1.

Volterra Equations 

1.1 
A Mechanical Problem Leading to an Integral Equation 

1.2 
Integral Equations and Algebraic Systems of Linear Equations 

1.3 
Volterra Equations 

1.4 
L subscript 2Kernels and Functions 

1.5 
Solution of Volterra Integral Equations of the Second Kind 

1.6 
Volterra Equations of the First Kind 

1.7 
An Example 

1.8 
Volterra Integral Equations and Linear Differential Equations 

1.9 
Equations of the Faltung type (Closed Cycle Type) 

1.10 
Transverse Oscillations of a Bar 

1.11 
Application to the Bessel Functions 

1.12 
Some Generalizations of the Theory of Volterra Equations 

1.13 
NonLinear Volterra Equations 
2. 
Fredholm Equations 

2.1 
Solution by the Method of Successive Approximations: Neumann's Series 

2.2 
An Example 

2.3 
Fredholm's Equations with PincherleGoursat Kernels 

2.4 
The Fredholm Theorem for General Kernels 

2.5 
The Formulae of Fredholm 

2.6 
Numerical Solution of Integral Equations 

2.7 
The Fredholm Solution of the Dirichlet Problem 
3. 
Symmetric Kernels and Orthogonal Systems of Functions 

3.1 
Introductory Remarks and a Process of Orthogonalization 

3.2 
Approximation and Convergence in the Mean 

3.3 
The RieszFischer Theorem 

3.4 
Completeness and Closure 

3.5 
Completeness of the Trigonometric System and of the Polynomials 

3.6 
Approximation of a General L subscript 2Kernel by Means of PGKernels 

3.7 
Enskog's Method 

3.8 
The Spectrum of a Symmetric Kernel 

3.9 
The Bilinear Formula 

3.10 
The HilbertSchmidt Theorem and Its Applications 

3.11 
Extremal Properties and Bounds for Eigenvalues 

3.12 
Positive KernelsMercer's Theorem 

3.13 
Connection with the Theory of Linear Differential Equations 

3.14 
Critical Velocities of a Rotating Shaft and Transverse Oscillations of a Beam 

3.15 
Symmetric Fredholm Equations of the First Kind 

3.16 
Reduction of a Fredholm Equation to a Similar One with a Symmetric Kernel 

3.17 
Some Generalizations 

3.18 
Vibrations of a Membrane 
4. 
Some Types of Singular or NonLinear Integral Equations 

4.1 
Orientation and Examples 

4.2 
Equations with Cauchy's Principal Value of an Integral and Hilbert's Transformation 

4.3 
The Finite Hilbert Transformation and the Airfoil Equation 

4.4 
Singular Equations of the Carleman Type 

4.5 
General Remarks About NonLinear Integral Equations 

4.6 
NonLinear Equations of the Hammerstein Type 

4.7 
Forced Oscillations of Finite Amplitude 
Appendix I. 
Algebraic Systems of Linear Equations 
Appendix II. 
Hadamard's Theorem 

Exercises; References; Index 
