A Pathway to Complex Analysis 2022/edition

This book is an honest attempt to establish a long-cherished belief that Complex Analysis is a fine meeting ground of analysis, geometry and topology. In the last five decades the curriculum has improved a lot and students in India have an exposure to real analysis, linear algebra, metric spaces and topology. Keeping the improved background of the students in mind, we develop the subject. The salient features of this book are: A careful treatment of argument and logarithms. Use of triangles and piecewise smooth paths in place of rectifiable paths. Geometric treatments of C-R equations, winding numbers and fractional linear transformations. A geometric treatment of analytic continuation explaining the subtleties to the beginner. The treatment of conformal property of holomorphic functions using the notion of oriented angles. Many insightful remarks which give a wider perspective to the readers and show them the `big picture. We emphasize the central point of the Cauchy theory, viz., the existence of local primitives and that of the local power series expansion. We point out how the properties of the holomorphic functions are the local/global manifestations of the corresponding results for power series. Many students approach the subject Complex Analysis with a sense of wonder, mystery and awe. They are mystified by the infinite differentiability of `analytic functions and so-called multivalued functions. The book will demystify them.