Operational Calculus

1
- §0. Operators
- Test-functions and Operators
- Convolution
- Entering Functions
- § 1. Perfect Operators
- Algebra of Perfect Operators
- The Operator of an Entering Function
- Translation Property
- K-functions
- Numbers and Operators
- Ratios of Perfect Operators

2
- §2. The Basic Facts
- Translation Property
- Numbers and Operators
- Functions Having no Jumps on [0, ?)
- Various Consequences
- Three Integral Equations
- Differential Equations: Procedure
- Higher Derivatives
- Continuity of the Convolution Integral
- § 3. Elementary Applications
- Table of Formulas
- Illustrative Examples
- § 4. Partial Fraction Decomposition
- Decompositions Obtained by Division
- Heaviside's Expansion
- Decomposition into Two Parts
- Repeated Linear Factors
- Heaviside Procedure
- Repeated Quadratic and Linear Factors
- Illustrative Examples

3
- §5. Further Applications
- Series of Translates
- Periodic Functions
- Advantages of Operational Calculus
- Use of Laplace Transform Tables
- Difference Equations
- Bibliographical Comments
- Notes and Further Comments
- § 6. Calculus of Operators
- Convergence
- Value of an Operator at a Point
- Integral of an Operator
- Function-operators
- Impulse and Dipole
- Distributions and Operators
- Two Examples
- Multiplication by t
- Static Deflection of Beams
- §7. Vectors
- Electric Circuits
- The General Case
- §8 Non-integrable Functions
- Convolution of Operators
- Finite Part of a Divergent Integral
- Indexed Functions
- Derivatives on the Open Interval (0, ?)
- Duhamel's Formula

4
- § 9. Partial Differential Equations
- Differentiation with Respect to x
- Vibrating String
- A Diffusion Problem
- Wave Problems
- The Infinite Interval
- Classical Solutions
- The Finite Interval
- § 10. Diffusion Problems
- The Infinite Interval
- Insulated Rod
- The Finite Interval
- Comparison with the Laplace Transformation

5
- § 11. Series of Operators
- Series of