Wikipedia:Reference desk/Archives/Mathematics/2012 January 19

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January 19

Mathematical elegance?

Hi. Suppose you've just completed the parts integral $\int {e^{x}\cos(x)dx}$, then you suddenly come across the integral $\int {e^{-x}\cos(x)dx}$. You could compute it but room for careless error would be created by the negative, since you have to apply parts twice. But by the simple, almost comic substitution u=-x, you have the exact same integral you had before but for a negative out front, because you have $-\int {e^{u}\cos(-u)dx}$ but cosine is an even function! In your opinion (not a usual question in math :), would this be considered an example of mathematical elegance, however simple? 84.208.197.187 (talk) 23:13, 19 January 2012 (UTC)

If we're computing an antiderivative, I suppose that would be the way to do it. If we were computing a definite integral, I'd sooner point out that $\int {e^{-x}\cos(x)dx}=\int {e^{-x}\cos(-x)dx}$, because $\cos$ is an even function as you pointed out, and that it geometrically follows that we can equate the area to a corresponding area under $e^{x}\cos(x)$, because that argument is clearer pictorially for me, personally. I might then point out that that observation is encoded in the substitution u=-x, but only to develop connections between multiple ideas, not to promote that as a method for solving this particular problem.

The question of elegance is, of course, a subjective one. I would certainly say that this is a nice example of the reason why I like mathematics; I enjoy the problems that appear dauntingly tedious or even impossible up front, but they turn out to have a nice trick that turns them nearly trivial. "Elegance" isn't necessarily the word I would have used to describe this particular problem, but if you're just asking whether it's the sort of thing a mathematician enjoys, my personal answer is yes. I hasten to add that I can guarantee you that there are mathematicians who disagree with my own personal bias here. --COVIZAPIBETEFOKY (talk) 23:44, 19 January 2012 (UTC)

We have an article on the Teakettle principle. Bo Jacoby (talk) 08:14, 20 January 2012 (UTC).

I hope you don't mean to imply that that principle describes every manifestation of the phenomenon I described. --COVIZAPIBETEFOKY (talk) 13:43, 20 January 2012 (UTC)