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Dear Anixx,
Mathematics is a very broad subject with many subfields.
The area of classical analysis that you work in is only a very small niche occupied by a very small fraction of mathematicians. In fact, given contemporary taste, it is quite far off the beaten track, so the fraction of mathematicians that know it is even smaller than for most subfields.
I would estimate less than 1% of mathematicians know what a polygamma function is. I haven't asked, but I would expect no mathematicians in my department of roughly 15 know what it is.
You also have to consider the population of MathOverflow. Algebraic geometry and logic are quite highly represented, whereas there are much fewer people from other areas. This means many readers of MathOverflow understand what a Deligne-Mumford stack is, considerably more than will understand what a polygamma function is.
@markvs:
The sum total of my analysis background is one semester of undergraduate real analysis, one semester of undergraduate complex analysis, and one semester of point set topology. I have never actually learned the definition of the gamma function (though I know it is some holomorphic function which specializes to factorial at integers and satisfies some simple functional equation that I have forgotten).
I never studied multivariable calculus; everything I know about the subject I learned while learning about differentiable manifolds. I never studied differential equations; everything I know about the subject comes from the half dozen or so times I have taught the introductory undergraduate course on the subject at various places.
I am not proud of my lack of background, but I am not particularly embarrassed by it. My lack of background is uncommon, but it is shared by a nontrivial number of US trained algebraists and combinatorialists. (Surely it says something that my then department chair let me teach differential equations the first time!)
In the other direction as it were, a university near mine with a mathematics PhD program offers only one graduate course in algebra. It has no algebraists on its faculty. I know this because they send their students here if they need or want to study more algebra. Students there regularly get PhDs without knowing what a module is. I believe some of these students go on to become mathematicians of good repute in their subfields without every learning what a module is.
Keep in mind that less than a third of my undergraduate education was on mathematics. In fact, at my previous job, students were forbidden from devoting more than 35% of their undergraduate studies to any single subject. (I think this is a good policy at least a certain types of schools.)
Well, I don't think Alexander, or indeed anyone, should be embarrassed by what they don't know. None of us knows more than a tiny portion of all the mathematics there is.
Being too embarrassed to admit one's ignorance is a major problem in mathematical culture. I would prefer to be supportive of people who say openly that they don't know stuff, rather than try to shame them. It takes bravery to be publicly honest about one's limitations.
Yes. Please define (or provide reference) for such things as $\sum_x f(x)$ when you use them in your question.
@Mark: Cut us a little slack.. At least we're not this guy: He-who-I-will-not-name
@David Speyer: Great comment. Thanks for taking the time to write it.
Gosh, I should turn in my analyst's badge. I never heard of the polygamma function before looking at this thread (during an afternoon off at an analysis conference in Banff) despite my 44 years in the profession. Now that I have learned what it is, I will no doubt forget the definition within an hour or so but hope that it resurfaces in my memory in case I should ever have need of it.
+! Bill Johnson. (link for you, Anixx)
Anixx, you should at least define symbols. That the polygamma function is denoted by a psi and not a gamma (or Gamma) would never have occurred to me without following a non-self-explanatory link and reading a paper/page (I didn't know you were linking to a wikipedia page, I thought it was an article). The FAQ asks that questions are pretty much self-contained. Merely saying at the beginning of the post "Let psi^(a)(x) be the polygamma function, where x takes values in (blah) and a takes values in (blah), please see this paper (link) for background." would have saved a lot of trouble. I hope that this helps you in further questions. I think it is unfortunate that people seemed to have jumped on some sort of down-voting bandwagon for the question that started this.
Bill: lol! I wonder if there are Fields Medalists out there who should be handing back their medals.
+1 David S
I'm pretty sure the State of Texas takes a chunk of Bill's salary in taxes, so that's already taken care of Mark. IMO it's best to contribute positively, rather than throwing out passive-aggressive insults. Mathematics is a big world and what is standard and important doctrine to one person might be something another person never pays attention to, and might even consider not very useful or even insightful. This isn't unusual. We live in a world where very different perspectives on the same thing are usually accepted, and even appreciated, I hope.
Ah. Well I imagine they get part of his salary back in things like sales taxes, then. Bill should go out and buy a beer. If my Googling skills are true, Texas taxes that at a 6.25% rate.
Well, I am in Canada right now, and it collected taxes on the wine I bought in town last night (as I got tired of drinking beer at the Banff Center), so Texas will have to wait a few days to collect more taxes from me. As for Mark, he is jealous because my title is longer than his.*
*Mark was just joking with someone (me) he knows, just as I am jesting now with him.
(May I suggest that friendly interchanges in the form of unfriendly interchanges be more clearly marked as such? The forum is read by many people who have absolutely no way of knowing which is which —or who knows who— and it would be good to make it clearer for them! This does reflect on the image of MO as a friendly place.)
@markvs: I did not know you and Mariano are friends!
Oh well.
I second Mariano's parenthetical remark. I have no idea who markvs is outside meta.MO and my impression of him (her?) here is that he (she?) does not hold back. So my reading of the initial exchange was that markvs was being quite rude to Bill. I'm pleased to hear that that is not the case.
I'm quite prepared to be told that I have no sense of humor, since I don't. I'm also, to return a little closer to the main topic, clearly an absolute ignoramus as I have no idea what the "polygamma" function is (is it to do with Greek parrots?) and only a hazy idea of the gamma function. I am not ashamed of this ignorance whatsoever: if I ever need to know, I'll look them up on the nLab. If they're not there, then clearly they aren't important.
As for whether or not the question should have explained these terms, that's up to the original poster. But someone who comes here to ask a question should remember that they come as a supplicant: they are trying to get someone to do something for them with no real reward. So when told, "It would help if you did X", it might be worth considering whether or not X will help get the question answered, or if not doing X will throw up such a furore that the actual question gets lost in the cross-fire. Then it might be worth just doing X instead of standing on some principle.
Regarding Anixx's last comment, and assuming for sake of argument that questions in algebraic functional analysis do not count as "math analysis questions",
http://mathoverflow.net/questions/85266/many-brownian-motions-moving-together/85297#85297
http://mathoverflow.net/questions/19079/what-is-the-status-of-the-gauss-circle-problem/20923#20923
http://mathoverflow.net/questions/43462/existence-of-a-smooth-function-with-nowhere-converging-taylor-series-at-every-poi/81479#81479 http://mathoverflow.net/questions/84958/is-sum-limits-n0-infty-xn-sqrtn-positive/85048#85048
just by skimming over the contributions of one expert.
It may be the shameless bully in me, but I do not think there is any point to this thread anymore.