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It is written in Riemann Surfaces (Oxford Graduate Texts in Mathematics) by Simon Donaldson, that:

"[t]he theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus"

Can someone please provide an articulated commentary on this statement.

Specifically, the statement suggests, [or seems to suggest], that Riemann surfaces were the logical / mathematical outcome of many years of careful development and refinement of traditional calculus. But: (i) what was / were the major milestones(s) in this road? and (ii) when the author uses the word 'culmination' what specifically is it the culmination of, and what problems / issues did the introduction of Riemann surfaces help to solve / clarify / etc.?

(This question was originally posted on Math SE, but I'm also posting it here because I'm seeking an expert's [in Riemann surface theory] feedback if possible.)

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closed as off topic by quid, Pete L. Clark, Qiaochu Yuan, Andy Putman, Felipe Voloch Sep 15 '11 at 18:41

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

If you consider traditional calculus to mean calculus in one variable, then that leads to complex analysis in one variable. At first it was done on C, but for more flexibility (particularly in relation to speaking about analytic continuation without awkward branch cuts) Riemann introduced the idea of doing complex analysis on a one-dimensional complex manifold, and those are essentially the same thing as Riemann surfaces. – KConrad Sep 15 '11 at 13:23
Vote to close. And -1, for again crossposting; three hours is a ridiculously short time to wait. – user9072 Sep 15 '11 at 16:01
@Ahmed, thank you for letting me know about Donaldson´s book. – James O Sep 15 '11 at 16:27
@Sadiq : mathematics-education is used for questions about teaching, not learning. I think the FAQ makes it clear that MO is intended for professional mathematicians. Also, the MO users who are interested in answering questions from undergraduates already read math.SE.. Finally, while it is true that questions about mathematical history are welcome here, your question is not really a serious math-history question. If you already knew about Riemann surfaces and complex analysis, the answer would be pretty clear. – Andy Putman Sep 15 '11 at 17:29
I'm not going to express an opinion about whether or not the question should be closed, but I want to give my interpretation of why others object. First, the mathematics-education tag is not appropriate for your question; it is meant for questions about mathematical pedagogy. (Actually, most questions with this tag would not be considered research level within the math education community, but I don't think there are enough experts in the field who frequent this site for the tag to be enforced.) Your question is about Riemann surfaces, not about education. – Paul Siegel Sep 15 '11 at 17:32

From the wording of your question it is possible you are asking someone to write an entire historical overview for you. So instead what I did was spend a few minutes on Ye Olde Google and found this:

The Concept of a Riemann Surface by Hermann Weyl. It is cheap and your local library might have it already it.

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This is more of a technical book. I was searching for an outline-type response. – Sadiq Ahmed Sep 15 '11 at 15:11

If you are looking for an outline, check Chapter III of "The Riemann legacy: Riemannian ideas in mathematics and physics" By Krzysztof Maurin

Here is the link to Google Books, where you can view the table of content

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