1. Infinite Series, Power Series.

The Geometric Series.

Definitions and Notation.

Applications of Series.

Convergent and Divergent Series.

Convergence Tests.

Convergence Tests for Series of Positive Terms.

Alternating Series.

Conditionally Convergent Series.

Useful Facts about Series.

Power Series; Interval of Convergence.

Theorems about Power Series.

Expanding Functions in Power Series.

Expansion Techniques.

Accuracy of Series Approximations.

Some Uses of Series.

2. Complex Numbers.

Introduction.

Real and Imaginary Parts of a Complex Number.

The Complex Plane.

Terminology and Notation.

Complex Algebra.

Complex Infinite Series.

Complex Power Series; Disk of Convergence.

Elementary Functions of Complex Numbers.

Euler’s Formula.

Powers and Roots of Complex Numbers.

The Exponential and Trigonometric Functions.

Hyperbolic Functions.

Logarithms.

Complex Roots and Powers.

Inverse Trigonometric and Hyperbolic Functions.

Some Applications.

3. Linear Algebra.

Introduction.

Matrices; Row Reduction.

Determinants; Cramer’s Rule.

Vectors.

Lines and Planes.

Matrix Operations.

Linear Combinations, Functions, Operators.

Linear Dependence and Independence.

Special Matrices and Formulas.

Linear Vector Spaces.

Eigenvalues and Eigenvectors.

Applications of Diagonalization.

A Brief Introduction to Groups.

General Vector Spaces.

4. Partial Differentiation.

Introduction and Notation.

Power Series in Two Variables.

Total Differentials.

Approximations using Differentials.

Chain Rule.

Implicit Differentiation.

More Chain Rule.

Maximum and Minimum Problems.

Constraints; Lagrange Multipliers.

Endpoint or Boundary Point Problems.

Change of Variables.

Differentiation of Integrals.

5. Multiple Integrals.

Introduction.

Double and Triple Integrals.

Applications of Integration.

Change of Variables in Integrals; Jacobians.

Surface Integrals.

6. Vector Analysis.

Introduction.

Applications of Vector Multiplication.

Triple Products.

Differentiation of Vectors.

Fields.

Directional Derivative; Gradient.

Some Other Expressions Involving V.

Line Integrals.

Green’s Theorems in the Plane.

The Divergence and the Divergence Theorem.

The Curl and Stokes’ Theorem.

7. Fourier Series and Transforms.

Introduction.

Simple Harmonic Motion and Wave Motion; Periodic Functions.

Applications of Fourier Series.

Average Value of a Function.

Fourier Coefficients.

Complex Form of Fourier Series.

Other Intervals.

Even and Odd Functions.

An Application to Sound.

Parseval’s Theorem.

Fourier Transforms.

8. Ordinary Differential Equations.

Introduction.

Separable Equations.

Linear First-Order Equations.

Other Methods for First-Order Equations.

Linear Equations (Zero Right-Hand Side).

Linear Equations (Nonzero Right-Hand Side).

Other Second-Order Equations.

The Laplace Transform.

Laplace Transform Solutions.

Convolution.

The Dirac Delta Function.

A Brief Introduction to Green’s Functions.

9. Calculus of Variations.

Introduction.

The Euler Equation.

Using the Euler Equation.

The Brachistochrone Problem; Cycloids.

Several Dependent Variables; Lagrange’s Equations.

Isoperimetric Problems.

Variational Notation.

10. Tensor Analysis.

Introduction.

Cartesian Tensors.

Tensor Notation and Operations.

Inertia Tensor.

Kronecker Delta and Levi-Civita Symbol.

Pseudovectors and Pseudotensors.

More about Applications.

Curvilinear Coordinates.

Vector Operators.

Non-Cartesian Tensors.

11. Special Functions.

Introduction.

The Factorial Function.

Gamma Function; Recursion Relation.

The Gamma Function of Negative Numbers.

Formulas Involving Gamma Functions.

Beta Functions.

Beta Functions in Terms of Gamma Functions.

The Simple Pendulum.

The Error Function.

Asymptotic Series.

Stirling’s Formula.

Elliptic Integrals and Functions.

12. Legendre, Bessel, Hermite, and Laguerre functions.

Introduction.

Legendre’s Equation.

Leibniz’ Rule for Differentiating Products.

Rodrigues’ Formula.

Generating Function for Legendre Polynomials.

Complete Sets of Orthogonal Functions.

Orthogonality of Legendre Polynomials.

Normalization of Legendre Polynomials.

Legendre Series.

The Associated Legendre Polynomials.

Generalized Power Series or the Method of Frobenius.

Bessel’s Equation.

The Second Solutions of Bessel’s Equation.

Graphs and Zeros of Bessel Functions.

Recursion Relations.

Differential Equations with Bessel Function Solutions.

Other Kinds of Bessel Functions.

The Lengthening Pendulum.

Orthogonality of Bessel Functions.

Approximate Formulas of Bessel Functions.

Series Solutions; Fuch’s Theorem.

Hermite and Laguerre Functions; Ladder Operators.

13. Partial Differential Equations.

Introduction.

Laplace’s Equation; Steady-State Temperature.

The Diffusion of Heat Flow Equation; the Schrodinger Equation.

The Wave Equation; the Vibrating String.

Steady-State Temperature in a Cylinder.

Vibration of a Circular Membrane.

Steady-State Temperature in a Sphere.

Poisson’s Equation.

Integral Transform Solutions of Partial Differential Equations.

14. Functions of a Complex Variable.

Introduction.

Analytic Functions.

Contour Integrals.

Laurent Series.

The Residue Theorem.

Methods of Finding Residues.

Evaluation of Definite Integrals.

The Point at Infinity; Residues of Infinity.

Mapping.

Some Applications of Conformal Mapping.

15. Probability and Statistics.

Introduction.

Sample Space.

Probability Theorems.

Methods of Counting.

Random Variables.

Continuous Distributions.

Binomial Distribution.

The Normal or Gaussian Distribution.

The Poisson Distribution.

Statistics and Experimental Measurements.

Miscellaneous Problems.

References.

Answers to Selected Problems.

Index.