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Theorems

November 4th, 2010 | | 422 views | Print This Page Print This Page | Leave a comment

Euler’s identity :

    \[ e^{i \pi} + 1 = 0 \]

Riemann zeta function evaluated at 2, Sum of p series when p = 2, Basel Problem and Euler’s product formula:

    \[ \zeta(2) = \left \displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2} = \prod_{p \text{ prime}} \frac{1}{1-p^{-2}} = \frac{\pi^2}{6} \]

Buffon’s needle problem:

If a short needle, of length l, is dropped on paper that is ruled with equally spaced lines of distance d \geq l, then the probability that the needle comes to lie in a position where it crosses one of the lines is exactly

    \[ \frac{2 l}{\pi d} \]