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Chapter 1. Fourier series

1.1 Basic concepts

1.2 Fourier series and Fourier coefficients

1.3 A mimimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem

1.4 Convergence of Fourier series

1.5 The Parseval formula

1.6 Determination of the sum of certain trigonemetric series

Chapter 2. Orthogonal systems

2.1 Integration of complex-valued functions of a real variable

2.2 Orthogonal systems

2.3 Complete orthogonal systems

2.4 Integration of Fourier series

2.5 The Gram-Schmidt orthogonalization process

2.6 Sturm-Liouville problems

Chapter 3. Orthogonal polynomials

3.1 The Legendre polynomials

3.2 Legendre series

3.3 The Legendre differential equation. The generating function of the Legendre polynomials

3.4 The Tchebycheff polynomials

3.5 Tchebycheff series

3.6 The Hermite polynomials. The Laguerre polynomials

Chapter 4. Fourier transforms

4.1 Infinite interval of integration

4.2 The Fourier integral formula: a heuristic introduction

4.3 Auxiliary theorems

4.4 Proof of the Fourier integral formula. Fourier transforms

4.5 The convention theorem. The Parseval formula

Chapter 5. Laplace transforms

5.1 Definition of the Laplace transform. Domain. Analyticity

5.2 Inversion formula

5.3 Further properties of Laplace transforms. The convolution theorem

5.4 Applications to ordinary differential equations

Chapter 6. Bessel functions

6.1 The gamma function

6.2 The Bessel differential equation. Bessel functions

6.3 Some particular Bessel functions

6.4 Recursion formulas for the Bessel functions

6.5 Estimation of Bessel functions for large values of x. The zeros of the Bessel functions

6.6 Bessel series

6.7 The generating function of the Bessel functions of integral order

6.8 Neumann functions

Chapter 7. Partial differential equations of first order

7.1 Introduction

7.2 The differential equation of a family of surfaces

7.3 Homogeneous differential equations

7.4 Linear and quasilinear differential equations

Chapter 8. Partial differential equations of second order

8.1 Problems in physics leading to partial differential equations

8.2 Definitions

8.3 The wave equation

8.4 The heat equation

8.5 The Laplace equation

Answers to exercises Bibliography Conventions Symbols Index

1.1 Basic concepts

1.2 Fourier series and Fourier coefficients

1.3 A mimimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem

1.4 Convergence of Fourier series

1.5 The Parseval formula

1.6 Determination of the sum of certain trigonemetric series

Chapter 2. Orthogonal systems

2.1 Integration of complex-valued functions of a real variable

2.2 Orthogonal systems

2.3 Complete orthogonal systems

2.4 Integration of Fourier series

2.5 The Gram-Schmidt orthogonalization process

2.6 Sturm-Liouville problems

Chapter 3. Orthogonal polynomials

3.1 The Legendre polynomials

3.2 Legendre series

3.3 The Legendre differential equation. The generating function of the Legendre polynomials

3.4 The Tchebycheff polynomials

3.5 Tchebycheff series

3.6 The Hermite polynomials. The Laguerre polynomials

Chapter 4. Fourier transforms

4.1 Infinite interval of integration

4.2 The Fourier integral formula: a heuristic introduction

4.3 Auxiliary theorems

4.4 Proof of the Fourier integral formula. Fourier transforms

4.5 The convention theorem. The Parseval formula

Chapter 5. Laplace transforms

5.1 Definition of the Laplace transform. Domain. Analyticity

5.2 Inversion formula

5.3 Further properties of Laplace transforms. The convolution theorem

5.4 Applications to ordinary differential equations

Chapter 6. Bessel functions

6.1 The gamma function

6.2 The Bessel differential equation. Bessel functions

6.3 Some particular Bessel functions

6.4 Recursion formulas for the Bessel functions

6.5 Estimation of Bessel functions for large values of x. The zeros of the Bessel functions

6.6 Bessel series

6.7 The generating function of the Bessel functions of integral order

6.8 Neumann functions

Chapter 7. Partial differential equations of first order

7.1 Introduction

7.2 The differential equation of a family of surfaces

7.3 Homogeneous differential equations

7.4 Linear and quasilinear differential equations

Chapter 8. Partial differential equations of second order

8.1 Problems in physics leading to partial differential equations

8.2 Definitions

8.3 The wave equation

8.4 The heat equation

8.5 The Laplace equation

Answers to exercises Bibliography Conventions Symbols Index

Library of Congress subject headings for this publication: Differential equations, Partial