Wilf H. Mathematics for the Physical Sciences 2006

Textbook in PDF format

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.
Vector Spaces and Matrices
Vector Spaces
Schwarz Inequality and Orthogonal Sets
Linear Dependence and Independence
Linear Operators on a Vector Space
Eigenvalues and Hermitian Operators
Unitary Operators
Projection Operators
Euclidean n-space and Matrices
Matrix Algebra
The Adjoint Matrix
The Inverse Matrix
Eigenvalues of Matrices
Diagonalization of Matrice
Functions of Matrices
The Companion Matrix
Bordering Hermitian Matrices
Definite Matrices
Rank and Nullity
Simultaneous Diagonalization and Commutativity
The Numerical Calculation of Eigenvalues
Application to Differential Equations
Bounds for the Eigenvalues
Matrices with Nonnegative Elements
Exercises
Orthogonal Functions
Orthogonal Polynomials
Zeros
The Recurrence Formula
The Christoffel-Darboux Identity
Modifying the Weight Function
Rodrigues' Formula
Location of the Zeros
Gauss Quadrature
The Classical Polynomials
Special Polynomials
The Convergence of Orthogonal Expansions
Trigonometric Series
Fejer Summa
Exercises
The Roots of Polynomial Equations
The Gauss-Lucas Theorem
Bounds for the Moduli of the Zeros
Sturm Sequences
Zeros in a Half-Plane
Zeros in a Sector; Erdos-Turan's Theorem
Newton's Sums, 100
Other Numerical Methods
Exercises
Asymptotic Expansions
Introduction; the 0, 0, ~ symbols
Sums
Stirling's Formula
Sums of Powers
The Functional Equation of
The Method of Laplace for Integrals
The Method of Stationary Phase
Recurrence Relations
Exercises
Ordinary Differential Equations
Equations of the First Order
Picard's Theorem
Remarks on Picard's Theorem; Wintner's Method
Numerical Solution of Differential Equations
Truncation Error
Predictor-Corrector Formulas
Stability
Linear Equations of the Second Order
Solution Near a Regular Point
Convergence of the Formal Solution
A Second Solution in the Exceptional Cas
The Gamma Function
Bessel Functions
Exercises
Conformal Mapping
Conformal Mapping
Univalent Functions
Families of Functions Regular on a Domain
The Riemann Mapping Theorem
A Constructive Approach
The Schwarz-Christoffel Mapping
Applications of Conformal Mapping
Analytic and Geometric Function Theory
Exercises
Extremum Problems
Functions of Real Variables
The Method of Lagrange Multipliers,
The First Problem of the Calculus of Variations
Some Examples
Distinguishing Maxima from Minima
Problems with Side Conditions
Several Unknown Functions or Independent Variables
The Variational Notation
The Maximization of Linear Functions with Co
On Best Approximation by Polynomials
Exercises
Solutions of the Exercises
Books Referred to in the Text