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This book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. Students should be familiar with the Cartesian representation of complex numbers and with the algebra of complex numbers, that is, they should know that (i**2= –1). A famiharity with multivariable calculus is also required, but here the fundamental ideas are reviewed. In fact, complex analysis provides a good training ground for multivariable calculus. It allows students to consohdate their understanding of parametrized curves, tangent vectors, arc length, gradients, line integrals, independence of path, and Green's theorem. The ideas surrounding independence of path are particularly difficult for students in calculus, and they are not absorbed by most students until they are seen again in other courses. The book consists of sixteen chapters, which are divided into three parts. The first part. Chapters I-VII, includes basic material covered in all undergraduate courses. With the exception of a few sections, this material is much the same as that covered in Cauchy's lectures, except that the emphasis on viewing functions as mappings reflects Riemann's influence. The second part. Chapters VIII-XI, bridges the nineteenth and the twentieth centuries. About half this material would be covered in a typical undergraduate course, depending upon the taste and pace of the instructor. The material on the Poisson integral is of interest to electrical engineers, while the material on hyperbolic geometry is of interest to pure mathematicians and also to high school mathematics teachers. The third part. Chapters XII-XVI, consists of a careful selection of special topics that illustrate the scope and power of complex analysis methods. These topics include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces. The final five chapters serve also to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis.
The Complex Plane and Elementary Functions.
Analytic Functions.
Line Integrals and Harmonic Functions.
Complex Integration and Analyticity.
Power Series.
Laurent Series and Isolated Singularities.
The Residue Calculus.
The Logarithmic Integral.
The Schwarz Lemma and Hyperbolic Geometry.
Harmonic Functions and the Reflection Principle.
Conformal Mapping.
Compact Families of Meromorphic Functions.
Approximation Theorems.
Some Special Functions.
Riemann Surfaces.
List of Symbols