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.
Review of Foundational Concepts.- Evaluation of Definite Integrals I: The Residue Theorem and Friends.- Evaluation of Definite Integrals II: Applications to Various Types of Integrals.- Cauchy Principal Value.- Integral Transforms.- Asymptotic Methods.
Ron Gordon earned a B.S. in Physics and B.S. in Mathematics from the University of Massachusetts, Amherst (1992), and a Ph.D. in Optics from the University of Rochester (1998). His doctoral thesis concerned mathematical properties of scanning light beams that optimize certain energy concentrations within specified regions near focus. Over the subsequent ten years, he worked for semiconductor manufacturing companies such as Motorola and IBM, developing mathematical models of light scatter from photomasks and of light propagation and diffraction through optical projection systems used in photolithography. One result of his research involved an analytical expression for a partially coherent diffraction image under certain conditions, derived by integrating about the boundary of a region defined by the intersection of three circles.
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