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# Analysis Real and Complex Analysis by Rudin

by micromass
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 Mentor P: 18,244 Author: Walter Rudin Title: Real and Complex Analysis Amazon Link: http://www.amazon.com/Complex-Analys.../dp/0070542341 Prerequisities: Baby Rudin Level: Grad Table of Contents:  Preface Prologue: The Exponential Function Abstract Integration Set-theoretic notatons and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0,\infty] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Positive Borel Measures Vector Spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises L^p-Spaces Convex functions and inequalities The L^p-spaces Approximation by continuous functions Exercises Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L^1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Complex Measures Total Variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on L^p The Riesz representation theorem Exercises Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L^1 Exercises Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Approximation by Rational Functions Preperation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class \mathcal{S} Continuity and the boundary Conformal mapping of an annulus Exercises Zeros of Holomorphic Functions Infinite products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szasz theorem Exercises Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises H^p-Spaces Subharmonic functons The spaces H^p and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index 
 P: 11 Would Spivak's Calculus and Pugh's Real Math Analysis be sufficient grounding for this text, at least for the "real" part of it? Also, what books and in what order would I need to work my way up from absolute nothing to this book's level in complex analysis? Order of growth towards this book's rigor seems straightforward when it comes to real analysis: regular calculus -> (maybe some intro to proofs) -> Calculus by Spivak like text -> (maybe some intro to proofs) -> baby Rudin... How does this work with complex variables? Does one need some sort of complex pre/calculus before attempting complex analysis? Thanks.
 Sci Advisor HW Helper P: 9,470 I own this book, have studied it from time to time in the past as a grad student, but have learned little from it. Experts consider it a classic, but as with other books by this author i find it unilluminating. I did find it an excellent source of practice problems for PhD prelims in analysis.
 P: 350 Real and Complex Analysis by Rudin Same reaction as mathwonk. I learned from Folland's book instead.
P: 828
 Quote by Vargo Same reaction as mathwonk. I learned from Folland's book instead.
OK, so I have very mixed feelings about Rudin and Folland. People say that Rudin makes more effort to impress the reader with his cleverness and that the proofs aren't enlightening (I once heard a professor describe it as being "pathelogically elegant".) But, Folland is nothing if not terse and there are a lot of steps that are skipped in proofs that the reader is assumed to a) know they are there and b)fill them in. Now, if you are learning, this isn't the worst thing that can happen, but it gets irritating.

My biggest gripe with Rudin is that he doesn't start abstract enough, whereas Folland starts out with premeasures on arbitrary sets and builds the theory from there and then says "hey, look at that, the 'dx' that you already know about is actually Lebesgue measure." Rudin abstracts a little at a time and it gets annoying.

However, I still enjoyed Rudin's book and learned a lot from it.

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