Graduate students and undergraduates who have taken an introductory course in real analysis are the target audience for this material on complex variables.
It is a significantly revised and updated edition of the well-known text by Robert B. Ash, delivering a succinct presentation that includes numerous problems and solutions as well as rigorous and thorough explanations.
Basic definitions for the topology of the plane, analytical functions, real-differentiability and the Cauchy-Riemann equations, and exponential and harmonic functions are presented in the introduction. The elementary theory, the general Cauchy theorem, and its applications—including singularities, residue theory, the open mapping theorem for analytical functions, linear fractional transformations, conformal mapping, and analytic mappings of one disk to another—are covered in the succeeding chapters. The factorization of analytical functions is discussed in detail together with the Riemann mapping theorem. The text concludes with a proof of the prime number theorem, which serves as an application of many of the concepts and findings presented in earlier chapters.
