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This one should be pretty easy to answer...

When mathematicians study "analysis", what are they actually studying?

(In particular, the phrases "real analysis" and "complex analysis" get banded around a lot. But what do they mean?)

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closed as not a real question by t.b., lhf, Pedro Tamaroff, robjohn, Pete L. Clark May 4 '12 at 5:45

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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Long answer: wikipedia –  The Chaz 2.0 May 2 '12 at 15:43
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Also, "should be pretty easy to answer" translates to "is pretty easy to google" in this case. –  The Chaz 2.0 May 2 '12 at 15:43
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@J.M. Which specific calculus do you mean? Differential and integral calculus? Or something else? –  MathematicalOrchid May 2 '12 at 15:43
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@MartinArgerami : I do not think it is the content of the question that people object to. It does not appear that the OP has made any effort(Googling/wikiing etc) before posting this questions. TheChaz stated this most eloquently above. And this,IMHO ,is what brought the downvotes. –  Ravi Donepudi May 2 '12 at 17:08
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@Martin, "looking for an opinion" is exactly what m.se is not here for. –  Gerry Myerson May 3 '12 at 6:43
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2 Answers 2

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In my view, the "one-sentence" answer is that Analysis is all the math that somehow deals with "limits", with "approaching things". Now, this is very broad, and it is possible that things fit that definition that I wouldn't call "analysis". In the end, I guess that as one goes through the stages of learning the trade, one gets a feeling of what "analysis" is. And, of course, this feeling will differ among people, that's why I would love to see several answers here.

Regarding the particular terms "real analysis" and "complex analysis", I would say that

1) "real analysis" stands for the study of the real numbers and their functions, with emphasis in continuity (and many might disagree with this definition, and that might even include me!)

2) "complex analysis" stands of the study of complex numbers and their functions, with emphasis on differentiability (this because differentiability of complex functions is a very strong property with wonderful consequences).

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To some extent it depends on where the mathematicians are. German undergraduates often have Analysis on their transcripts as a first-year course, where Americans (say) would have Calculus. The German first-year courses may be more rigorous than the American, but I don't think they rise to the level of Analysis on an American transcript, which would typically be a third-year course.

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