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CHAPTER 1．Some Preliminary Topics 1

1．Equicontinuous Families of Functions 1

2．The Weierstrass Approximation Theorem 4

3．The Fourier Integral 9

4．The Laplace Transform 13

5．Ordinary Differential Equations 17

6．Lebesgue Integration 25

7．Dini＇s Theorem 27

CHAPTER 2．Partial Differential Equations of First Order 28

1．Linear Equations in Two Independent Variables 28

2．Quasi-linear Equations 33

3．The General First-order Equation 36

CHAPTER 3．The Cauchy Problem 42

1．Classification of Equations with Linear Principal Parts 42

2．Characteristics 44

3．Canonical Forms 46

4．The Cauchy Problem for Hyperbolic Equations 48

5．The One-dimensional Wave Equation 53

6．The Riamann Function 55

7．Classification of Second-order Equations in Three or More Independent Variables 58

8．The Wave Equation in Two and Three Dimensions 60

9．The Legendre Transformation 65

CHAPTER 4．The Fredholm Alternative in Banach Spaces 69

1．Linear Spaces 69

2．Normed Linear Spaces 71

3．Banach Spaces 74

4．Linear Functionals and Linear Operators 76

5．The Fredholm Alternative 82

CHAPTER 5．The Fredholm Alternative Hilbert Spaces 90

1．Inner-product Spaces 90

2．Hilbert Spaces 95

3．Projections,Linear Functionals,Adjoint Operators 99

4．Hermitian and Completely Continuous Operators 104

5．The Fredholm Alternative 111

6．Integral Equations 118

7．Hermitian Kernels 121

8．Illustrative Example 127

CHAPTER 6．Elements of Potential Theory 130

1．Introduction 130

2．Laplace＇s Equation and Theory of Analytic Functions 131

3．Fundamental Solutions 133

4．The Mean-value Theorem 135

5．The Maximum Principle 136

6．Formulation of the Dirichlet Problem 138

7．Solution of the Dirichlet Problem for the Disc 139

8．The Converse of the Mean-value Theorem 146

9．Convergence Theorems 149

10．Strengthened Form of the Maximum Principle 152

11．Single and Double Layers 152

12．Poisson＇s Equation 157

CHAPTER 7．The Dirichlet Problem 167

1．Subharmonic Functions 167

2．The Method of Balayage 170

3．The Perron-Remak Method 176

4．The Method of Integral Equations 179

5．The Dirichlet Principle 183

6．The Method of Finite Differences 199

7．Conformal Mapping 211

CHAPTER 8．The Heat Equation 217

1．The Initial-value Problem for the Infinite Rod 217

2．The Simplest Problem for the Semi-infinite Rod 221

3．The Finite Rod 256

CHAPTER 9．Green＇s Functions and Separation of Variables 232

1．The Vibrating String 232

2．The Green＇s Function of the Operator d2/dx2 235

3．The Green＇s Function of a Second-order Differential Operator 237

4．Eigenfunction Expansions 239

5．A Generalized Wave Equation 241

6．Extension of the Definition of Green＇s Functions 243

SOLUTIONS TO SELECTED EXERCISES 253

SUGGESTIONS FOR FURTHER STUDY 267

INDEX 269