$\newcommand{\mt}[4]{\begin{bmatrix}#1\\#3\end{bmatrix}}$ Via jupytext this file can be shown as a jupyter notebook.
Chudnovsky family of $\frac{1}{\pi}$ formulas¶
This file computes Ramanujan-Sato series for $\frac{1}{\pi}$ of the Chudnovsky family 1, i.e. for Sato triples of the form \begin{align} (N, \gamma_N, \tau_N) &= \left( N, \mt{-1}{-1}{1}{0}, \frac{-1+i\sqrt{4N-1}}{2N} \right) \end{align} and of the Chudnovsky family 2, i.e. for Sato triples of the form \begin{align} (N, \gamma_N, \tau_N) &= \left( N, \mt{0}{-1}{1}{0}, \frac{i\sqrt{N}}{N} \right). \end{align}
In [1]:
)cd ..
)read input/jfricas-test-support.input )quiet
The current FriCAS default directory is /home/hemmecke/backup/git/qeta All user variables and function definitions have been cleared. All )browse facility databases have been cleared. Internally cached functions and constructors have been cleared. )clear completely is finished. The current FriCAS default directory is /home/hemmecke/backup/git/qeta/tmp
In [ ]:
)set output formatted off
)set output algebra on
Setup¶
In [2]:
-------------------------------------------------------------------
--setup
-------------------------------------------------------------------
In [3]:
)set stream calculate 2
C ==> QQ
MZZ ==> Matrix ZZ
QCMODPOL ==> QEtaClassicalModularPolynomial
AN ==> AlgebraicNumber
SATOTRI ==> Record(fnn: PP, fgamma: MZZ, ftau: AN, ftaufd: AN)
QETAAUX ==> QEtaAuxiliaryPackage
Px ==> UP('x, ZZ)
In [4]:
digits 100;
In [5]:
)read projectdir.input )quiet
phindir := PROJECTDIR "/data/phiN";
In [6]:
psiFactors(nn) ==> (_
phiN := classicalModularPolynomial(nn,phindir)$QCMODPOL;_
phiNxx := eval(phiN, 'y='x)::Px;_
psiN := squareFreePart phiNxx;_
[fac.factor for fac in factors factor psiN])
In [7]:
gammaFD tau ==> toFundamentalDomain(tau, 1)
In [8]:
oopiC(nn,gammaN,tauN) ==> (_
gammafd := gammaFD tauN;_
taufd := moebiusTransform(gammafd, tauN);_
psiNfactors := psiFactors nn;_
jAtTauN := kleinJAt(1, taufd, psiNfactors);_
zN := simplifyRadicals(1/jAtTauN);_
basedir := PROJECTDIR "/data/oneoverpi/chudnovsky";_
chudnovskyOneOverPiFormula(nn,gammaN,tauN,zN,basedir))
In [9]:
oopiC1(nn) ==> (_
gammaN := matrix [[-1,-1],[1,0]];_
tauN := (-1+sqrt(1-4*nn))/(2*nn);_
oopiC(nn, gammaN, tauN))
In [10]:
oopiC2(nn) ==> (_
gammaN := matrix [[0,-1],[1,0]];_
tauN := sqrt(-nn)/nn;_
oopiC(nn, gammaN, tauN))
In [11]:
-------------------------------------------------------------------
--endsetup
-------------------------------------------------------------------
Chudnovski family 1¶
In [12]:
oopiC1 2
-- approximate with m:=1 -- abfcands:=[[63, 8, sqrt(15)/75], [63, -8, sqrt(15)/75], [63, 8, -sqrt(15)/75], [63, -8, -sqrt(15)/75]] -- approximate with m:=2 -- abfcands:=[[63, 8, sqrt(15)/75], [63, -8, -sqrt(15)/75]]
Out[12]:
\[
\frac{1}{\pi }=\frac{\sqrt{15}}{75}\, \left(\sum_{n=0}^{\infty }{{\left(-\frac{1}{3375}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(63\, n+8\right)}\right)
\]
In [13]:
oopiC1 3
Out[13]:
\[
\frac{1}{\pi }=\frac{\sqrt{2}}{64}\, \left(\sum_{n=0}^{\infty }{{\left(-\frac{1}{32768}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(154\, n+15\right)}\right)
\]
In [14]:
oopiC1 5
Out[14]:
\[
\frac{1}{\pi }=\frac{\sqrt{6}}{192}\, \left(\sum_{n=0}^{\infty }{{\left(-\frac{1}{884736}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(342\, n+25\right)}\right)
\]
In [15]:
oopiC1 7
Out[15]:
\[
\frac{1}{\pi }=\frac{3\, \sqrt{30}}{1600}\, \left(\sum_{n=0}^{\infty }{{\left(-\frac{1}{12288000}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(506\, n+31\right)}\right)
\]
In [16]:
oopiC1 11
Out[16]:
\[
