Wikipedia:Reference desk/Archives/Mathematics/2015 April 22

= April 22 =

Euclid's theorem
Go to Euclid's theorem. I think it would be interesting if someone could add a proof that uses the fact that ln(0) is negative infinity. In other words, it contains a formula that would show that ln(0) is a finite negative number if there were a finite number of primes. Georgia guy (talk) 00:20, 22 April 2015 (UTC)

Boxed expression in LaTeX
See what I discovered:
 * $$\begin{array}{|c|} \hline EXPRESSION \\ \hline \end{array}$$

See 'A box' section in Talk:Minkowski's question mark function for some details. :) CiaPan (talk) 11:57, 22 April 2015 (UTC)


 * Hm... why doesn't \fbox (or \framebox) work in math mode on WP? I get parse errors when I try it here, but on my home installation, I can use it both within and outside of math mode, and even nest in various ways, e.g.  SemanticMantis (talk) 16:04, 22 April 2015 (UTC)
 * Because wiki math markup is only a fairly small subset of LaTeX. Maybe not even a subset &mdash; I'm not at all sure that everything that works in wikimath also works in LaTeX. --Trovatore (talk) 23:37, 22 April 2015 (UTC)
 * Fair enough. Part of me wonders why it can't be full LaTeX, or why not at least a proper subset, but I suspect the technical details might not be that interesting :) SemanticMantis (talk) 14:36, 23 April 2015 (UTC)

zeroes of infinite sums
Can someone refer me to some material on how to find the zeroes of functions that are defined as infinite sums? RJFJR (talk) 23:18, 22 April 2015 (UTC)
 * I don't think there is any general method. Can you give a little more context?  Are you looking for exact solutions, or approximations within some tolerance, or approximations within an arbitrarily small tolerance?  Are the sums power series or trigonometric series or something else? --Trovatore (talk) 23:27, 22 April 2015 (UTC)


 * It seems like you try to decide Riemann Hypothesis, don't you? HOOTmag (talk) 23:33, 22 April 2015 (UTC)


 * Assuming that you have uniform estimates on the truncation error (e.g., for a uniformly convergence series), you can use the method of bisection to locate individual zeros. In more favorable cases, standard numerical methods like Newton's method can be used, but one needs to be careful that the series converges in an appropriately strong sense.   Sławomir Biały  (talk) 23:39, 22 April 2015 (UTC)


 * A power series can be treated like this. Truncate the series and compute the zeroes of the resulting polynomial by the the Durand-Kerner method. Bo Jacoby (talk) 09:17, 23 April 2015 (UTC).


 * That result is probably not useful for the full series, since analytic functions in general need not have roots at all, such as with the exponential function, nor have a uniform limit for the position of the zeros.--Jasper Deng (talk) 09:08, 24 April 2015 (UTC)
 * The result is useful when the series converges around a root. Bo Jacoby (talk) 07:02, 24 April 2015 (UTC).

I was hoping for an answer like "read chapter 5 of book XYZZY for a general overview". I'm not sure what is knowable. They tell me there is a zero to zeta around .5 + 14.***i (or something) but I don't know how they found it. And when I try to think about finding the zeroes of complex valued functions of a complex argument my intuition gets confused. I think you pointed me in the right direction: I need to go back and look in my old numerical analysis book about how if you truncate the series you can bound how close the zero is. Thank you. RJFJR (talk) 17:47, 23 April 2015 (UTC)


 * For the zeta function specifically, the complex argument of zeta is under control on the critical line, so the problem of finding zeros is effectively reduced to finding zeros of a single real function. See Titchmarsh "The Zeros of the Riemann zeta-function", 1935, Proceedings of the Royal Society, Vol 151, No 873 pp. 234-255.  Parts of that paper are  readable.   Sławomir Biały  (talk) 20:50, 23 April 2015 (UTC)
 * Our Z function and Riemann–Siegel formula and links therein may help, and they cite how people found such zeros. The Z function is rigged to be real on the critical line by the functional equation, so intuition won't get so confused. :-)John Z (talk) 06:05, 25 April 2015 (UTC)

Thank you, everyone. (I think I need to get out my old text books and review some.) RJFJR (talk) 22:54, 25 April 2015 (UTC)

Need help estimating the radius of a planet from its gravity
So I've been trying to figure out a way to estimate certain variables about a planet (primarily its radius) from the force of its gravity, and I seem to be getting it wrong. (And no, this is not a homework problem. Just something I'm trying to figure out / trying to remember from high school physics in my spare time.) Could someone help me out?

So I start out with this standard take on Newton's law of universal gravitation (b stands for the hypothetical Planet B, since that's as good a letter as any):

$$g_{b} = \frac{GM_{b}}{{r_{b}}^{2}}$$

Since we know what the planet's g will be, solve for radius:

$${r_{b}}^{2} = \frac{GM_{b}}{g_{b}}$$

Here's where things might be getting dicey -- since this is going to be a rocky planet, like Earth, I assume that its mass and radius will be proportional to Earth's mass and gravity (i.e. if planet is bigger than Earth, its mass will be greater). Meaning:

$$\frac{M_{b}}{r_{b}} = \frac{M_{e}}{r_{e}}$$

Solve for Planet B's mass:

$$M_{b} = \frac{r_{b}M_{e}}{r_{e}}$$

Then introduce that back into the other equation.

$${r_{b}}^{2} = \left(\frac{G}{g_{b}}\right)\left(\frac{r_{b}M_{e}}{r_{e}}\right)$$

$$r_{b} = \frac{GM_{e}}{g_{b}r_{e}}$$

So it looks like it should work. But the calculations don't seem to be working out. Can anyone tell me where I'm going wrong here? Thanks. --Brasswatchman (talk) 23:51, 22 April 2015 (UTC)
 * For planets of the same density (all rocky planets should have roughly the same average density), the mass is proportional to the cube of the radius. That makes a pretty big difference. --Trovatore (talk) 23:54, 22 April 2015 (UTC)
 * It would, wouldn't it? :) Okay. Let's see if I can work out the rest of this, then...


 * $${r_{b}}^{2} = \left(\frac{G}{g_{b}}\right)\left(\frac{M_{e}{r_{b}}^{3}}{{r_{e}}^{3}}\right)$$


 * $${r_{b}}^{-1} = \frac{GM_{e}}{g_{b}{r_{e}}^{3}}$$


 * $$\frac{1}{r_{b}} = \frac{GM_{e}}{g_{b}{r_{e}}^{3}}$$


 * $$r_{b} = \frac{g_{b}{r_{e}}^{3}}{GM_{e}}$$


 * Does that look right? (Numbers at least are looking more in line with what I'd expect.) --Brasswatchman (talk) 00:15, 23 April 2015 (UTC)
 * Yes, or simpler still $$ r_b = \frac{g_b}{g_e} r_e. $$. Abecedare (talk) 00:38, 23 April 2015 (UTC)
 * Excellent. Thank you both. --Brasswatchman (talk) 00:42, 23 April 2015 (UTC)
 * And while you are at it, if the densities don't match but somehow you have an estimate of the planet's density: $$r_b = \frac{g_b}{g_e} \cdot \frac{\rho_e}{\rho_b} r_e.$$ Abecedare (talk) 00:49, 23 April 2015 (UTC)

Measure the gravitational acceleration on the ground (g at r) and on the top of a tall building (g+dg at r+dr). Then the radius of the earth is r = − 2 g dr / dg. Bo Jacoby (talk) 06:44, 23 April 2015 (UTC).