Mathematics for physicists/
Philippe Dennery, Andre Krzywicki.
- New York: Dover, 1967.
- xiii, 384 p. : ill. ; 24 cm.
Preface; Chapter I The Theory of Analytic Functions; 1. Elementary Notions of Set Theory and Analysis; 1.1 Sets; 1.2 Some Notations of Set Theory; 1.3 Sets of Geometrical Points; 1.4 The Complex Plane; 1.5 Functions; 2. Functions of a Complex Argument; 3. The Differential Calculus of Functions of a Complex Variable; 4. The Cauchy-Riemann Conditions; 5. The Integral Calculus of Functions of a Complex Variable; 6. The Darboux Inequality; 7. Some Definitions; 8. Examples of Analytic Functions; 8.1 Polynomials; 8.2 Power Series. 8.3 Exponential and Related Functions9. Conformal Transformations; 9.1 Conformal Mapping; 9.2 Homographic Transformations; 9.3 Change of Integration Variable; 10. A Simple Application of Conformal Mapping; 11. The Cauchy Theorem; 12. Cauchy's Integral Representation; 13. The Derivatives of an Analytic Function; 14. Local Behavior of an Analytic Function; 15. The Cauchy-Liouville Theorem; 16. The Theorem of Morera; 17. Manipulations with Series of Analytic Functions; 18. The Taylor Series; 19. Poisson's Integral Representation; 20. The Laurent Series. 21. Zeros and Isolated Singular Points of Analytic Functions21.1 Zeros; 21.2 Isolated Singular Points; 22. The Calculus of Residues; 22.1 Theorem of Residues; 22.2 Evaluation of Integrals; 23. The Principal Value of an Integral; 24. Multivalued Functions; Riemann Surfaces; 24.1 Preliminaries; 24.2 The Logarithmic Function and Its Riemann Surface; 24.3 The Functions f(z) = z1/n and Their Riemann Surfaces; 24.4 The Function f(z) = (z2 -- 1)1/2 and Its Riemann Surface; 24.5 Concluding Remarks; 25. Example of the Evaluation of an Integral Involving a Multivalued Function. 26. Analytic Continuation27. The Schwarz Reflection Principle; 28. Dispersion Relations; 29. Meromorphic Functions; 29.1 The Mittag-Leffler Expansion; 29.2 A Theorem on Meromorphic Functions; 30. The Fundamental Theorem of Algebra; 31. The Method of Steepest Descent; Asymptotic Expansions; 32. The Gamma Function; 33. Functions of Several Complex Variables. Analytic Completion; Chapter II Linear Vector Spaces; 1. Introduction; 2. Definition of a Linear Vector Space; 3. The Scalar Product; 4. Dual Vectors and the Cauchy-Schwarz Inequality; 5. Real and Complex Vector Spaces; 6. Metric Spaces. 7. Linear Operators8. The Algebra of Linear Operators; 9. Some Special Operators; 10. Linear Independence of Vectors; 11. Eigenvalues and Eigenvectors; 11.1 Ordinary Eigenvectors; 11.2 Generalized Eigenvectors; 12. Orthogonalization Theorem; 13. N-Dimensional Vector Space; 13.1 Preliminaries; 13.2 Representations; 13.3 The Representation of a Linear Operator in an N-Dimensional Space; 14. Matrix Algebra; 15. The Inverse of a Matrix; 16. Change of Basis in an N-Dimensional Space; 17. Scalars and Tensors; 18. Orthogonal Bases and Some Special Matrices; 19. Introduction to Tensor Calculus.