Synopsis
Using the familiar software Microsoft ® Excel, this book examines the applications of complex variables. Implementation of the included problems in Excel eliminates the “black box” nature of more advanced computer software and programming languages and therefore the reader has the chance to become more familiar with the underlying mathematics of the complex variable problems. This book consists of two parts. In Part I, several topics are covered that one would expect to find in an introductory text on complex variables. These topics include an overview of complex numbers, functions of a complex variable, and the Cauchy integral formula. In particular, attention is given to the study of analytic complex variable functions. This attention is warranted because of the property that the real and imaginary parts of an analytic complex variable function can be used to solve the Laplace partial differential equation (PDE). Laplace's equation is ubiquitous throughout science and engineering as it can be used to model the steady-state conditions of several important transport processes including heat transfer, soil-water flow, electrostatics, and ideal fluid flow, among others. In Part II, a specialty application of complex variables known as the Complex Variable Boundary Element Method (CVBEM) is examined. CVBEM is a numerical method used for solving boundary value problems governed by Laplace's equation. This part contains a detailed description of the CVBEM and a guide through each step of constructing two CVBEM programs in Excel. The writing of these programs is the culminating event of the book. Students of complex variables and anyone with interest in a novel method for approximating potential functions using the principles of complex variables are the intended audience for this book. The Microsoft Excel applications (including simple programs as well as the CVBEM program) covered will also be of interest in the industry, as these programs are accessible to anybody with Microsoft Office.
About the Authors
Bryce graduated from West Point in 2017 earning a Bachelor's degree in Mathematical Sciences. His first Army assignment was as a graduate student at MIT where he earned a Master's degree in Mechanical Engineering as a Draper Fellow. He then attended the Air Defense Artillery Basic Officer Leader Course and the Terminal High Altitude Area Air Defense (THAAD) School at Fort Sill, OK. He was then assigned as a Tactical Control Officer for the THAAD battery in Guam where he was for 6 months before being reassigned as a graduate student at Carnegie Mellon University (CMU). He is currently pursuing a Master's degree in Computational Data Science at CMU in Pittsburgh, PA where he is part of the inaugural class of Army Artificial Intelligence Scholars.
Dr Theodore V. Hromadka II is a Distinguished Professor of Mathematics at the Department of Mathematical Sciences at the United States Military Academy. He has been the Principal Engineer and Hydrologist for nearly 20 years at the consulting firm Hromadka & Associates. He holds several professional licenses, including Civil Engineer, Geoscientist, and Geologist. Furthermore, he holds professional certifications as a Hydrologist for both surface water and groundwater. He has over 40 years of professional engineering experience and has served in various academic and research positions for over 35 years. He is a Diplomate of the American Academy of Water Resources Engineers and serves on the Board of Directors of the Wessex Institute of Technology in Great Britain. He is Professor Emeritus at the California State University, having served in the Departments of Mathematics, Environmental Studies, and Geological Sciences.
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