Improper Riemann integrals (Q2869820)

The need to calculate improper integrals arises in numerous theoretical as well as practical problems. In many situations, it is enough to find the value of the integral using a computer algebra system, or to consult a standard reference such as [\textit{I. S. Gradshteyn} and \textit{I. M. Ryzhik}, Table of integrals, series, and products. Amsterdam: Elsevier/Academic Press (2007; Zbl 1208.65001)]. However, it might be much more difficult to verify the corresponding result by hand calculation. The present book is unique in that it presents a systematic overview of various techniques for the evaluation of improper Riemann integrals, and contains a large amount of solved problems as well as exercises.NEWLINENEWLINENEWLINEChapter 1 introduces the topic of improper Riemann integrals. It discusses various cases such as integrals over unbounded intervals, integrals where the function is undefined at the endpoints or at an interior point. Principal values of improper integrals are introduced, too. The whole book deals almost exclusively with integration of functions which are either continuous, or discontinuous at finitely many points. The first chapter is concluded with some basic criteria for the existence of improper integrals.NEWLINENEWLINENEWLINEChapter 2 focuses on various real-analytic techniques, such as differentiation under the integral sign, interchange of integration and summation, or the double integral technique. It introduces the real gamma and beta functions, and presents integrals which can be evaluated in terms of these special functions. The final section is devoted to the Laplace transform and its properties, as well as calculation of various Laplace-type integrals.NEWLINENEWLINENEWLINEChapter 3 is the most extensive one and discusses the evaluation of real integrals and infinite series via complex-analytic techniques, especially by means of the residue theorem. The Fourier transform is introduced here, and the author shows how to calculate numerous Fourier-type integrals. Chapter 4 contains a list of 194 integrals and 85 series evaluated in the preceding chapters.NEWLINENEWLINEThe text is easily accessible to undergraduate students. In order to avoid more advanced mathematics such as Lebesgue's integration theory, the author prefers to formulate the necessary theorems with simple and easy to verify assumptions instead of trying to provide results in their most general form.NEWLINENEWLINEMoreover, the author has tried to make the text self-contained; as a consequence, there is some overlap with the material usually covered in real and complex analysis courses. For example, the third chapter has over 150 pages devoted to an introduction to complex analysis. The crucial fact that holomorphic and analytic functions coincide is derived in a nonstandard way: The author starts by observing that the real and imaginary parts of a holomorphic function satisfy Laplace's equation, and then uses Poisson's integral formula to get analyticity.NEWLINENEWLINEOn the other hand, the book contains a wealth of interesting material which is rarely found elsewhere, such as the Frullani integrals, or the Fourier transform with complex arguments. Also, the part devoted to the calculation of real integrals by complex-analytic methods goes far beyond the usual complex analysis courses.NEWLINENEWLINEThroughout the book, the author shows how improper integrals appear in various applications -- in physics, probability theory, ordinary and partial differential equations, or geometry.NEWLINENEWLINETo sum up, this is a useful companion to real and complex analysis textbooks. Although the standard tables of integrals contain many more results, this book actually teaches the reader how to calculate the integrals and provides plenty of exercises to practice the theory.