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Ref: The Road to Reality: a complete guide to the laws of the universe, (Vintage, 2005) by Roger Penrose [Chap. 7: Complex-number calculus and Chap. 8: Riemann surfaces and complex mappings]

I'm searching for an easily readable and understandable book (or resource of any kind; but preferably textbook with many worked-out problems and solutions and problem sets) to learn complex analysis and basics of Riemann surfaces - and applications to theoretical physics. (Particularly: material geared towards / written with undergraduate-level physics / theoretical physics students in view).

Any suggestions?

My math background: I have a working knowledge of single- and multivariable calculus, linear algebra, and differential equations; also some rudimentary real analysis.

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    I liked Howie's _Complex Analysis_, but I have yet to find a decent Riemann surfaces text...2011-09-14
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    [Crosspost from MO](http://mathoverflow.net/questions/75366/primer-on-complex-analysis-and-riemann-surfaces-for-undergraduate-physics-theor)2011-09-14
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    I've looked through the posts with that tag. I found mostly questions similar to mine. ... I think the posts with that tag are different for the two sites. ... (Need clarification here.)2011-09-17
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    Ref: http://math.stackexchange.com/questions/tagged/education ... Plus, the description field is empty. ...2011-09-17
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    Ok. Thanks. - regards2011-09-17

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I don't think you'll be able to read a book on Riemann surfaces before you study complex analysis. Once you've finished learning the rudiments of complex analysis, I recommend Rick Miranda's book "Algebraic Curves and Riemann Surfaces". It probably is the gentlest introduction I know.

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    I've cursorily glanced through Weyl's "The Concept of a Riemann Surface". But not sure it might be the right fit for my needs- I'm searching for a work to allow me to learn about Riemann surfaces as hassle-free as possible [from a physics / theoretical physics point of view] - without the complications of pure mathematics; and possibly with substantial example questions-and-solutions and problems sets at ends of chapters.2011-09-14
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    (Plus, Weyl's book seems to be outdated for some reason.)2011-09-14
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    You definitely should not try to read Weyl's book. It's a great historical document (for instance, the original edition of it contained the first modern definition of a manifold in terms of charts and atlases). However, it is not an easy read. I'm not sure what you mean by a book "without the hassles of pure mathematics". Do you mean a book that doesn't prove anything? If you don't like the proofs, skip them. In any case, Miranda's book has tons and tons of examples and good (but pretty easy) exercises at the end of each section.2011-09-14
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    "... without the hassles of pure mathematics" - a book which provides a good 'feel' / intuition for the subject-content, ie. isn't too focused on being rigorous and [math] theory-oriented. (Plus, it's a definite plus if it included many examples from physics, but it's not absolutely required.)2011-09-14
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    I don't think that you'll find a book like the one you describe, but again you might like Miranda's book. At the very least, it spends a lot of time working out very explicit examples.2011-09-14
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    (While we're on this specific topic: was wondering what made Jost's "Compact Riemann surfaces" such a good 'introduction' to modern mathematics? .. I've been - and kind of still am - under the impression that there's general divisions in pure maths - and within each there is a body of well developed theory (with, no doubt, connections with others): but why specifically is "Compact Riemann surfaces" unique the way the book seems to suggest?2011-09-14
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    (i.e. was 'classical' maths mostly geometry, algebra, and mostly real analysis, while complex analysis - and the introduction of Riemann surfaces - the "transition" stage to 'modern' mathematics?)2011-09-14
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    (Incidentally, Penrose's book sort of delves cursorily into this: so was curious where the 'classical'-'modern' transition lies - in terms of historical period and mathematical content, and what / how Riemann surfaces have to do with it.)2011-09-14
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    If you look at the preface to Jost's book, he is not claiming that his book will give students a broad overview of "modern mathematics", whatever that is. Instead, he points out that the subject of compact Riemann surfaces is connected to a huge number of topics in mathematics while remaining relatively accessible, so it is a good first place to see the tools one would learn early in one's graduate education combined and used to prove some deep theorems. I don't think he is making any claim about "modern" vs "pre-modern" mathematics or anything like that.2011-09-14
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    Okay. Thanks for the note. But I'm still a bit curious (mystified) regarding the supposedly 'classical' vs. 'modern' maths: was it in the course of the gradual development of the various subject-areas: i.e. the emergence of non-Euclidean geometry via the 5th axiom of Euclid being overturned [Geometry]; the steady progress of algebra (from insolubility of quintic equation onwards, to using complex analytic ideas for the proof of the Fundamental Theorem of Algebra [Algebra]); and the development of complex analysis as a progression from calculus / real analysis [Analysis]... (cont'd)...2011-09-14
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    ... which led finally to the mathematical ideas that are now referred to as 'modern' maths?2011-09-14
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    ie. was Riemann the first synthesizer of these streams, and the one who brought forth their essential unity via his use of the "Riemann" surface ideas? ...2011-09-14
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    I'm skeptical that one can draw a sharp distinction between "classical" and "modern" math except for the trivial observation that a lot of math has been developed over the past 100 years. Certainly there is no "definition" of what makes math modern other than the fact that it was developed recently. You can shovel the bullsh*t around, but I don't think there is much value in that.2011-09-14
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    Many popular books on history of maths refer to Riemann as a "towering" figure of mathematics. But I don't understand what makes a "towering" figure in maths: all mathematicians do math. They're all doing the same thing. So what made him so 'towering' from the rest of his contemporaries? [I know what I wrote above comes of a bit pretentious, but I'm not trying to be pretentious - just trying to understand what made him so important vs. other mathematicians of his time.]2011-09-14
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    Isn't that kind of a silly question? He's important because he proved a lot of great theorems and introduced a number of concepts that subsequently became central to mathematics. The same way anyone doing anything becomes a "towering figure" -- by doing great work.2011-09-14
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    @UGPhysics: Among many results in real and complex analysis,differential geometry and number theory(the greatest as-yet unsolved problem of number theory bears his name),Riemann was responsible for the abstract definition of a surface,which lead directly to the concept of a topological manifold.He was also a famed physicist, responsible for many results in mechanics and fluid dynamics. See Morris Kline's classic history of mathematics for details of the remarkable life and career of Riemann.2011-09-18
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Theodore Gamelien's Complex Analysis requires only basic calculus, is very geometric and covers just about everything a mathematics undergraduate or graduate student needs to know about functions of a complex variable.It includes an excellent and very basic introduction to Riemann surfaces. For all those reasons,it's a must have for any mathematics or physics student at either level.

