Improper Riemann Integrals is the first book to collect classical and modern material on the subject for undergraduate students. The book gives students the prerequisites and tools to understand the convergence, principal value, and evaluation of the improper/generalized Riemann integral. It also illustrates applications to science and engineering problems.
The book contains the necessary background, theorems, and tools, along with two lists of the most important integrals and sums computed in the text. Numerous examples at various levels of difficulty illustrate the concepts and theorems. The book uses powerful tools of real and complex analysis not only to compute the examples and solve the problems but also to justify that the computation methods are legitimate.
Enriched with many examples, applications, and problems, this book helps students acquire a deeper understanding of the subject, preparing them for further study. It shows how to solve the integrals without exclusively relying on tables and computer packages.
Table of Contents
Improper Riemann Integrals
Definitions and Examples
Cauchy Principal Value
Some Criteria of Existence
Real Analysis Techniques
Integrals Dependent on Parameters
Commuting Limits with Integrals and Derivatives
Double Integral Technique
The Real Gamma and Beta Functions
A Brief Overview of Laplace Transform
Complex Analysis Techniques
Basics of Complex Variables
Power Series, a Quick Review
Limits, Continuity and Derivatives
Line Integrals in the Complex Plane
Cauchy-Goursat Theorem and Consequences
Roots, Singularities, Residues
Contour Integration and Integrals
Definite Integrals with Sines and Cosines
List of Non-Elementary Integrals and Sums in Text
List of Non-Elementary Integrals
List of Non-Elementary Sums
"When one wants to calculate an improper Riemann integral one is facing various pitfalls. This is why an enormous amount of examples, exercises, and problems is la raison d’etre of this book, which is devoted to the calculation of improper Riemann integrals. Moreover, the author does his best to make the book as accessible as possible. … I can recommend this book, which shows to students, among other things, the power of ‘elementary techniques.’"
—Mathematical Reviews, January 2015
"By collecting a lot of important improper integrals and sums used in various domains, and presenting their calculation in an accessible but rigorous way, the book will be of great use to students in mathematics and related areas and for applied scientists (statisticians, engineers, physicists) as well."
—Studia Universitatis Babeş-Bolyai Mathematica, 59, 2014