The scope of this book is the improper or generalized Riemann integral and infinite sum (series). The reader will study its convergence, principal value, evaluation and application to science and engineering. Improper Riemann integrals and infinite sums are interconnected. In the new edition, the author has involved infinite sums more than he did in the first edition. Apart from having computed and listed a large number of improper integrals and infinite sums, we have also developed the necessary theory and various ways of evaluating them or proving their divergence. Questions, problems and applications involving various improper integrals and infinite sums (series) of numbers emerge in science and application very often. Their complete presentations and all rigorous proofs would require taking the graduate-level courses on these subjects. Here their statements are adjusted to a level students of all levels can understand and use them efficiently as powerful tools in a large list of problems and applications.
1 Improper Riemann Integrals, Definitions, Criteria
1.1 Definitions and Examples
1.2 Applications
1.3 Problems
1.4 Cauchy Principal Value
1.5 A Note on the Integration by Substitution
1.6 Problems
1.7 Some Criteria of Existence
1.8 Problems
1.9 Three Important Notes on Chapter 1
1.10 Uniformly Continuous Functions
2 Calculus Techniques
2.1 Normal Distribution Integral
2.2 Applications
2.3 Problems
3 Real Analysis Techniques
3.1 Integrals Dependent on Parameters
3.2 Problems
3.3 Commuting Limits and Integrals
3.4 Commuting Limits and Derivatives
3.5 Problems
3.6 Double Integral Technique
3.7 Problems
3.8 Frullani Integrals
3.9 Problems
3.10 The Real Gamma Functions
3.11 The Beta Function
3.12 Applications
3.13 Problems
3.14 Appendix
3.15 Problems
4 Laplace Transform
4.1 Laplace Transform, Definitions, Theory
4.2 Problems
4.3 Inverse Laplace Transform
4.4 Applications
4.5 Problems
Bibliography
Index
Biography
Professor Ioannis Markos Roussos was born on November 5, 1954, at the village Katapola of the island of Amorgos, Greece. After primary and secondary education, he studied mathematics at the National and Kapodistrian University of Athens and received his BSc Degree (1972–1977). Then, he studied graduate mathematics and computer sciences at the University of Minnesota and received his Masters and PhD degrees (1977–1986). His specialization in mathematics was in Differential Geometry and Analysis. He has taught mathematics at the University of Minnesota (1977–1987), University of South Alabama (1987–1990) and Hamline University (1990–2022). Besides this book, he has published 17 research papers, ten expository papers and the book Basic Lessons on Isometries, Similarities and Inversions in the Euclidean Plane. He has participated in meetings and has refereed papers and promotions of other professors. Other interests are classical music, history, international relations and travelling.