A bit more sophisticated but equally wonderful is Singerman and Jones Complex Functions:A Geometric and Algebraic Approach-an incredibly rich and sophisticated second course for students who have already had an "epsilon-delta" type complex variables course and need to learn about the less analytic aspects of the subject. There's a terrific introduction to Riemann surfaces and meromorphic functions.

There's LOTS of others-the already mentioned book by Needham is a treasure,if you're willing to overlook its sometimes loose approach.That should be more then enough to get you started!

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    I totally agree! I like Jones and Singerman book. However, Gamelin's Complex Analysismight be more readable.2011-10-04
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You may enjoy Needham's Visual Complex Analysis.

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    I have his text. Penrose especially recommends it, but for Riemann surfaces I'm still searching for a good starter. ... Any suggestions there?2011-09-14
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On the reference on Riemann surfaces: Forster's book "Lectures on Riemann Surfaces" is a great book that is easy to read and has many exercises (of course assuming knowledge of single variable complex analysis).

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As a beginning theoretical physics graduate student, incidentally I am these days going through the book "Introduction to Algebraic Curves" by Phillip Griffiths and it does seem to be a nice book for Riemann surfaces and such stuff. Also Jost's book on the same topic is great but focused in a different direction. For various things in complex analysis that I am once in a while getting stuck with, I seem to be able to pick them up from the book by Eliash Stein and Ramishakarchi. Somehow this combination of books does seem to work for me at least till now.

As a physics student one has to enter mathematics "laterally". Thats part of the challenge and thrill of physics whether or not the mathematical "purists" cringe about it :)