American Mathematical Soc., 2011 - 305 Seiten
The text covers a broad spectrum between basic and advanced complex variables on the one hand and between theoretical and applied or computational material on the other hand. With careful selection of the emphasis put on the various sections, examples, and exercises, the book can be used in a one- or two-semester course for undergraduate mathematics majors, a one-semester course for engineering or physics majors, or a one-semester course for first-year mathematics graduate students. It has been tested in all three settings at the University of Utah. The exposition is clear, concise, and lively. There is a clean and modern approach to Cauchy's theorems and Taylor series expansions, with rigorous proofs but no long and tedious arguments. This is followed by the rich harvest of easy consequences of the existence of power series expansions. Through the central portion of the text, there is a careful and extensive treatment of residue theory and its application to computation of integrals, conformal mapping and its applications to applied problems, analytic continuation, and the proofs of the Picard theorems. Chapter 8 covers material on infinite products and zeroes of entire functions. This leads to the final chapter which is devoted to the Riemann zeta function, the Riemann Hypothesis, and a proof of the Prime Number Theorem.
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analytic function analytically continued boundary bounded centered at z0 closed curve coefficients compact subsets complement completes the proof complex numbers complex variable component conformal equivalence conformal map connected open set constant continuous function converges uniformly Corollary deﬁned deﬁnition differentiable Dirichlet problem domain Dr z0 entire function equation Example Exercise Set f is analytic Find formula Fourier transform function defined function element function f harmonic function hence Im(z implies inverse function Lemma Let f limit linear fractional transformation log function meromorphic function modulus neighborhood of z0 non-vanishing non-zero open disc open set open subset parameter interval plane point z0 pole polynomial power series power series expansion previous exercise previous theorem principal branch Prove radius of convergence Re(z real numbers residue result Riemann sequence simply connected Solution triangle unit circle unit disc upper half-plane Weierstrass