Wikipedia:Reference desk/Archives/Mathematics/2009 December 6

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December 6

A Related Problem In Base Four

I know this is actually a reference section rather than a recreational mathematics forum, but it's the best thing I've got to work with. I'm going to ask another question, closely related to the other recent ones, but not quite the same and in base four.

The natural analogue of the 1 December question in any base is to ask for the deterministic sequence obtained by prepending the smallest number with only the leading digit non-zero. If one relaxes it so that one has a choice of leading digits, one gets a different set of questions. In base four, consider choosing from among whatever choices give relative primality as before at each new length. I would expect that eventually one is stuck choosing a number that is divisible by 3, but just how large a number may one get before then? I expect a tree with a lot of doublings for a while, but then eventually the doublings should become pruned away faster than they are replaced. This seems like a nice slightly difficult problem in computational mathematics, and I hope it's found to be worth looking at. I apologize if it seems out of place.Julzes (talk) 05:16, 6 December 2009 (UTC)

Answer: 113 digits in base four. It took a while to write the program (including a @#$^$@&%$ power outage), but about 10 seconds to run it.Julzes (talk) 07:09, 9 December 2009 (UTC)

finite differentiation

The factorial and the primorial have values only for integer arguments, and so cannot be operated on by infinitesmal difference operators (i.e., differentiation). However, I presume that they can be operated on by finite difference operators - if so, what are the results?ImJustAsking (talk) 11:59, 6 December 2009 (UTC)

Just jumping in here to mention that the definition of the factorial can in fact be expanded to all the real numbers (indeed all the complex numbers) excluding the negative integers. The result is the famous Gamma function, which can of course be differentiated. If you're after closed form expressions for finite differences of n! and !p, you could easily simply write down the expression for (n+1)! - n! and simplify/factorise the result. Similarly for the primorial. Being an integer difference means the denominator of the expression is of course 1. I don't see anything complicated in your questions, or am I missing something? Zunaid●for your great great grand-daughter 12:39, 6 December 2009 (UTC)

Indeed, Γ(n+1) = n! for all n ∈ N. So the function C → C given by z ↦ Γ(z+1) is an extension of the factorial function. The derivative, with respect to z, is given by Ψ(z+1)Γ(z+1) where Ψ is the digamma function. ~~ Dr Dec (Talk) ~~ 14:18, 6 December 2009 (UTC)

Maybe I.J.A. wants the k-fold iterated finite differences of the sequence $F(n):=n!,$ say $(\delta F)(n)=(n+1)!-n!,$ $(\delta ^{2}F)(n)=(n+2)!-2(n+1)!+n!$ and $\scriptstyle \delta ^{k}F(n)=\sum _{j=0}^{k}(-1)^{k+j}{k \choose j}(n+j)!,$ that may also be written $\scriptstyle \int _{0}^{\infty }x^{n}(x-1)^{k}e^{-x}\,dx.$ Note that the numbers $(\delta ^{k}F)(n)$ have a combinatorial interpretation as the number of permutations of $\scriptstyle \{1,\dots ,n+k\}$ that leave fixed no j ≤ k. In particular $(\delta ^{k}F)(0)$ counts the number of permutations of $\scriptstyle \{1,\dots ,k\}$ with no fixed points, sometimes named derangements. --pma (talk) 15:12, 6 December 2009 (UTC)

Integral

Resolved

I am reading a proof and am told that the value of an integral is 0 and that this is obvious. I believe I can see that it is 0 but at the very least it took a few minutes of calculations to get that. So, I wonder if there is some "obvious" thing I am missing or if perhaps obvious just means you just need some basic complex analysis and some basic calc 2 stuff.

$\int _{-\infty }^{\infty }{\frac {1}{(t+i)^{2}}}\,dt$.

Thanks StatisticsMan (talk) 16:30, 6 December 2009 (UTC)

Well, if you change variable with $\textstyle \tau =-1/t\,$ you get indeed

$\int _{-\infty }^{\infty }{\frac {1}{(t+i)^{2}}}\,dt=-\int _{-\infty }^{\infty }{\frac {1}{(\tau +i)^{2}}}\,d\tau .$

--pma (talk) 16:42, 6 December 2009 (UTC)

Very nice. Michael Hardy (talk) 08:02, 7 December 2009 (UTC)

• The integrand has a simple antiderivative. Recall that ∫ x⁻² dx = k − x⁻¹ where k is the constant of integration. It follows that

$\lim _{a\to \infty }\int _{-a}^{a}{\frac {dt}{(t+i)^{2}}}=\lim _{a\to \infty }\left[{\frac {-1}{t+i}}\right]_{-a}^{a}=\lim _{a\to \infty }{\frac {-2a}{a^{2}+1}}=0.$ ~~ Dr Dec (Talk) ~~ 19:27, 6 December 2009 (UTC)

It's also pretty much immediate from Cauchy's theorem. Algebraist 19:56, 6 December 2009 (UTC)

Hmm, so it is obvious. Thanks all. StatisticsMan (talk) 20:55, 6 December 2009 (UTC)

may we say trivial? ;-) pma (talk) 23:48, 6 December 2009 (UTC)

I forgot that I can just integrate like normal in the complex numbers. By the way, Dr Dec, by definition the integral is the double limit as both a goes to $-\infty$ and b go to $\infty$ of $\int _{a}^{b}...$. If we know the integral converges, then I think the way you did it is going to give the same answer, and I guess we probably do know it converges because it is a second power in the denominator, so I guess we can just do what you did! So, yea, trivial seems good. StatisticsMan (talk) 01:44, 7 December 2009 (UTC)

Well, if my informal approach doesn't suit, and you want to be totally rigorous then you will find that the double limit approach of which you speak doesn't work either! We have to abandon Riemannian integration altogether. See this subsection. ~~ Dr Dec (Talk) ~~ 00:29, 8 December 2009 (UTC)

If you're going to say there's some difference between the limit as a grows, of the integral from −a to a, and the "double limit" of which you speak, then you should remember that in those cases where they may be different, there may also be a difference between whether you first take the limit as b → ∞ and then the limit as a → −∞, or vice versa, and you may also get different limits by letting the point (a, b) approach (−∞, +∞) along different paths. Michael Hardy (talk) 21:31, 7 December 2009 (UTC)

The double limit obviously goes to zero anyway, since each endpoint goes to zero separately. Algebraist 01:51, 7 December 2009 (UTC)