Complex Made Simple
This book has three parts. The first part may be thought of as a first year course in complex analysis. In the second part, proofs of some more advanced results are given. The third part is formed of appendices. The book starts with a definition of holomorphic functions and proofs of basic theorems, followed by a proof of the Cauchy theorem, the residue theorem and the open mapping theorem. The next chapter gives an application of what has been done so far; it is about Euler's formula for sin(z). The next two chapters are devoted to inverses of holomorphic and conformal mappings, followed by two chapters covering the Riemann mapping theorem and the relation between the theory of holomorphic and harmonic functions. The following chapters lead to a classification of elements of the automorphism group of the unit disk. The first part of the book ends with an analytic continuation of functions and the proof of the Picard theorems. Part two is devoted to Abel's theorem, a characterization of Dirichlet domains, more advanced versions of the maximal modulus theorem, properties of the Gamma function and its analytic continuation, universal covering spaces and Cauchy's theorem for nonholomorphic functions. The book can be used as a textbook of basics of complex analysis. There are many exercises and the exposition is very nice.