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The Laplace Transformation I – General Theory

Complex Functions Theory a-4

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Language:  English
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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 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
About the Author

Leif Mejlbro

Leif Mejlbro was educated as a mathematician at the University of Copenhagen, where he wrote his thesis on Linear Partial Differential Operators and Distributions. Shortly after he obtained a position at the Technical University of Denmark, where he remained until his retirement in 2003. He has twice been on leave, first time one year at the Swedish Academy, Stockholm, and second time at the Copenhagen Telephone Company, now part of the Danish Telecommunication Company, in both places doing research.

At the Technical University of Denmark he has during more than three decades given lectures in such various mathematical subjects as Elementary Calculus, Complex Functions Theory, Functional Analysis, Laplace Transform, Special Functions, Probability Theory and Distribution Theory, as well as some courses where Calculus and various Engineering Sciences were merged into a bigger course, where the lecturers had to cooperate in spite of their different background. He has written textbooks to many of the above courses.

His research in Measure Theory and Complex Functions Theory is too advanced to be of interest for more than just a few specialist, so it is not mentioned here. It must, however, be admitted that the philosophy of Measure Theory has deeply in uenced his thinking also in all the other mathematical topics mentioned above.

After he retired he has been working as a consultant for engineering companies { at the latest for the Femern Belt Consortium, setting up some models for chloride penetration into concrete and giving some easy solution procedures for these models which can be applied straightforward without being an expert in Mathematics. Also, he has written a series of books on some of the topics mentioned above for the publisher Ventus/Bookboon.