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



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Contents page

1 A quick historical recap
2 Introduction to hypergeometric series
 2.1 Definition
 2.2 A few properties and simple examples
3 The function Psi : Basis of formulae of kind Machin or BBP
 3.1 Definiton
 3.2 Differential equations
4 Formula of type Machin
5 BBP formulae: The technic
 5.1 s = 1,  v = 0  : The polylogarithms
  5.1.1 Definition
  5.1.2 Remarquable values
  5.1.3 Formula of duplication
  5.1.4 Euler's and Landen's formulae for the dilogarithm
  5.1.5 Landen's formula for the trilogarithm
  5.1.6 Particular values
  5.1.7 A few classicals integrals
 5.2 Links between integrals, BBP formulae and polylogarithm
  5.2.1 Notations
 5.3 Integrals and BBP formuale
  5.3.1 Integrals equivalent to BBP series
  5.3.2 More general integrals and BBP formulae
  5.3.3 Calculation of integrals
 5.4 Function Y  and polylogarithms
 5.5 Interest in BBP formulae
6 BBP formulae in base 2 : s  (-  N  ,v = p
    q  , x =-1n
   2  in Y
 6.1 The considered integrals
 6.2 The method
 6.3 Formulae for p  , ln(2)  , ln(3)  and ln(5)
  6.3.1 Application to BBP formulae for p
  6.3.2 BBP formulae for ln (2), ln (3)  and ln(5)
 6.4 Cases of polylogarithms of order 2  : Formulae of order 2
  6.4.1 The classicals expressions : I(1),I(1),I(1),I(1)
 1   2  3   4  and I(1)
5
  6.4.2 Calculation of  (1)  (1)
I7 ,I8  and  (1)
I9
  6.4.3 Calculion of  (1)
I10 ,  relation between  (1)
I6  and  (1)
I11
  6.4.4 Applications of the determanition of BBP formulae
  6.4.5 A few composite formulae
 6.5 Cases of polylogarithms of order 3
  6.5.1 Calculation of  (2)
I4
  6.5.2 Calculation of I(82)
  6.5.3 Relationship bewteen I(52)  and I(92),  value of I(12)0
  6.5.4 Calculation of I(2)
 7
  6.5.5 Application to the determinaion of BBP formulae
 6.6 Cases of polylogarithms of order 4
  6.6.1 The relationship
  6.6.2 Application to the determination of BBP formulae
  6.6.3 Formulae for p4, p2ln2(2)  and ln4(2)
 6.7 Case of polylogarithm of order 5
  6.7.1 The relationship
  6.7.2 Application to the determination of BBP formulae
  6.7.3 Formulae for z(5),p4ln(2),p2ln3(2)  and ln5(2)
  6.7.4 Simplifications of those formulae
7 s  fixed integres,v = p
    q  , x = -1n
    3  : BBP in base 3
 7.1 Formulae for   V~ -
p  3
 7.2 Formulae of order 2 : Integrals with ln(y)
 7.3 Formulae of order 3 : Integrals with ln2
 (y)
8 And the other bases then ? ?
9 Polygamma and Clausen
 9.1 The polygamma function
 9.2 The function digamma
 9.3 Polygamma of order m > 1
  9.3.1 Links between the integrals and the BBP formulae
  9.3.2 A graphical approach
  9.3.3 Analytical translation
 9.4 Combinations by Kölbig
  9.4.1 Clausen's functions
  9.4.2 Kölbig's results.
10 Introduction to factorials and combinations
 10.1 A first example
 10.2 Umbral calculus
11 Central binomial series
 11.1 Inversion of combinations
 11.2 Useful developpments
 11.3 First direct formulae
 11.4 Formulae of greater order
  11.4.1 A first formulae with demo
  11.4.2 Calculation of a derivation of tpcoth(at)
        2
  11.4.3 Uses of F1
  11.4.4 Uses of F2
  11.4.5 Other formulae
12 Other binomial coefficients
 12.1 Primitive formulae
 12.2 Polynomials factorial formulae
 12.3 So, what does all this mean ?
 12.4 Proofs
 12.5 Fast combinations: BBP factorial formulae
 12.6 So, what does all this mean ?
 12.7 Typical proof
  12.7.1 The method
  12.7.2 Application : The proof of the first formula
  12.7.3 Proof of the formulae in C2n7n
  12.7.4 Prediction of formulae
 12.8 Product of combinations
13 Harmonics series
 13.1 Proximity of harmonic series and of polylogarithms
 13.2 Study of  k      + sum  oo  Hkn n
fp(x) = n=1 np x  and of  k     + oo  sum   Hkn   n+1
gp(x) = n=1 (n+1)px
  13.2.1 Definitons, remarquable links
  13.2.2 Calculation of certain functions
 13.3 Applications of calculation of certain series
  13.3.1 With x = 1
  13.3.2 With x = 12
  13.3.3 With x = -1
  13.3.4 With x = i
  13.3.5 With          V~ -
x = 12 + i23
  13.3.6 Withx = - 1+ i V~ 3
      2   2
  13.3.7 With     1   i
x = 2 + 2
 13.4 Generalisations
  13.4.1 Euler's sum
  13.4.2 A formula combining Harmonic and combination
  13.4.3 An other formula
  13.4.4 Harmonics of harmonics!
14 Series of greater factorial: Ramanujan, Borwein, Chudnovsky....
 14.1 Squared central binomial coefficients: Elliptical formulae
 14.2 Cubic central binomial ceofficients: Ramanujan's Identity
15 Other hypergeometric formulae concerning p


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