\frac{1}{\pi }=\frac{\sqrt{15}}{3200}\, \left(\sum_{n=0}^{\infty }{{\left(-\frac{1}{884736000}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(5418\, n+263\right)}\right)
\]
In [17]:
oopiC1 13
Out[17]:
\[
\frac{1}{\pi }=\frac{\sqrt{102}\, \sqrt{-31\, \sqrt{17}+295}}{613632}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{1519\, \sqrt{17}-6263}{14155776}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(33558\, n-400\, \sqrt{17}+3145\right)}\right)
\]
In [18]:
oopiC1 17
Out[18]:
\[
\frac{1}{\pi }=\frac{\sqrt{330}}{580800}\, \left(\sum_{n=0}^{\infty }{{\left(-\frac{1}{147197952000}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(261702\, n+10177\right)}\right)
\]
In [19]:
oopiC1 19
Out[19]:
\[
\frac{1}{\pi }=\frac{\sqrt{330}\, \sqrt{-165393\, \sqrt{5}+3777190}}{356835776}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{165393\, \sqrt{5}-369830}{5887918080}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(92158\, n-1600\, \sqrt{5}+6965\right)}\right)
\]
In [20]:
oopiC1 23
Out[20]:
\[
\frac{1}{\pi }=\frac{\sqrt{1326}\, \sqrt{71\, \sqrt{13}-146}}{721344}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{-1623699\, \sqrt{13}+5854330}{4346707968}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(18018\, n-320\, \sqrt{13}+1755\right)}\right)
\]
In [21]:
oopiC1 29
Out[21]:
\[
\frac{1}{\pi }=\frac{\sqrt{330}\, \sqrt{-34827\, \sqrt{5}+105310}}{100943040}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{21627567\, \sqrt{5}-48360710}{29439590400}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(359766\, n-10176\, \sqrt{5}+33433\right)}\right)
\]
In [22]:
oopiC1 41
Out[22]:
\[
\frac{1}{\pi }=\frac{\sqrt{10005}}{4270934400}\, \left(\sum_{n=0}^{\infty }{{\left(-\frac{1}{262537412640768000}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(545140134\, n+13591409\right)}\right)
\]
Chudnovski family 2¶
In [23]:
oopiC2 2
Out[23]:
\[
\frac{1}{\pi }=\frac{\sqrt{5}}{25}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{1}{8000}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(28\, n+3\right)}\right)
\]
In [24]:
oopiC2 3
Out[24]:
\[
\frac{1}{\pi }=\frac{2\, \sqrt{15}}{25}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{1}{54000}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(11\, n+1\right)}\right)
\]
In [25]:
oopiC2 5
Out[25]:
\[
\frac{1}{\pi }=\frac{\sqrt{55}\, \sqrt{-5967\, \sqrt{5}+21650}}{126445}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{884\, \sqrt{5}-1975}{2129600}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(836\, n-15\, \sqrt{5}+93\right)}\right)
\]
In [26]:
oopiC2 7
Out[26]:
\[
\frac{1}{\pi }=\frac{18\, \sqrt{255}}{7225}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{1}{16581375}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(133\, n+8\right)}\right)
\]
In [27]:
oopiC2 11
Out[27]:
\[
\frac{1}{\pi }=\frac{\sqrt{5423}\, \sqrt{\left(-3152822448369087\, \sqrt{33}+1671938620951599\right)\, {\sqrt[3]{12673551\, \sqrt{33}+20882819556991}}^{2}+\left(86821860140346056544\, \sqrt{33}+46039822427087764704\right)\, \sqrt[3]{12673551\, \sqrt{33}+20882819556991}+8099067989505947112945220}}{81859590447596465431233446636633588}\, \left(\sum_{n=0}^{\infty }{\frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(\left(-2003538790\, \sqrt{33}-102284495730\right)\, {\sqrt[3]{12673551\, \sqrt{33}+20882819556991}}^{2}+\left(55174819686085\, \sqrt{33}-2816694143119125\right)\, \sqrt[3]{12673551\, \sqrt{33}+20882819556991}+2261057708396291474516\, n+263632521437104356540\right)\, {\left(\frac{\left(17165366663342807\, \sqrt{33}-9102776936292039\right)\, {\sqrt[3]{12673551\, \sqrt{33}+20882819556991}}^{2}+\left(-472696794097439641184\, \sqrt{33}-250661255436366718944\right)\, \sqrt[3]{12673551\, \sqrt{33}+20882819556991}+13805596914850663516307472}{100052103114133896994767941376}\right)}^{n}}\right)
