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I have been told by different people that probably no subject can unify algebra, analysis and geometry better than Riemann surfaces. Regardless of how true it is, I'm looking for a textbook that explains the basic ideas and theorems of Riemann surfaces with a fairly reasonable background which includes undergraduate algebra, undergraduate analysis and undergraduate geometry.

In other words, the audience of the book should be advanced undergrad students. Since I want it for self-study, I'd really like to find a book that has solutions. If not, then a textbook with graphics, drawings or intuitive explanations would suit me the best.

I'm tagging this question as 'reference-request' and 'soft-question'. I will really appreciate it if you share with me your pedagogical experience or your own troubles when you wanted to get introduced to Riemann surfaces. Any piece of advice about how to approach the subject is welcome and highly appreciated

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    $\begingroup$ @GNUSupporter8964民主女神 地下教會 Thanks for the edit and adding the relevant tags. $\endgroup$ – stressed out Mar 2 at 6:28
  • $\begingroup$ It doesn't cover Riemann surfaces, but you might take a look at "Visual Complex Analysis" by Needham as a preliminary. $\endgroup$ – awkward Mar 2 at 13:05
  • $\begingroup$ @awkward Thanks for the suggestion. I love that book. It's been one of my goals for a long time to finish reading it. $\endgroup$ – stressed out Mar 2 at 16:14
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    $\begingroup$ It is not what you are asking for, but according to your background and interests, I recommend you Geometry of surfaces by J. Stillwell $\endgroup$ – Dante Grevino Mar 8 at 1:05
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Take a look at the last chapter of Carleson and Gamelin "Complex Analysis". The book is aimed at UCLA undergraduate students and it has exercises.

One more option is:

Narasimhan, Nievergelt, "Complex Analysis in One Variable". It is aimed a bit higher than Carleson and Gamelin (and is aimed at 1st year graduate students), but it covers more material.

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  • $\begingroup$ Thanks (+1). I checked the book. Could you please tell me which chapters are necessary to understand the last chapter? Do I have to read the book thoroughly or can I just ignore some chapters and still understand the last chapter? $\endgroup$ – stressed out Mar 7 at 4:22
  • $\begingroup$ @stressedout How much complex analysis do you already know ? $\endgroup$ – Moishe Kohan Mar 7 at 4:39
  • $\begingroup$ I have Complex Functions this semester but I have already self-studied the material up to Cauchy's integral theorem and Rouche's theorem. I know the Residue theorem and Laurent series, for example. $\endgroup$ – stressed out Mar 7 at 4:43
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    $\begingroup$ @stressedout: Then read chapters 5, 9, 10, 11, 15 and then 16 (Riemann surfaces) of Carleson-Gamelin. You consult other chapters when needed. One thing to know: their book covers only one of the two cornerstones of Riemann surfaces but not the other: They do not discuss the Riemann-Roch theorem. For that I do not know of any undergraduate-level treatment. $\endgroup$ – Moishe Kohan Mar 7 at 16:22
  • $\begingroup$ Thank you. If I manage to understand these chapters and finish chapter 16, I will probably read Otto Foster's Lectures on Riemann surfaces or Jurgen Jost's Compact Riemann Surfaces: An Introduction to Contemporary Mathematics. I have heard good things about the later one. Do you think if I understand chapter 16 of T Gomelin's book I will be ready to read and understand those two books I mentioned? Or if you don't know those two books, what do you think my next step should be? $\endgroup$ – stressed out Mar 8 at 6:34

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