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Graduate Mathematical Physics with Mathematica Supplements
This up-to-date textbook on mathematical methods of physics is designed for a one-semester graduate or two-semester advanced undergraduate course. The formal methods are supplemented by applications that use MATHEMATICA to perform both symbolic and numerical calculations.
The book is written by a physicist lecturer who
knows the difficulties involved in applying mathematics to real problems. As many as 40 exercises are included at the end of each chapter. A student CD includes a basic introduction to MATHEMATICA, notebook files for each chapter, and solutions to selected exercises.
Free solutions manual available for lecturers at college@wiley-vch.de.
Preface.
Note to the Reader.
1 Analytic Functions.
1.1 Complex Numbers.
1.2 Take Care with Multivalued Functions.
1.3 Functions as Mappings.
1.4 Elementary Functions and Their Inverses.
1.5 Sets, Curves, Regions and Domains.
1.6 Limits and Continuity.
1.7 Di.erentiability.
1.8 Properties of Analytic Functions.
1.9 Cauchy–Goursat Theorem.
1.10 Cauchy Integral Formula.
1.11 Complex Sequences and Series.
1.12 Derivatives and Taylor Series for Analytic Functions.
1.13 Laurent Series.
1.14 Meromorphic Functions.
Problems for Chapter 1.
2 Integration.
2.1 Introduction.
2.2 GoodTricks.
2.3 Contour Integration.
2.4 Isolated Singularities on the Contour.
2.5 Integration Around a Branch Point.
2.6 Reduction to Tabulated Integrals.
2.7 Integral Representations for Analytic Functions.
2.8 Using MATHEMATICA to Evaluate Integrals.
Problems for Chapter 2.
3 Asymptotic Series.
3.1 Introduction.
3.2 Method of Steepest Descent.
3.3 Partial Integration.
3.4 Expansion of an Integrand.
Problems for Chapter 3.
4 Generalized Functions.
4.1 Motivation.
4.2 Properties of the Dirac Delta Function.
4.3 Other Useful Generalized Functions.
4.4 Green Functions.
4.5 Multidimensional Delta Functions.
Problems for Chapter 4.
5 Integral Transforms.
5.1 Introduction.
5.2 FourierTransform.
5.3 Green Functions via Fourier Transform.
5.4 Cosine or Sine Transforms for Even or Odd Functions.
5.5 Discrete Fourier Transform.
5.6 LaplaceTransform.
5.7 Green Functions via Laplace Transform.
Problems for Chapter 5.
6 Analytic Continuation and Dispersion Relations.
6.1 Analytic Continuation.
6.2 Dispersion Relations.
6.3 Hilbert Transform.
6.4 Spreading of a Wave Packet.
6.5 Solitons.
Problems for Chapter 6.
7 Sturm–Liouville Theory.
7.1 Introduction: The General String Equation.
7.2 Hilbert Spaces.
7.3 Properties of Sturm–Liouville Systems.
7.4 Green Functions.
7.5 Perturbation Theory.
7.6 Variational Methods.
Problems for Chapter 7.
8 Legendre and Bessel Functions.
8.1 Introduction.
8.2 Legendre Functions.
8.3 Bessel Functions.
8.4 Fourier–BesselTransform.
8.5 Summary.
Problems for Chapter 8.
9 Boundary-Value Problems.
9.1 Introduction.
9.2 Green’s Theorem for Electrostatics.
9.3 Separable Coordinate Systems.
9.4 Spherical Expansion of Dirichlet Green Function for Poisson’s Equation.
9.5 Magnetic Field of Current Loop.
9.6 Inhomogeneous Wave Equation .
Problems for Chapter 9.
10 Group Theory.
10.1 Introduction.
10.2 Finite Groups.
10.3 Representations.
10.4 Continuous Groups.
10.5 Lie Algebra.
10.6 Orthogonality Relations for Lie Groups.
10.7 Quantum Mechanical Representations of the Rotation Group.
10.8 UnitarySymmetries inNuclear andParticlePhysics.
Problems for Chapter 10.
Bibliography 459
Index.
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