Complex Analysis Howie
Dallas Querry
Howie's book is a gem. I want to use it the next time I teach complex analysis. Not only do Howie's selection of topics and their sequence correspond perfectly to what I believe to be the ideal approach to this gorgeous subject, the writing style is (again) wonderful...I think this is a terrific book. I'm going to use it the first chance I get. And I recommend it very, very highly.
Howie has written an outstanding book on complex variables...The readability of the book is improved by more than 80 figures and numerous examples. Also included are 140 exercises with complete solutions in an appendix. All this makes the book ideal for self-study. Summing up: Highly recommended.
"This book takes account of the varying needs and backgrounds and provides a self-study text for students in mathematics, science and engineering. ... Clear and careful explanations are backed up with worked examples and more than 100 exercises, for which full solutions are provided." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 49 (3-4),2003)
I'm out of college, and trying to self-learn complex analysis. I'm finding Ahlfors' text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.)
Lang - Complex Analysis (typical Lang style with concise proofs, altough it starts quite slowly, a nice coverage of topological aspects of contour integration, and some advanced topics with applications to analysis and number theory in the end)
I like Conway's Functions of one complex variable I a lot. It is very well written and gives a thorough account of the basics of complex analysis. And a section on Riemann's $\zeta$-function is also included.
I second the answer by "wildildildlife" but specially the book by Freitag - "Complex Analysis" and the recently translated second volume to be published this summer. It is the most complete, well-developed, motivated and thorough advanced level introduction to complex analysis I know. The first volume starts out with complex numbers and holomorphic functions but builds the theory up to elliptic and modular functions, finishing with applications to analytic number theorem proving the prime number theorem. The second volume develops the theory of Riemann surfaces and introduces several complex variables and more modular forms (of huge importance to modern number theory). They are filled with interesting exercises and problems most of which are solved in detail at the end!
You just need a good background in undergraduate analysis to manage. Moreover, I think they should be your next step after a softer introduction to complex analysis if you are interested in deepening your knowledge and getting a good grasp at the different aspects and advanced topics of the whole subject.
This is a self-contained, very accessible, comprehensive, and masterfully written textbook that I do find very suitable for the serious self-taught possessing the rare mathematical maturity, and being in command of a quite modest (but non-negligible) background.
Among its many competitors, this work distinguishes itself by being, by far, the most modern in scope and means, since it introduces in a very harmonious way and from the very beginning, mainly from scratch, key ideas from homological algebra, algebraic topology, sheaf theory, and the theory of distributions, together with a systematic use of the Cauchy-Riemann $\bar\partial$-operator. So for instance, once you're going to tackle Cauchy's integral theorem, you'll be fully equipped to prove it in its full generality, and without the typical "hand-weaving" most texts rely on and hide behind.
A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka is a well written free online textbook. It is available in PDF format from San Francisco State University at this authors website.
I don't think it has the zeta function or the PNT (I could be wrong, it has been a long time since I looked at it), but "Invitation to Complex Analysis" by Ralph P. Boas is really nice, and suitable for self study because it has about 60 pages of solutions to the texts problems.
For a good introduction i referred "A First Course in complex Analysis by Dennis G.Zill" and for little advanced case i would like to refer "Complex Analysis by Dennis G. Zill and Patrick Shanahan".
I've taught a few times from Churchill's book, and used it as an undergrad. I'm liking it less all the time. I would probably switch to Marsden/Hoffman next time. At a more advanced level, I like Nevanlinna and Paatero, "Introduction to Complex Analysis." It has a chapter on the Riemann zeta function within which there is a discussion of the distribution of primes. I used this in the beginning grad course in complex, along with Hille's "Analytic Function Theory," which I liked very much.
"Schaum's Outline of Complex Variables, Second Edition" by Murray Spiegel.This has plenty of solved and unsolved exercises ranging from the basics on complex numbers, to special functions and conformal mappings. This has a note on the zeta function.
