Mathematics for the physical sciences 物理科学中的数学

This book offers advanced undergraduates and graduate students in physics, engineering, and other natural sciences a solid foundation in several fields of mathematics. Clear and well-written, it assumes a previous knowledge of the theory of functions of real and complex variables, and it is ideal for classroom use, self-study, or as a supplementary text. Topics include vector spaces and matrices; orthogonal functions; the roots of polynomial equations; asymptotic expansions; ordinary differential equations; conformal mapping; and extremum problems. Exercises appear at the end of each chapter, along with solutions at the back of the book. 1962 ed.

Chapter 1 Vector Spaces and Matrices
　1.1　Vector Spaces
　1.2　Schwarz's Inequality and Orthogonal Sets
　1.3　Linear Dependence and Independence
　1.4 Linear Operators on a Vector Space
　1.5　Eigenvalues and Hermitian Operators
　1.6　Unitary Operators
　1.7　Projection Operators
　1.8　Euclidean n-space and Matrices
　1.9　Matrix Algebra
　1.10 The Adjoint Matrix
　1.11 The Inverse Matrix
　1.12 Eigenvalues of Matrices
　1.13 Diagonalization of Matrices
　1.14 Functions of Matrices
　1.15 The Companion Matrix
　1.16 Bordering Hermitian Matrices
　1.17 Definite Matrices
　1.18 Rank and Nullity
　1.19 Simultaneous Diagonalization and Commutativity, 33
　1.20 The Numerical Calculation of Eigenvalues, 34
　1.21 Application to Differential Equations, 36
　1.22 Bounds for the Eigenvalues
　1.23 Matrices with Nonnegative Elements, 39
　Bibliography
　Exercises
Chapter 2 Orthogonai Functions
　2.1　Introduction
　2.2　Orthogonal Polynomials
　2.3　Zeros, 51
　2.4　The Recurrence Formula
　2.5　The Christoffel-Darboux Identity
　2.6　Modifying the Weight Function
　2.7　Rodrigues' Formula, 58
　2.8　Location of the Zeros, 59
　2.9　Gauss Quadrature, 61
　2.10 The Classical Polynomials
　2.11 Special Polynomials, 67
　2.12 The Convergence of Orthogonal Expansions
　2.13 Trigonometric Series, 72
　2.14 Fejer Summability, 75
　Bibliography, 79
　Exercises, 79
Chapter 3 The Roots of Polynomial Equations
　3.1　Introduction, 82
　3.2　The Gauss-Lucas Theorem
　3.3　Bounds for the Moduli of the Zeros
　3.4　Sturm Sequences, 90
　3.5　Zeros in a Half-Plane, 95
　3.6　Zeros in a Sector; Erdos-Turfin's Theorem
　3.7　Newton's Sums
　3.8　Other Numerical Methods
　Bibliography
　Exercises
Chapter 4 Asymptotic Expansions
　4.1　Introduction; the O symbols
　4.2　Sums
　4.3　Stirling's Formula
　4.4　Sums of Powers
　4.5　The Functional Equation ofs(s)
　4.6　The Method of Laplace for Integrals
　4.7　The Method of Stationary Phase
　4.8　Recurrence Relations
　Bibliography
　Exercises
Chapter 5 Ordinary Differential Epuateons
Chapter 6 Conformal Mapping
Chapter 7 Extremum Problems
Solutions of the Exercises
Boods Referrde to in the Text
Oringinl Works Cited in the Text
Index