Improper Riemann Integrals

Improper Riemann Integrals

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Features

  • Describes various methods from advanced calculus and real and complex analysis for computing improper integrals
  • Provides the necessary theorems and results
  • Offers justification for many steps by using the tools of advanced analysis, a feature often omitted from other books
  • Contains detailed examples that illustrate the tools for checking convergence of improper integrals and evaluating their exact value when possible
  • Presents many of the same examples in different ways, highlighting the interconnection among topics, how a problem can be viewed in various contexts, and how different methods can solve the same problem
  • Includes problems in every section, several projects and applications, and an extensive bibliography at the back of the book
  • Lists all the major integrals and finite and infinite sums computed in the text

Summary

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
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

Bibliography

Index

 
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