"Geometric Function Theory: Explorations in Complex Analysis" by Steven Krantz. This is good for more advanced topics in classic function theory, probably suitable for advanced UG/PG. It covers classic topics, such as the Schwarz lemma and Riemann mapping theorem, and moves onto topics in harmonic analysis and abstract algebra.
"Complex Analysis in Number Theory" by Anatoly Karatsuba.This book contains a detailed analysis of complex analysis and number theory (especially the zeta function). Topics covered include complex integration in number theory, the Zeta function and L-functions.
Note: I only mean this answer to be an addendum to all the other answers. In particular, the following books are probably not the best books for someone at an "intermediate sophistication level for an undergrad."However, I also think these (very good) books will be of help to future readers. Also, they were not mentioned in the other duplicate posts (here and here).
Since there were a few other graduate level books mentioned above, I thought this answer is also appropriate. Perhaps this book is best for a second course on complex analysis. The first two chapters are content from standard undergraduate complex analysis.
This (very old) book is good if you want to learn to do hard calculations. It is hard to read, but personally, I think it is a very rewarding book. Same with Schlag's book, this may not be a good first course in complex analysis, but it may be good once you have learnt the basics after reading more basics books such as Stein and Shakarchi.
Sorry I can't offer too many details, it's been a long time. Let's see, standard stuff like Laurent series, complex numbers, Cauchy's theorem, Goursat on the way to Cauchy, Euler's formula etc. Not in that order.
(One more incidentally, I know it's a bit much, but for what it's worth, I was able to get a marginal pass on the complex analysis QUAL at UCLA before starting grad school there, based mainly on what I learned from the course.)
Lots of good recommendations here-but for self study,you can't beat Complex analysis by Theodore W. Gamelin. It's highly geometric, has very few prerequisites and reaches very near the boundaries of research by the end.
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In the past few years genome-wide association (GWA) studies have uncovered a large number of convincingly replicated associations for many complex human diseases. Genotype imputation has been used widely in the analysis of GWA studies to boost power, fine-map associations and facilitate the combination of results across studies using meta-analysis. This Review describes the details of several different statistical methods for imputing genotypes, illustrates and discusses the factors that influence imputation performance, and reviews methods that can be used to assess imputation performance and test association at imputed SNPs.
In the first year we discussed the basics - Linear Algebra, Ordinary Differential Equations, Real Analysis and Probability. In the second year we built on those basics, studying Metric Spaces, the Riemann Integral, Group Theory and calculus on Vector Spaces.
In the third year of a four-year masters-level course, especially one with an applied focus that will be of interest to quants, we need to begin thinking about more abstract concepts that will prepare us for study of Stochastic Calculus, Probabilistic Machine Learning and Bayesian Econometrics.
Both of these courses contain ideas that underlie Probability Theory, Time Series Analysis and some aspects of Machine Learning. Measure Theory teaches us about generalising the Riemann Integral to the Lebesgue Integral, while Linear Functional Analysis discusses function spaces, many of which are necessary for solutions to certain Partial Differential Equations.
At this level there are less video lectures available, since the content becomes quite complex. However, there are still plenty of accessible textbooks and lecture notes, many of which contain questions and solutions to test your knowledge.
Complex numbers are a generalisation of real numbers motivated by the need to define the concept of $\mathbbi=\sqrt-1$. This comes about because of the solution to particular equations, such as the familiar quadratic equation from highschool algebra, which possesses complex roots when $b^2 - 4ac \lt 0$ in the solution to the equation $ax^2 + bx + c = 0$.
Complex analysis may be seen to be quite abstract and not immediately applicable to "real world" situations that are often modelled to take place in $\mathbbR^n$. However it is a highly applicable subject in various areas including quantum mechanics (the Schrodinger equation), fluid dynamics (via conformal mappings) and electrical engineering (signals analysis/Fourier transform, control theory etc).
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