Synopses & Reviews
Coverage of sequences and series are central to the author’s approach, and as such, analyticity is defined using convergence of power series throughout. The result is that readers view analytic functions fundamentally different than differentiable functions from calculus, and in addition, the differences between complex and real analysis become apparent. Sequences and series of functions are foundational to understanding complex analysis, and uniform coverage of compact sets is included to interchange certain limit processes. While this is often glossed over in competing books, the author presents this discussion with rigor. Since power series is discussed initially, the definitions of exponential and trigonometric functions are delayed, allowing readers’ definitions to be in terms of series. This approach emphasizes that definitions are natural extensions of their real counterparts. Each section ends with a “Summary and Notes” essay that gives readers an overview of the topics discussed as well as the historical background. Topical coverage includes: complex numbers; complex functions and mappings; analytic functions; Cauchy’s integral theory; the Residue Theorem; harmonic functions and Fourier series; sets and functions; and results from advanced calculus.
Review
Muir presents an undergraduate textbook on complex analysis thatemphasizes the logically and complete and consistent development of topics rather than glossing over details to get to the good parts. Heassumes that students have completed an undergraduate sequence of calculus courses, but highlights how complex analysis differs fromcalculus. His topics are the complex numbers, complex functions and mappings, analytic functions, Cauchy's integral theory, the residue theorem, and harmonic functions and Fourier series.Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)