Week of 10/29

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Revision as of 15:17, 31 October 2007

A few examples of the importance of Gaussian distributions

It turns out that the FWHM is nontrivial: this link shows how to compute it

the answer turns out to be $2 \sqrt{2 \ln 2} \sigma \approx 2.3548 \sigma$

The fundamental property of the delta function: you must remember this

$\int _ a ^ b \delta (t - t_0) f(t) dt = f(t_0)$ if $t_0 \in [a,b] \,$ .

$\int _ a ^ b \delta (t - t_0) f(t) dt = 0$ if $t_0 \notin [a,b] \,$

Remember, in the former case, you evaluate $f \,$ at the point where the argument of the delta function is zero. In this case when $t = t_0. \,$

Two nice links on Sinc functions: from Mathworld and Wikipedia

The following two figures are from Barbara Romanowicz, a seismologist at the University of California, Berkeley.

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 The fundamental property of the delta function: you must remember this
-<math>\int _ a ^ b \delta (t - t_0) f(t) dt = f(t_0)</math> if <math>t_0 \in [a,b] \,</math>.  Remember, you evaluate <math>f \,</math> at the point where the argument of the delta function is zero.  In this case when<math> t = t_0. \,</math>
+<math>\int _ a ^ b \delta (t - t_0) f(t) dt = f(t_0)</math> if <math>t_0 \in [a,b] \,</math>.
 <math>\int _ a ^ b \delta (t - t_0) f(t) dt = 0</math> if <math>t_0 \notin [a,b] \,</math>
+Remember, in the former case,
+you evaluate <math>f \,</math> at the point where the argument of the delta function is zero.  In this case when<math> t = t_0. \,</math>
 [http://www-history.mcs.st-and.ac.uk/history/Biographies/Dirac.html Paul Dirac's bio]