¼
The constant
The constant p ¼
3 Ramanujan type formulae
In 1914, the extraordinary mathematician from India, Srinivasa Ramanujan
published a set of 14 new formulae ([1]), one of them is famous:

1 p

= 
2Ö2 9801


¥ å
k = 0


(4k)! (k!)^{4}4^{4k}


(1103+26390k) 99^{4k}

(*) 

The convergence is geometric but very impressive, let's take a look to the
first iterates:



3.141592(7300133056603139961890252155185995816071...) 
 

3.141592653589793(8779989058263060130942166450293...) 
 

3.14159265358979323846264(90657027588981566774804...) 
 

3.1415926535897932384626433832795(552731599742104...) 
 

3.141592653589793238462643383279502884197(6638181...) 

 

Each term adds about 8 digits to the result.
In his article, Ramanujan gave very little details of how he found such
formulae. To find those relations, he needed modular equations and
hypergeometric functions. It's possible to find a complete demonstration of
those series in ([2]). In this book the following theorem is given:

1 p

= g_{N} 
¥ å
k = 0


(1/4)_{k}(1/2)_{k}(3/4)_{k} (k!)^{3}

( a_{N}+kb_{N})g_{N}^{2k}, N > 2 

where (a_{N,}b_{N,}g_{N}) are algebraic numbers whenever N is rational and (a)_{k} is the rising factorial : (a)_{k} = a(a+1)(a+2)...(a+k1). The expression of (a_{N,}b_{N,}g_{N}) as functions of N is rather complex... For example g_{N} is given by :
g_{N} = 
2 g_{N}^{12}+g_{N}^{12}



where g_{N} is a Ramanujan's invariant.
Ramanujan's formula is given for N = 58, for which g_{58}^{2} = (5+Ö[29])/2 so g_{58} = 1/99^{2} and a_{58}+kb_{58} = 2Ö2(1103+ 26390k) ([2], [4]).
Other interesting series are found with (N = 6,10,18,22,58), all given by
Ramanujan in his article. Here are the value of g_{N} and g_{N}
for the corresponding values of N:



 

(2+Ö5)^{1/6},g_{10} = 1/9 
 

(5+2Ö6)^{1/6},g_{18} = 1/49 
 

(1+Ö2)^{1/2},g_{22} = 1/99 
 

((5+ 
 __ Ö29

)/2)^{1/2},g_{58} = 1/99^{2} 

 

In the same article, Ramanujan also gave:

1 p

= 
¥ å
k = 0


æ ç
è

2k
k

ö ÷
ø

3


(42k+5) 2^{12k+4}



This series adds roughly 2 digits per term. A proof was published
in [2].
Much later, a few years ago [3], David and Gregory
Chudnovsky found a very powerful series of this type :

1 p

= 12 
¥ å
k = 0

(1)^{k} 
(6k)! (3k)!(k!)^{3}


(13591409+545140134k) 640320^{3k+3/2}



Each term of this series adds about 15 digits...
3.1 Record of computation with the Ramanujan like formulae :
1985 : Gosper found 17,500,000 decimal digits with formula (*).
1994 : The Chudnovsky brothers found 4,044,000,000 digits with their formula.
References
 [1]
 S. Ramanujan, Modular equations and
approximations to p, Quart. J. Pure Appl. Math., (1914), vol. 45, p.
350372
 [2]
 J.M. Borwein and P.B. Borwein, Pi and the AGM  A study
in Analytic Number Theory and Computational Complexity, A
WileyInterscience Publication, New York, (1987)
 [3]
 D. V. Chudnovsky and G. V. Chudnovsky, Approximations and complex multiplication according to Ramanujan, in Ramanujan Revisited, Academic Press Inc., Boston, (1988), p. 375396 & p.
468472.
 [4]
 P. Eymard and J. P. Lafon, Autour du nombre p,
Paris, Hermann, (1999)
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The constant pi
File translated from T_{E}X by T_{T}H, version 2.32.
On 17 Aug 2000, 21:44.