Wikipedia:Reference desk/Archives/Mathematics/2016 February 8

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February 8

"Opposite" of Normal Distribution

What is the equation of the normal distribution turned up-down? Does this distribution have some name in the literature? עברית (talk) 08:38, 8 February 2016 (UTC)

If you mean a distribution with an inverted bell shape, it can't be that simple, because its integral would be infinite. —Tamfang (talk) 08:54, 8 February 2016 (UTC)

Oopss.. Thanx! עברית (talk) 10:41, 8 February 2016 (UTC)

As for where it might occur, I can imagine the penumbra under a moderately high object, like a flag, would reach a minimum brightness near the center. StuRat (talk) 18:28, 8 February 2016 (UTC)

Elementary Proof of an Integral Identity involving Bessel functions

It is well known that, for positive values of a, $\int _{0}^{\frac {\pi }{2}}\cos(a\cos x)dx=\int _{0}^{\infty }\sin(a\cosh x)dx={\frac {\pi }{2}}J_{0}(a).$ I was wondering whether it is possible to prove the first half of the identity in an elementary manner, without any explicit recourse to Bessel functions and their various properties.

I've tried writing them both as $\int _{0}^{1}{\frac {f_{1,2}(ax)}{\sqrt {1-x^{2}}}}~dx,$ with $f_{1}(t)=\cos(t)$ and $f_{2}(t)={\frac {\sin(1/t)}{t}},$ and then expand ${\frac {1}{\sqrt {1-x^{2}}}}$ into its binomial series, and reverse the order of summation and integration, but the general terms of the two series are not equal (not to mention the fact that each is expressed in terms of incomplete gamma functions and/or exponential integrals of imaginary argument). — 79.118.187.240 (talk) 13:11, 8 February 2016 (UTC)

It's not too hard to show that both satisfy the same second order differential equation (which turns out to he the Bessel equation). Sławomir
Biały 13:34, 8 February 2016 (UTC)

Grams in milliliters

What is 100g plain flour in milliliters? 2A02:8084:9360:3780:141:A29C:2CFA:4D6F (talk) 14:58, 8 February 2016 (UTC)

It depends on the flour, how packed it is, and if it is from wheat, rice, barley, or other. There should be an equivalence on your package of flour for volume to weight, that you can use. Based on mine, 100g is about 0.195l or 195 milliliters. It will vary. Dhrm77 (talk) 16:13, 8 February 2016 (UTC)

One other factor is how dry the flour is. StuRat (talk) 17:25, 8 February 2016 (UTC)

That question (probably) doesn't belong on this page. This is about Math, not measurements. Dhrm77 (talk) 16:15, 8 February 2016 (UTC)

I disagree. Conversion of units involves math, and not much else, so this is the place for it. StuRat (talk) 17:24, 8 February 2016 (UTC)

It does involve more than math, it involves actual volume/weight ratios and unit conversions. The question might be better located here: Reference desk/Miscellaneous or in places like ask.com or quora.com. Also, googling the question gives various answers.Dhrm77 (talk) 18:11, 8 February 2016 (UTC)

Volume/weight ratios and unit conversions are maths. They're the practical maths which make everyday life possible. DuncanHill (talk) 18:24, 8 February 2016 (UTC)

I have a cooks dry measure which gives 100g of flour as a shade under 200 ml. DuncanHill (talk) 16:17, 8 February 2016 (UTC)

As Dhrm77 correctly said above, this is a science question about the density of flour. This site gives 593 kilograms per cubic metre as the typical density of wheat flour (density will depend, of course, on the material from which the flour is ground, and, to some extent, on the degree of fineness of the grind). That figure converts to only 169 ml for 100g. I haven't checked it by experiment. I suspect that another variable will be dampness (as mentioned above) and compression (a cubic metre of flour will be firmly compressed at the bottom). Dbfirs 21:08, 9 February 2016 (UTC)

... later ... This site says only 528 grams per litre for sifted white wheat flour, so that value of 189 ml per 100g corresponds more closely to the answers above. I suspect that sifting introduces air which reduces the density. Dbfirs 21:44, 9 February 2016 (UTC)