Wikipedia:Reference desk/Archives/Mathematics/2011 March 2

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March 2

"Common" log?

Why is the base-10 log called the "common log" if it seems that the base-e log is much more "common"? —Preceding unsigned comment added by 72.128.95.0 (talk) 02:54, 2 March 2011 (UTC)

Before the advent of electronic calculators, base-10 logarithms were used extensively used to make pen-and-paper calculations tractable. Base 10 had clear practical advantages in the pre-computer age; you only need to tabulate a single decade in order to find logarithm for numbers of any size. Back then it was certainly more "common" to meet base-10 logs than natural ones. –Henning Makholm (talk) 03:15, 2 March 2011 (UTC)

Yes, I can recall (many years ago) thinking, when I first met "natural" logs (having previously used common logs for years, including to check every simple multiplication and division in my "O-level" examination papers), that they seemed very "unnatural", rather like measuring in base "e", but I soon got used to the idea of these "uncommon" logs. Dbfirs 01:37, 3 March 2011 (UTC)

Our article titled common logarithm answers this question. Michael Hardy (talk) 18:30, 3 March 2011 (UTC)

Weird Latex behavior

For some reason, Latex has started numbering the equations in my document using negative numbers (eg 2.-1, 2.-28), with every equation on the same page being given the same number and no proper order to how it's being done. I can't figure out what I changed to make this happen. Has anyone else experienced this? —Preceding unsigned comment added by 118.208.21.155 (talk) 13:35, 2 March 2011 (UTC)

No. Could you post some of the code you're using to get the equations, please? — Fly by Night (talk) 15:59, 2 March 2011 (UTC)

The \setcounter command might help you find the bug, or at least override it. See description here [1]. Set the equation counter to 0 at the start of the section or document, and see if it increments properly from there. SemanticMantis (talk) 16:24, 2 March 2011 (UTC)

My first troublesshooting steps would be to make a temporary copy of the .tex and experimentally remove everything from it that you can remove without making the bug go away, until I had a minimal example showing the bad behavior. Remove irrelevant text first, then packages one by on. It's often more efficient to attempt find irrelevant things by starting at the end of the file. The culprit may be a package or feature that has nothing to do with equation numbering -- TeX is such a horrible programming language that any buggy macro has the potential to randomly break anything else. –Henning Makholm (talk) 16:38, 2 March 2011 (UTC)

TeX is like democracy. Perhaps horrible, but still the best game in town :) SemanticMantis (talk) 16:55, 2 March 2011 (UTC)

Fourier Transform

I know that the F.T of Delta Function is 1 and denoted as delta(w), but how did it make the integration of e^-jwt=2*PI*delta(w)41.95.12.44 (talk) 17:31, 2 March 2011 (UTC)

What is it that you'd like us to explain? Do you want us to show that

$\int _{-\infty }^{\infty }\delta (w)e^{-jwt}\,\operatorname {d} w=1\,?$

The Dirac delta function has the fundamental property that

$\int _{-\infty }^{\infty }f(w)\delta (w-a)\,\operatorname {d} w=f(a)\,$

The result follows from letting ƒ(w) = e^–jwt and a = 0. — Fly by Night (talk) 16:06, 3 March 2011 (UTC)

I think he is asking the converse: why and how we can say that $\textstyle \int _{-\infty }^{\infty }e^{-iwt}dt=2\pi \delta (w)$. I'm not sure that there is a really satisfying fundamental reason for this -- the improper integral certainly doesn't converge. For $w\neq 0$, the value of 0 for the integral can probably be justified as a Cauchy principal value. But for $w=0$ it's not clear that the divergence of $\textstyle \int _{-\infty }^{\infty }1dt$ is "just fast enough" that the value of the entire integral ought to be taken as precisely $2\pi \delta (0)$. I tend to think of it as simply a convention, justified by the fact that it makes the self-duality of the Fourier transform work for deltas as well as it does for ordinary functions. –Henning Makholm (talk) 16:48, 3 March 2011 (UTC)

You get that equality (in the sense of distributions) by proving that $\textstyle \int _{-\infty }^{\infty }f(t)\int _{-\infty }^{\infty }e^{i\omega t}\,\mathrm {d} \omega \,\mathrm {d} t=2\pi f(0)$ (for all f with compact support, or whatever other test function restriction). I think you can do that with integration by parts, but I don't know the details immediately. --Tardis (talk) 17:19, 3 March 2011 (UTC)

Ah, ω, of course. I did goranully wonder what was up with the w. –Henning Makholm (talk) 17:56, 3 March 2011 (UTC)

I think the person who posted the question meant $\omega$ but wrote w. In electrical engineering, it is standard to use the letter $\omega$ for frequency, and j rather than i for the imaginary unit. Michael Hardy (talk) 20:21, 3 March 2011 (UTC)

---

A complete proof requires the following Fourier inversion theorem:

• If f and F are both absolutely integrable functions, then the Fourier inversion formula holds:

$f(t)=\int _{-\infty }^{\infty }e^{i\omega t}F(\omega )\,d\omega ,\quad 2\pi F(\omega )=\int _{-\infty }^{\infty }e^{-i\omega t}f(t)\,dt.$

See any textbook with a rigorous account of the Fourier transform. Rudin's Functional analysis, for instance. Under the hypothesis that f is absolutely integrable, F is continuous and the pointwise values of the integral in the second equation make sense, so evaluating this equation at $\omega =0$ gives

$2\pi F(0)=\int _{-\infty }^{\infty }f(t)\,dt.$

Assuming this result, by $\int _{-\infty }^{\infty }e^{i\omega t}\,dt=2\pi \delta (\omega )$, we mean that

$\lim _{M\to \infty }\int _{-M}^{M}e^{i\omega t}\,dt=2\pi \delta (\omega )$

where the limit is understood as the limit in the sense of distributions. All this means that in order to verify the identity, we just need to check that for any compactly supported smooth function F, we have

$\lim _{M\to \infty }\int _{-\infty }^{\infty }F(\omega )\int _{-M}^{M}e^{i\omega t}\,dt\,d\omega =2\pi F(0).$

Since F is smooth and compactly supported, its inverse Fourier transform f decays at infinity faster than any rational function (Schwartz function), and so is absolutely integrable. So interchanging the two integrals (justified by Fubini's theorem) and then applying dominated convergence gives

$\lim _{M\to \infty }\int _{-M}^{M}\int _{-\infty }^{\infty }F(\omega )e^{i\omega t}\,dt\,d\omega =\int _{-\infty }^{\infty }f(t)\,dt=2\pi F(0)$

as required. Sławomir Biały (talk) 14:19, 4 March 2011 (UTC)