Timeline for What are some fundamental "sources" for the appearance of pi in mathematics?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Mar 15, 2010 at 5:11 | comment | added | Terry Tao | pi naturally arises as a normalisation constant in any integral involving an exponential (see: Laplace's method, or the principle of stationary phase), thanks ultimately to the identity $\int_R e^{-\pi x^2}\ dx = 1$ (which is not a bad definition of pi, really). The factorial is a special case of the Gamma function, which is an integral involving an exponential. | |
| Mar 14, 2010 at 19:26 | comment | added | Douglas Zare | That should have been the left endpoint rule on $[1/n,1]$ with subintervals of width $1/n$, or you can also use $[1,n]$ with subintervals of width $1$. | |
| Mar 14, 2010 at 18:21 | comment | added | Douglas Zare | The presence of $e$ is easily explained by the geometric mean of $[0,1]$ which is $1/e$. The $\sqrt n$ factor comes from the difference between the trapezoid rule and right endpoint rule for integrating $\log x$ on $[0,1]$. | |
| Mar 14, 2010 at 18:08 | comment | added | Qiaochu Yuan | The appearance of e is much less mysterious. Knowing only that (1 + 1/n)^n \le e for all n you can already deduce that n! \ge (n/e)^n. What's really mysterious is the extra work you have to do to get that last factor in. | |
| Mar 14, 2010 at 17:59 | comment | added | Michael Lugo | While we're on this subject, what's e doing there? | |
| Mar 14, 2010 at 17:20 | history | answered | Robin Chapman | CC BY-SA 2.5 |