User talk:GFauxPas/Archive1

Change to MathWorld citation template
I noticed (based on One-to-One and Strictly Between) that some pages on MathWorld are credited to different authors from Eric Weisstein, and so require that author to be included in the citation.

I have fixed the template (which is now "MathWorld" not "Mathworld", that's just me tidying up) so as to be able to include the author (which, if not given, defaults to the "Weisstein, Eric W." format as per normal).

What you need to do is add "author=author-name" and "authorpage=author-pagename" where "author-name" is the displayname of the author and "author-pagename" is the name of the html file on MathWorld (not including the full path, not including the extension).

An example:

which gives:
 *  


 * {{MathWorld|One-to-One|One-to-One|author=Barile, Margherita|authorpage=Barile

If the page is given as written by "Weisstein, Eric W." then you should not add the "author" and "authorpage" tags.

I have included this info in the usage section of the Template:MathWorld page itself, but I'm bringing it to your attention because I know you've been active in using it.

Chx. --prime mover 02:55, 31 December 2011 (CST)

Source Review
Just a quick heads up to draw your attention to Definition:Random Variable, which has an open SourceReview call listed on your name. &mdash; Lord_Farin (talk) 11:42, 5 April 2014 (UTC)


 * I have that book packed away; I'll have to dig it up. --GFauxPas (talk) 11:20, 8 April 2014 (UTC)

Mathjax
Is there a way to take the code of a given page and copy paste it into some external program or into some website, and end up with a pdf?

--GFauxPas (talk) 14:43, 8 April 2014 (UTC)

zeta 2
If you've been watching my sandbox, you might have seen me work on this:


 * $\displaystyle \int_{\to 0}^{\to 1} \ln x \ln \left({1-x}\right) \, \mathrm dx = 2 - \zeta\left({2}\right)$

Should I put this up as a theorem, or is it too specific an exercise to be worth sharing?

--GFauxPas (talk) 11:42, 9 April 2014 (UTC)


 * I think it's worth sharing, specially after all the hard work you put into it. The name needs to be considered carefully. Might want to put it into an "/Examples" subpage of either Logarithms or Integral Calculus (or both) ... once I've reached that part of my Spiegel handbook where there are hundreds of examples of integrations I will be in there to flesh it out, but it's something I haven't got to yet. --prime mover (talk) 04:26, 10 April 2014 (UTC)


 * Or maybe under Basel Problem... --GFauxPas (talk) 11:03, 10 April 2014 (UTC)


 * No, rather not under Basel problem -- the latter is a specific problem with a specific solution (i.e. the one Euler came up with, which explains its name), this just happens to have a result which includes the same zeta instance. --prime mover (talk) 16:43, 10 April 2014 (UTC)

Real and Imaginary Parts
Are these theorems?


 * $\displaystyle \int \operatorname{Im}\,\left({f\left({s}\right)}\right)\, \mathrm ds = \operatorname{Im}\,\left({\int

f\left({s}\right)\,\mathrm ds}\right)$


 * $\displaystyle \int \operatorname{Re}\,\left({f\left({s}\right)}\right)\, \mathrm ds = \operatorname{Re}\,\left({\int f\left({s}\right)\,\mathrm ds}\right)$

for real $s$? For complex $s$?

--GFauxPas (talk) 12:14, 5 May 2014 (UTC)


 * If $f$ is real then it's true trivially. If $f$ is complex then I believe (although I would have to go back to my source works to prove it) that it's not.


 * The question for whether it is true for real or complex $s$ (as opposed to $f$) I'm not sure how much sense it makes, but consider the fact that while $s$ may be real then it may not be the case that so is $f(s)$.


 * I don't know without thinking about it for more than 5 seconds but, as I say, I don't believe so. --prime mover (talk) 12:52, 5 May 2014 (UTC)

Gamma function Proof
Regarding the proof for:


 * $\displaystyle \mathcal L \left \{ {t^p} \right \} = \frac {\Gamma \left({p+1}\right)} { s^{p+1} }$

for $\operatorname{Re}\left({p}\right) > -1$, some sites are doing this. Is it justified?


 * $\displaystyle \int_0^{\to +\infty} t^p e^{-st} \, \mathrm dt = \int_{0}^{?} t^p e^{-u} \frac {\mathrm dt}{\mathrm du}\, \mathrm du$

where $u = st$, and then doing various stuff with the limits of integration. But it seems to me like a mistake; if $u$ is complex, then how can the limits of integration be complex? If $u$ is real, how can we replace a complex variable with a real variable?

