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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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    Short answer: it's calculus, but with way more rigor.2012-05-02
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    [Long answer: wikipedia](http://en.wikipedia.org/wiki/Mathematical_analysis)2012-05-02
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    Also, "should be pretty easy to answer" translates to "is pretty easy to google" in this case.2012-05-02
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    @J.M. Which specific calculus do you mean? Differential and integral calculus? Or something else?2012-05-02
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    I don't understand the negative votes. Sounds like a good question to me.2012-05-02
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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.2012-05-02
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    @FortuonPaendrag: I understand that. But that means that The Chaz and others seem to think that the Wikipedia article is good enough, which is not that obvious to me. I would actually be very happy to see what short answers to the question can be given by users of this site; I often see refreshingly different points of view here, and I feel that would fit this question.2012-05-02
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    @Martin: What's there to understand? Two of the three criteria explicit in the popup for a downvote are (paraphrased) "no research effort" and "not useful". The potential for great answers doesn't make this (even) a good question.2012-05-02
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    @Martin: downvote popup, not "vote to close" popup2012-05-02
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    @TheChaz: my bad. I'll erase that comment, as it is based on a false premise. I still disagree on the the downvoting, though, as "no research effort" implies that the asker is looking for a concrete response, while this question sounds to me like someone looking for an opinion. And I still think that "useful" in a place where most questions are about pure math and there is a tag "terminology", is a tricky thing to define.2012-05-02
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    @Martin, "looking for an opinion" is exactly what m.se is *not* here for.2012-05-03
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    I read the Wikipedia article, and felt unenlightened. I thought it was a reasonable question to ask here... Apparently nobody else thinks so. Oh well.2012-05-03
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    @GerryMyerson: are you implying that there is a single, canonical, exact answer to what "analysis" means, and that such an answer is the one given in Wikipedia?2012-05-03
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    @Martin, I can't imagine how you got that from my comment. You wrote, "this question sounds to me like someone looking for an opinion." One of the reasons we are permitted to close questions is "This question is not a good fit to our Q&A format. We expect answers to generally involve facts, references, or specific expertise; this question will likely solicit opinion, debate, arguments, polling, or extended discussion." See? Questions that solicit opinion are "not a good fit" to the m.se format, so we close them.2012-05-03
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    @Gerry: as well as I wrote "looking for an opinion" I could have written "looking for expertise", and then it would have been a good fit? You even wrote an answer which I feel contributes to the topic. Why would the question be wrong, then?2012-05-03
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    @Martin, if you had written "looking for expertise" instead of "looking for an opinion," I would not have pointed out to you that "looking for an opinion" is grounds for closure. I don't know why you are asking me why the question would be "wrong", as I have done nothing to indicate that I have that view.2012-05-03
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    I voted to close the question because in my opinion it is way too broad. This site is for *focused* questions. If you want to get a taste of what "analysis" is, you could look at the wikipedia article. If you don't like that, you can look in the references it gives. If you don't have access to them you can google "real analysis online text". And so forth. If *none* of these things are helpful, why do you think that someone writing a few paragraphs on the fly will be helpful?2012-05-04
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    Also, you haven't provided any information about your own background, interests, specific concerns, etc., so how is someone supposed to tailor an answer to you? Without this information, we're just rewriting a little bit of the wikipedia article, and that seems like a waste of everyone's time.2012-05-04

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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.