Complex AnalysisThe present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. Somewhat more material has been included than can be covered at leisure in one term, to give opportunities for the instructor to exercise his taste, and lead the course in whatever direction strikes his fancy at the time. A large number of routine exercises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc. ) and I would recom mend to anyone to look through them. More recent texts have empha sized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex anal ysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. The systematic elementary development of formal and convergent power series was standard fare in the German texts, but only Cartan, in the more recent books, includes this material, which I think is quite essential, e. g. , for differential equations. I have written a short text, exhibiting these features, making it applicable to a wide variety of tastes. The book essentially decomposes into two parts. |
Contents
CHAPTER I | 3 |
3 Complex Valued Functions | 12 |
5 Complex Differentiability | 28 |
Copyright | |
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a₁ absolute value algebraic analytic continuation analytic function analytic isomorphism assume automorphism b₁ boundary bounded calculus Cauchy's theorem Chapter closed disc closed path coefficients compact set complex numbers constant term contained continuous function converges absolutely converges uniformly curve defined derivative differentiable disc of radius entire function equal equations Example EXERCISES exists Figure finite number formal power series fractional linear map function f given Hence holomorphic function integral inverse isomorphism Lemma Let f Let f(z Let ƒ limit maximum modulus principle meromorphic non-zero open disc open set point of accumulation poles polynomial positive integer power series expansion proves the theorem quotient radius of convergence real numbers rectangle residue say that ƒ sequence Show subset Suppose Theorem 1.2 unit circle unit disc upper half plane w₁ whence write z₁ zeros ду дх