I asked someone about this and he suggested that maybe they're making the integral into a contour integral, but that doesn't help me make it look like $\Gamma$... or does it? --GFauxPas (talk) 14:19, 26 May 2014 (UTC)


 * It's justified. You can prove by using the contour of two rays going to infinity, connected by an ever larger arc, that the "direction" of the infinity does not influence the value of the integral. This is because the integral over the arc goes to zero, because of the exponential. 't May be beyond the current scope of PW, though. &mdash; Lord_Farin (talk) 17:46, 26 May 2014 (UTC)


 * Once I get to it, I intend this *not* to be beyond the current scope. It's all in my complex analysis source notes -- but these are all buried way down in the substrata of the intellectual sedimentary rock that my office resembles ... and anyway, I'm mucking about with primitives at the moment. --prime mover (talk) 19:35, 26 May 2014 (UTC)


 * Ah, so, it's not $u = st$ so much as $u = rt$ where $s = re^{i\theta}$? And with $\displaystyle \int_C t^p e^{-u}\, \mathrm du$, $C$ is a curve connecting any two rays connected by an arc, as the length of the rays go to $+\infty$? and $\arg s$ doesn't affect anything? --GFauxPas (talk) 11:09, 27 May 2014 (UTC)


 * Yes, I think that's about it. &mdash; Lord_Farin (talk) 20:48, 27 May 2014 (UTC)

Periodic Functions
Is there such a theorem, that if a function is periodic, its derivative is also periodic, with the same period? --GFauxPas (talk) 01:19, 18 July 2014 (UTC)


 * Not that I know of. Not on ProofWiki anyway. Have a go. --prime mover (talk) 07:02, 18 July 2014 (UTC)

On deleting content
Please remember, before you delete the content and thereby trash a redirect of any page, please check first to see there are no pages which link to the page you're marking for deletion. If there are any, then feel free to change those links to the new page. As it is we were in danger of losing all the links to Linear Combination of Laplace Transforms which would have been a suboptimal situation. Thankz. --prime mover (talk) 13:57, 20 July 2014 (UTC)

Modeling question
So I'm taking a class on modeling. What's the explanation or justification of this process?:

Given a sequence $\langle a \rangle : \N \to \R$, we create a function $f:\R \to \R$ such that $f \restriction_{\N} = \langle a \rangle$.

The objective is to find $\sup (\langle a \rangle)$ or $\inf(\langle  a \rangle)$.

To do so, we find the local extrema of $f$ (say at $f(c)$), and the assumption is that $\langle a \rangle$ attains its supremum/infimum at either $a_i$ or $a_{i+i}$, where $i \le c \le i + 1$.

Why and when is this method kosher? --GFauxPas (talk) 22:57, 11 September 2014 (UTC)


 * Very quickly, thinking as I type ...


 * If you've identified *all* the local extrema of $f$ (assuming that there are a finite number), then it can intuitively seen that one of the points either side of the *most* extreme one is going to be "as close as you can get" to it. This of course requires that the function does not have great big peaks at the $f \left({n + \dfrac 1 2}\right)$ points. Consider $f = \sin \left({\pi x}\right)$ as an awkward case which does not fit this intuitive notion but which *does* fulfil the conditions of the statement. --prime mover (talk) 05:29, 12 September 2014 (UTC)

R Studio
Is there an easy way to implement the eqn template in R studio? --GFauxPas (talk) 12:47, 24 September 2014 (UTC)

Homework Help
I'm stumped!

The experiment is picking a random point in the disk $x^2 + y^2 \le 4$.

Let $X$ and $Y$ be the $x$ and $y$ coordinate of the sampled point.

Find $f_{X,Y}$, the probability density function for $X$ and $Y$.

So I have to find a function whose area is $1$ over this disk.

But what now?

I know how to find the area of the region (if I pretended I didn't know the area of a circle), I would do:


 * $\displaystyle \int_0^{2\pi} \int_{0}^{2} r \, \mathrm dr \, \mathrm d\theta$

but that doesn't seem helpful.

--GFauxPas (talk) 01:46, 29 September 2014 (UTC)


 * Consider dividing by a well-chosen constant. Deduce what $f_{X,Y}$ is. &mdash; Lord_Farin (talk) 17:21, 29 September 2014 (UTC)


 * I consulted the professor. The problem is written poorly in that it omits that $\left({X,Y}\right)$ has a uniform distribution, after which the problem is easy. --GFauxPas (talk) 19:33, 29 September 2014 (UTC)