Wikipedia:Reference desk/Archives/Mathematics/2013 July 29

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July 29

Inconsistency in definition of figurate numbers.

There seems to be an inconsistency in the manner by which figurate numbers are defined. Triangle, Square and Hexagonal are generated by determining the figure that matches their respective grids with a minimum side equal to the figurate number and then the other figurate numbers, 5, and 7 up seem to be generated so they sort of match the hexagon (center entry and then wrap perimeters of the shape n times until you get to the size. This means that for the figurate shapes larger than 4, they are basically equal to x*Tsubn +1 where x is the number of sides and Tsubn is the nth triangular number (in the current sense). (or no center entry and they are just x*Tsubn)

However this would be viewed as very odd for Triangular and Square. The triangular numbers would go 1, 4, 10, 19, 31, etc and the square numbers would go 1, 5, 13, 25, 41. Is there any way to get any real consistency here other than to define the triangle, square and hexagon ones in the current manner and just ignoring the rest? This isn't really dealt with in Figurate numbers.Naraht (talk) 15:56, 29 July 2013 (UTC)

There are two separate families of two-dimensional figurate numbers - polygonal numbers and centered polygonal numbers. Is it possible you are confusing the two types ? For example, the sequence of centered triangular numbers does indeed start 1, 4, 10, 19, 31 ... Gandalf61 (talk) 16:18, 29 July 2013 (UTC)

I think that covers the difference, however from an external standpoint, I think the polygonal numbers "feel more right" for the triangle and square, and the centered polygonal number for a hexagon. For example, the hexagonal numbers (which are "cornered") just look odd to me.Naraht (talk) 16:57, 29 July 2013 (UTC)

Differentiation under the integral sign without absolute convergence

From the article Differentiation under the integral sign#Other problems to solve there's a clever little trick:

$\int _{0}^{\infty }\;{\frac {\sin \,x}{x}}\;dx\to \int _{0}^{\infty }\;e^{-\alpha \,x}\;{\frac {\sin \,x}{x}}\;dx$

The idea is that differentiating under the integral sign gives an easier integral, and then setting α = 0 lets you evaluate the original integral. But as for the validity of this method, the article just says "The first integral is absolutely convergent for positive α but only conditionally convergent when α is 0. Therefore differentiation under the integral sign is easy to justify when α > 0, but proving that the resulting formula remains valid when α is 0 requires some careful work." Unfortunately the article doesn't give a lot of clues about when differentiating under the integral sign is valid. In fact this sentence is the only reference to "absolute convergence" in the entire article, so even the requirements for the more basic case are not expounded at any point. In the more complicated case of α = 0, what is the "more careful work" that the article is hinting at? (This is related to an earlier question on which I got a helpful response but I think is sufficiently different to justify a new question.) ManyQuestionsFewAnswers (talk) 21:50, 29 July 2013 (UTC)

There are actually two limits that need to be commuted with the integral. First, for $\alpha >0$, you need to justify that differentiation under the integral sign is justified. This can be shown by the dominated convergence argument, and requires that (for each fixed $\alpha$) the difference quotients

${\frac {1}{h}}{\frac {\sin x}{x}}(e^{-(\alpha +h)x}-e^{-\alpha x})$

be bounded by an absolutely integrable function of x (which in this case is true, so differentiation under the integral is justified). The second limit needed is that

$\int _{0}^{\infty }{\frac {\sin x}{x}}\,dx=\lim _{\alpha \to 0^{+}}\int _{0}^{\infty }e^{-\alpha x}{\frac {\sin x}{x}}\,dx.$

Since the first integral doesn't converge absolutely, dominated convergence doesn't work and you have to use a direct $\epsilon \delta$ argument (or emulate the proof of Abel's theorem). Sławomir Biały (talk) 23:01, 29 July 2013 (UTC)

This is a very helpful reply thanks. I will go away and try to put some flesh on this skeleton argument to see if I have absorbed it correctly - would be too greedy of me to ask others to fill it in! I do think the Differentiation under the integral sign could really be improved by inclusion of some of this material, especially the dominant convergence argument. 23:21, 29 July 2013 (UTC) — Preceding unsigned comment added by ManyQuestionsFewAnswers (talk • contribs)