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 noneuclidian
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 arithmeticogeometric
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(x^{2}) 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 (1i)^{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 .
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