This Complex Functions Theory a-4 text is the fourth e-book in a series which has previously characterized analytic functions by their complex differentiability and proved Cauchy’s Integral Theorem, provided alternative proofs which show that locally, every analytic function is described by its Taylor series, shown the connection between analytic functions and geometry, and reviewed conformal maps and their importance in solving Dirichlet problems. Complex Functions Theory a-4 builds on these previous texts, focusing on the general theory of the Laplace Transformation Operator. This e-book and previous titles in the series can be downloaded for free here.

All theorems are accompanied by their proofs, and all equations are explained and demonstrated in detail. A comprehensive index follows the text.

Readers interested in a full overview of complex analytic functions should refer to the related titles in this series, all of which are available for free download on bookboon.com: Elementary Analytic Functions - Complex Functions Theory a-1, Calculus of Residua - Complex Functions Theory a-2, Stability, Riemann Surfaces, and Conformal Mappings: Complex Functions Theory a-3.

1. The Lebesgue Integral
   1. Null sets and null functions
   2. The Lebesgue integral
2. The Laplace transformation
   1. Definition of the Laplace transformation using complex functions theory
   2. Some important properties of Laplace transforms
   3. The complex inversion formula I
   4. Convolutions
   5. Linear ordinary differential equations
3. Other transformations and the general inversion formula
   1. The two-sided Laplace transformation
   2. The Fourier transformation
   3. The Fourier transformation on L1(R)
   4. The Mellin transformation
   5. The complex inversion formula II
   6. Laplace transformation of series
   7. A catalogue of methods of finding the Laplace transform and the inverse Laplace transform
4. Tables
5. Index