"Mathematics for the Physical Sciences" by Herbert S. Wilf

Advanced undergraduates and graduate students in the natural sciences receive a solid foundation in several fields of mathematics with this text. Topics include vector spaces and matrices; orthogonal functions; polynomial equations; asymptotic expansions; ordinary differential equations; conformal mapping; and extremum problems. Includes exercises and solutions.

Contents
Chapter 1 Vector Spaces and Matrices
1.1 Vector Spaces
1.2 Schwarz 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 Matrice
1.14 Functions of Matrices
1.15 The Companion Matrix, 25
1.16 Bordering Hermitian Matrices
1.17 Definite Matrices
1.18 Rank and Nullity
1.19 Simultaneous Diagonalization and Commutativity
1.20 The Numerical Calculation of Eigenvalues
1.21 Application to Differential Equations
1.22 Bounds for the Eigenvalues
1.23 Matrices with Nonnegative Elements
Bibliography
Exercises
Chapter 2 Orthogonal Functions
2.1 Introduction
2.2 Orthogonal Polynomials
2.3 Zeros
2.4 The Recurrence Formula
2.5 The Christoffel-Darboux Identity
2.6 Modifying the Weight Function
2.7 Rodrigues' Formula
2.8 Location of the Zeros
2.9 Gauss Quadrature
2.10 The Classical Polynomials
2.11 Special Polynomials
2.12 The Convergence of Orthogonal Expansions
2.13 Trigonometric Series
2.14 Fejer Summa
Exercises
Chapter 3 The Roots of Polynomial Equations
3.1 Introduction
3.2 The Gauss-Lucas Theorem
3.3 Bounds for the Moduli of the Zeros
3.4 Sturm Sequences
3.5 Zeros in a Half-Plane
3.6 Zeros in a Sector; Erdos-Turan's Theorem
3.7 Newton's Sums, 100
3.8 Other Numerical Methods
Bibliography
Exercises
Chapter 4 Asymptotic Expansions
4.1 Introduction; the 0, 0, ~ symbols
4.2 Sums
4.3 Stirling's Formula
4.4 Sums of Powers
4.5 The Functional Equation of
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 Equations
5.1 Introduction
5.2 Equations of the First Order
5.3 Picard's Theorem
5.4 Remarks on Picard's Theorem; Wintner's Method
5.5 Numerical Solution of Differential Equations
5.6 Truncation Error
5.7 Predictor-Corrector Formulas
5.8 Stability
5.9 Linear Equations of the Second Order
5.10 Solution Near a Regular Point
5.11 Convergence of the Formal Solution
5.12 A Second Solution in the Exceptional Cas
5.13 The Gamma Function
5.14 Bessel Functions
Bibliography
Exercises
Chapter 6 Conformal Mapping
6.1 Introduction
6.2 Conformal Mapping
6.3 Univalent Functions
6.4 Families of Functions Regular on a Domain
6.5 The Riemann Mapping Theorem
6.6 A Constructive Approach
6.7 The Schwarz-Christoffel Mapping
6.8 Applications of Conformal Mapping
6.9 Analytic and Geometric Function Theory
Bibliography
Exercises
Chapter 7 Extremum Problems
7.1 Introduction
7.2 Functions of Real Variables
7.3 The Method of Lagrange Multipliers,
7.4 The First Problem of the Calculus of Variations
7.5 Some Examples
7.6 Distinguishing Maxima from Minima
7.7 Problems with Side Conditions
7.8 Several Unknown Functions or Independent Variables
7.9 The Variational Notation
7.10 The Maximization of Linear Functions with Co
7.12 On Best Approximation by Polynomials
Bibliography
Exercises
Solutions of the Exercises
Books Referred to in the Text
Index