Theorem
90(4)
4.3 Evaluation of Integrals by Residue Method
94(10)
4.4 Cauchy's Principal Value
104(3)
4.5 Exercises
107(4)
PART II Functional Spaces
111(146)
5 Vector Spaces
113(24)
5.1 Vector Spaces
113(3)
5.2 Metric, Norm and Scalar Product
116(8)
5.3 Complete Spaces
124(1)
5.4 Finite-Dimensional Hilbert Spaces
125(3)
5.5 Hilbert Spaces
128(4)
5.6 The Orthogonal Complement
132(2)
5.7 Exercises
134(3)
6 Spaces of Functions
137(16)
6.1 Different Norms and Different Notions of Convergence
138(2)
6.2 The Space L1ω(Omega;)
140(4)
6.3 The Hilbert Space L2ω(Omega;)
144(2)
6.4 Hilbert Basis and Fourier Expansion
146(5)
6.5 Exercises
151(2)
7 Distributions
153(22)
7.1 Test Functions
153(3)
7.2 Distributions
156(7)
7.3 Limits of Distributions
163(2)
7.4 Operations on Distributions
165(8)
7.5 Exercises
173(2)
8 Fourier Analysis
175(26)
8.1 Fourier Series
176(8)
8.2 The Fourier Transform
184(13)
8.3 Exercises
197(4)
9 Linear Operators in Hilbert Spaces I: The Finite-Dimensional Case
201(22)
9.1 Linear Operators in Finite Dimension
201(6)
9.2 Spectral Theory
207(13)
9.3 Exercises
220(3)
10 Linear Operators in Hilbert Spaces II: The Infinite-Dimensional Case
223(34)
10.1 Operators in Normed Spaces
224(5)
10.2 Operators in Hilbert Spaces
229(8)
10.3 Eigenvalues and Spectral Theory
237(16)
10.4 Exercises
253(4)
PART III Appendices
257(2)
Appendix A Complex Numbers, Series and Integrals
259(10)
A.1 A Quick Review of Complex Numbers
259(2)
A.2 Notions of Topology, Sequences and Series
261(3)
A.3 The Lebesgue Integral
264(5)
Appendix B Solutions of the Exercises
269(52)
Bibliography 321(2)
Index 323
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