Complex Integration: A Compendium of Smart and Little-Known Techniques for Evaluating Integrals and Sums

Description

Integrals and sums are not generally considered for evaluation using complex integration. This book proposes techniques that mainly use complex integration and are quite different from those in the existing texts. Such techniques, ostensibly taught in Complex Analysis courses to undergraduate students who have had two semesters of calculus, are usually limited to a very small set of problems. Few practitioners consider complex integration as a tool for computing difficult integrals. While there are a number of books on the market that provide tutorials on this subject, the existing texts in this field focus on real methods. Accordingly, this book offers an eye-opening experience for computation enthusiasts used to relying on clever substitutions and transformations to evaluate integrals and sums. The book is the result of nine years of providing solutions to difficult calculus problems on forums such as Math Stack Exchange or the author's website, residuetheorem.com. It serves to detail to the enthusiastic mathematics undergraduate, or the physics or engineering graduate student, the art and science of evaluating difficult integrals, sums, and products. Preface Why Complex Integration? Goals Prerequisites A Note About Nomenclature Acknowledgements Prologue Contents 1 Review of Foundational Concepts 1.1 Sequences and Series 1.2 Integrals 1.3 Two Dimensions 1.4 Analytic Functions 2 Evaluation of Definite Integrals I: The Residue Theorem and Friends 2.1 Poles 2.1.1 Cauchy's Theorem in the Presence of a Pole 2.1.2 The Residue Theorem is a Shortcut 2.2 Branch Points 2.3 Poles and Branch Points 3 Evaluation of Definite Integrals II: Applications to Various Types of Integrals 3.1 Integrands Defined Over [0,infty) 3.2 Integration Over the Unit Circle 3.3 Residue Reduction Contours 3.4 Definite Integrals Evaluated Between Branch Points 3.5 Sums 4 Cauchy Principal Value 4.1 Removable Singularities 4.2 Poles on Contours 4.3 The Hilbert Transform 5 Integral Transforms 5.1 The Fourier Transform 5.2 Laplace Transforms 5.3 Two-Sided Laplace and Mellin Transforms 6 Asymptotic Methods 6.1 Asymptotic Expansions 6.2 The Monotonic h-Transform and a New Complex Integration Technique 6.3 Euler-Maclurin Summation Appendix Epilogue