Formulas giving Pi
based on analytical methods
(By chronological order of authors or people who inspired them)
Lord
Brounker (1620 - 1684)
Other continued fractions
Newton : (1642
- 1727)
suite similaire :
Leibniz
(1646 - 1716)
Katahiro
(1664 - 1739)
Machin
(1680 - 1751)
Moivre/Stirling (1667 - 1754) / (1692 - 1770)
Euler
(1707 - 1783)
1) *1739*
and more generally, we have
where *Ber*_{2i} is the Bernoulli's number with index *2i*
(see appendix: Numbers
of Bernoulli)
2)
3) et 4) (*1737*)
Buffon (1707 - 1788)
If a needle of length *2a* is dropped on a parquet formed of floorboard of
width *2b*,
the probability that the needle cuts one of the lines of this parquet is
Gauss (1777 - 1855)
Cesaro (1859 - 1906)
The probability that two integers ramdomly chosen
are prime between them is ...
Ramanujan (1887 - 1920)
Writing (x)_{n} the value
:
we can write:
Gosper
We have a general formula for x less than 1:
where _{2}*F*_{1} is a hypergeometric seie, which distract us from our topic, so I won't
mention it....
For *x=1/2*, we get :
which has a convergence of *2n*
For we can write :
of convergence *3.39n*
William Gosper is also used to formulae that are a bit wierd making use
of , donc ask me of their use!
For example :
and by generalising :
Sums of Reynolds :
G
and D. Chudnovsky
Borwein
with **A**=63365028312971999585426220+28337702140800842046825600*5^{1/2}
+ 384*5^{1/2}(10891728551171178200467436212395209160385656017 + 4870929086578810225077338534541688721351255040*5^{1/2})^{1/2}
**B**=7849910453496627210289749000+3510586678260932028965606400 + 2515968*3110^{1/2}(6260208323789001636993322654444020882161
+ 2799650273060444296577206890718825190235*5^{1/2})^{1/2}
**C**=-214772995063512240-96049403338648032*5^{1/2}-1296*5^{1/2}(10985234579463550323713318473
+ 4912746253692362754607395912*5^{1/2})^{1/2}
Plouffe
Brown
So, lets take a natural integer *n *different from 0. For example *10*. So far everything is all right !
Now let's considerthe nearest "multiple" bigger than or equal to *n-1*.
In our case, we find *18* because it is a "multiple" of *9=10-1* and bigger than *10*.
Let us repeat this considering the nearest "multiple" bigger or equal to *n-2*, here *24*, then of *n-3* (*28*), of *n-4* (*30*) and so on for *n-k* unto we get to *k=n-1*. We call *f(n)* the result (*f(10)=34*)
Well, funnily enough,
Woon
Bellard
with
Mandelbrot/Bolle
let us consider the point c=(-0.75,X) of the
complex plane, that is a point straight over the "neck" of the Mandelbrot
set.
Let n be the number of iterations from which the characteristic quadratic sequence of the Mandelbrot
set Zn+1=Zn2+c with Z0=-0 diverges (*Zn2*). With X being smaller
and smaller we have:
Remainders theorem - E. Estenave - C. Frétigny
I can not put them all down, there's too
many of them ! You will discover them as you read this page.
Nevertheless here are a few examples:
and also:
...
Anonymous (see the attic)
From the sums of Riemann
Triangle of c(n,k)
Some sums with factorials (see Gosper)
(1) from Euler and (2) from
Comtet (1974)
Capes 1994
(an national exam in France to become teacher)
Let *D* be the disc of center *z*_{0} and radius *r* :
*D(z*_{0},r)={zC z-z_{0}r}
and *r*_{k}=min{r>0, z_{0}C, card(Z[i]D(z_{0},r))k}
then
Chebyshev Polynômials
A first example :
If we know a relation of type : then we can construct the sequence *U*_{n}^{k} and the following sequence :
You want something different to *10* as the denominator? Sure,
here is a even more general formula for *p*^{2}>1,
still starting from the relation with the *Arctan* :
with *T(n,x)* Chebyshev's polynomials.
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