This function is defined for all x e (-1. 1). and its range consists of all y. The arctanh x is an odd, nonperiodic. unbounded function that crosses the coordinate axes at the origin x = 0, y = 0. This is an increasing function in its domain with no points ofextremum and an inflection point at the origin. It has two vertical asymptotes: x = ±1. The graph of the function y = arctanh x is given in Fig. . | xxviii Preface formula 5 in Subsection . At the end of each chapter we present a list of main and additional literature sources containing more detailed information about topics of interest to the reader. Special font highlighting in the text cross-references an extensive table of contents and an index help the reader to find the desired information. We would like to express our deep gratitude to Alexei Zhurov for fruitful discussions and valuable remarks. We also appreciate the help of Vladimir Nazaikinskii and Grigorii Yosifian for translating several chapters of this book and are thankful to Kirill Kazakov and Mikhail Mikhin for their assistance in preparing the camera-ready copy of the book. The authors hope that this book will be helpful for a wide range of scientists university teachers engineers and students engaged in the fields of mathematics physics mechanics control chemistry biology engineering sciences and social and economical sciences. Some sections and examples can be used in lectures and practical studies in basic and special mathematical courses. Andrei D. Polyanin Alexander V. Manzhirov Main Notation Special symbols equal to identically equal to not equal to approximately equal to of same order as used in comparisons of infinitesimals or infinites less than a less than b is written as a b or equivalently b a less than or equal to a less than or equal to b is written as a b much less than a much less than b is written as a b greater than a greater than b is written as a b or equivalently b a greater than or equal to a greater than or equal to b is written as a b much greater than a much greater than b is written as a b plus sign the sum of numbers a and b is denoted by a b and has the property a b b a - minus sign the difference of numbers a and b is denoted by a - b multiplication sign the product of numbers a and b is denoted by either ab or a b sometimes a X b and has the property ab ba the inner product of vectors a and b is denoted by a
