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Boris Gourévitch
The world of Pi - V2.57
modif. 13/04/2013



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Formula of type Machin

Compared to the function Y  , it's the case when it's parameter are s = 1,v = 12  , and x = pq  varies.

The Arctan function is expressed this way with the function Y  . Hence, we get

            oo                  (       )         (         )
arctan(x) =  sum  (-1)kx2k+1-= x-Y  -x2,1, 1 =  x.F    12,1 ,-x2
           k=0   2k+ 1      2         2     2 2 1   32
(27)

This means that the Arctan formula of type Machin is expressed as a linear combination of the function Y  . For example, we have according to Euler's formula p = 20arctan(1-)+ 8arctan (3-):
             7           79

        (        )       (          )
    10     -1   1    12     --9-   1
p =  7 Y - 49,1,2  + 79Y   -6241,1,2
(Euler)

Here we are dealing with rational series which made up the principle method to calculate Pi between their "official" discovery by Machin (1705) and those of modern algorithm like Brent-Salamin / Borwein at the end of the 70s.

This formula correspond in hypergeometric notation to the form

    10    (  1      1)   12    (  1      9  )
p = -- .2F1   2,31 ,- --  + --.2F1   2,31,- ----
     7        2    49    79        2    6241
(28)

The proof is done by using the formula

      (1)         (  1  )        (     q     )
arctan  -  = arctan  ----- + arctan  -2--------
       p            p+ q           p + pq+ 1
(29)

several time. It's long, might as well say it! A simple method was done by the Borwein brother's in 1987, which consist to show that a formula

 sum n        (  )
   akarctan  -1- = arctan(1) = p-
k=1          uk               4
(30)

is equivalent to the complex expression

 prod n (     )ak
    uk-+-i   = 1
k=1 uk - i
(31)

Starting from here, the proof of the arctan formula is as simple as the product of complex number !


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