www.pi314.net Boris Gourévitch The world of Pi - V2.57 modif. 13/04/2013 Pi-Day in
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Carl Friedrich Gauss
(1777 - 1855)

A few arctan formulae

On his life

Ah, Gauss, without any doubts one of the greatest mathematicians of all time... He was born in 1777 in Brunswick and claimes that his genuis came from his motehr. He was in fact a prodigy child and his capacity are quickly reconised. He loved to tell how he was the only one in class, one day the teacher wanted to be left in peace, to have found the solution to the problem that was given, to work out the sum of the first 100 integers. Since 1+100=2+99=3+98=...=101, Gauss deduced that the result must be 50*101=5050 and put his slate board on his teacher desk before he finished speaking.... Gauss was not yet 10 years old... He would love to tell later on those anecdotes worried about his fame... Was he afraid of running out of it? He wasn't 16 when he imagined a method to calculate the orbite of planets and to solve linear system, his famous Gauss pivot....
He became director of Göttingen's observatory after his work on celestial bodies, he publish in 1801 Disquisitiones arithmeticae in which he creates congruences and studies quadratic shapes and various properties of algebra. Then working on non-euclidian geometry he starts being more and more interested in physics and accept (the only time!) the collaboration of Wilhelm Weber until 1837. The domains that catch his interest are magnetism, optics and electricity...
Toward the end of his life he forms brilliants students like Riemann, Dedekind, Eisentein. He said, propably speaking too early, about the last one, surprised by his capacity, that there was only 3 mathematicians that shaped their era: Archimède, Newton and Eisenstein... Unfortunatly for mathematics, he died prematurly...
Gauss, as for him, pass away in 1855 after a long full life...

Around

Unfortunatly, if his relation with maths was completly fundamental, he left very few formulae using . It was true that he was more specialise in arithmetics and geometry, but its a bit dissapointing!!! However, let us not forget that it was him that lead the studies of arithmetico-geometric sequences used by Brent/Salamin without, for once, seing all the possible problems concerning the calculation of .
Note that the result of the integral of  R on curves of type exp(-x2) like the Gauss Bell were discovered for the first time by Abraham de Moivre and so the proof is on his page.

Proofs

There's no need to clutter this page to fill it with equalities when the principle is very simple. All that is needed is to use the rules given in the proof part on the page dedicated to Machin. Rule 4) is for example verified for the first formula above with k=1, m=12, a=18, n=8, b=57, and we add a p=5, and c=-239 on the same model. We do as expected get (1-i)k(a+i)m(b+i)n(c+i)p=-144996366690679927359016418457031250000000 ! and the imaginary part is null....
Due to laziness, I'm not recopying the calculation for the seconde formula which is perfectly similar....
Ah, let's move onto the trials !

Trial

First serie : (infinity is replaced by n in the serie)

 n=0 3,144 (2) n=3 3,1415926535629 (10) n=10 27 decimals correct

Second série :

 n=0 3,1420 (2) n=3 3,141592653589759 (13) n=10 35 decimals correct

We notice a convergence of roughly 2.5n+2 for the first serie and 3.2n+3 for the second which is very interesting. The first formula has, in fact, been used to check the calculation done for the record of 51 billions decimals of .