1st Edition

Improper Riemann Integrals

By Ioannis Roussos Copyright 2014
    690 Pages 36 B/W Illustrations
    by Chapman & Hall

    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.

    Improper Riemann Integrals
    Definitions and Examples
    Cauchy Principal Value
    Some Criteria of Existence

    Real Analysis Techniques
    Calculus Techniques
    Integrals Dependent on Parameters
    Commuting Limits with Integrals and Derivatives
    Double Integral Technique
    Frullani Integrals
    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




    Ioannis Roussos

    "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