Árpád Kurusa
mathematician, associate professor
 Department of Geometry
Bolyai Institute
Faculty of Science
University of Szeged
english
Árpád Kurusa

Maths-blog

Day of π

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Published on 2012. március 14. szerda, 11:48

On the day of $\pi$ the number $e$ should also be celebrated.

The beautiful formula

$ e^{i\pi}+1=0 $

of Leonhard Euler (1707–1783) says it all as the fact behind it is $e^{i\varphi}=\cos\varphi+i\sin\varphi$ for all real $\varphi$.

David Wells surveyed some readers of the Mathematical Intelligencer in 1988 to choose the most beautiful formula. Euler's above mentioned formula won the glory, but the number $\pi$ got a role in the fifth, the eights and the forteenth formula as well:

$$ \frac{\pi^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2}+\cdots \tag{5} $$ $$ \frac{\pi-3}{4}=\frac{1}{2\cdot3\cdot4}+\frac{1}{4\cdot5\cdot6}+\cdots+\frac{1}{2k\cdot(2k+1)\cdot(2k+2)}+\cdots \tag{14} $$

(The $\pi$ appears also in the eights statement, that "only" establishes that $\pi$ is transcendental.)

An infinite product due to Euler in which $\pi$ is crucial:

$$ \sin x= x\cdot\left(1-\frac{(x/\pi)^2}{1}\right)\cdot\left(1-\frac{(x/\pi)^2}{4}\right)\cdots\left(1-\frac{(x/\pi)^2}{k^2}\right)\cdots $$

Bailey, Borwein és Plouffe proved that

$$ \pi=\sum_{k=0}^{\infty}\left(\frac4{8k+1} - \frac2{8k+4} - \frac1{8k+5} - \frac1{8k+6}\right)16^{-k} $$

which served as a bases for them to prove that the $n$-th digit in the hexadecimal form of $\pi$ can be computed without calculating previous digits and within a timeperiod depending on $n$ linearly.

 

 

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