\]
In [28]:
oopiC2 13
Out[28]:
\[
\frac{1}{\pi }=\frac{\sqrt{4485}\, \sqrt{-41\, \sqrt{13}+14230}}{22178325}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{4428\, \sqrt{13}-15965}{2628072000}\right)}^{n}\, \frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(20124\, n-125\, \sqrt{13}+1339\right)}\right)
\]
In [29]:
oopiC2 17
Out[29]:
\[
\frac{1}{\pi }=\frac{\sqrt{3995}\, \sqrt{\left(-226005120\, \sqrt{34}+931842981\, \sqrt{2}\right)\, \sqrt{470141877665\, \sqrt{17}+1938444620639}-14510556\, \sqrt{17}+999402460}}{4845184193802350}\, \left(\sum_{n=0}^{\infty }{\frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, {\left(\frac{\left(33482240\, \sqrt{34}-138050812\, \sqrt{2}\right)\, \sqrt{470141877665\, \sqrt{17}+1938444620639}+2149712\, \sqrt{17}-9148295}{240038776000}\right)}^{n}\, \left(3\, \sqrt{799}\, \sqrt{\left(-53642250544845\, \sqrt{34}+221172617181575\, \sqrt{2}\right)\, \sqrt{470141877665\, \sqrt{17}+1938444620639}-74806228618500\, \sqrt{17}+529868253638764}+20643609592\, n\right)}\right)
\]
In [30]:
oopiC2 19
Out[30]:
\[
\frac{1}{\pi }=\frac{\sqrt{41287}\, \sqrt{\left(-10929292898297887503\, \sqrt{57}+46790959723152066565\right)\, {\sqrt[3]{1389621159\, \sqrt{57}+2160326138439465613}}^{2}+\left(14128579635589648189198368\, \sqrt{57}+60487883615609351909424352\right)\, \sqrt[3]{1389621159\, \sqrt{57}+2160326138439465613}+42606624779792709583748590794916}}{1972509196675587550031322499834356639105499452}\, \left(\sum_{n=0}^{\infty }{\frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(\left(-2619609510330\, \sqrt{57}-136303612260410\right)\, {\sqrt[3]{1389621159\, \sqrt{57}+2160326138439465613}}^{2}+\left(3386437097617067535\, \sqrt{57}-176203205918294828165\right)\, \sqrt[3]{1389621159\, \sqrt{57}+2160326138439465613}+5999263612395157391363472204\, n+674613885598536071619092260\right)\, {\left(\frac{\left(98363636084680987527\, \sqrt{57}-421118637508368599085\right)\, {\sqrt[3]{1389621159\, \sqrt{57}+2160326138439465613}}^{2}+\left(-127157216720306833702785312\, \sqrt{57}-544390952540484167184819168\right)\, \sqrt[3]{1389621159\, \sqrt{57}+2160326138439465613}+1407496543855661362660114138674064}{3094772256357919054123135315671316224}\right)}^{n}}\right)
\]
In [31]:
oopiC2 23
Out[31]:
\[
\frac{1}{\pi }=\frac{2\, \sqrt{67045}\, \sqrt{\left(-8570263017626724577073184\, \sqrt[3]{2}\, \sqrt{69}-22213533635460486281457792\, \sqrt[3]{2}\right)\, {\sqrt[3]{3292165368405969\, \sqrt{69}+3223144068915206797209803}}^{2}+\left(1004794831396743845776257973990320\, {\sqrt[3]{2}}^{2}\, \sqrt{69}-2604359316146865623377463141321040\, {\sqrt[3]{2}}^{2}\right)\, \sqrt[3]{3292165368405969\, \sqrt{69}+3223144068915206797209803}+51746583480088207642557566381137442539068275}}{116688986454332740763903262419365247358874974514821305446249416875}\, \left(\sum_{n=0}^{\infty }{\frac{\left(6\, n\right)!}{{n!}^{3}\, \left(3\, n\right)!}\, \left(\left(-1298518765388007400864\, \sqrt[3]{2}\, \sqrt{69}-248449532355399201452932\, \sqrt[3]{2}\right)\, {\sqrt[3]{3292165368405969\, \sqrt{69}+3223144068915206797209803}}^{2}+\left(152240974726504358893023916090\, {\sqrt[3]{2}}^{2}\, \sqrt{69}-29128721591303152721423324392030\, {\sqrt[3]{2}}^{2}\right)\, \sqrt[3]{3292165368405969\, \sqrt{69}+3223144068915206797209803}+304058026085873403634730669660907287947135\, n+23750938979670453744469722082492817838760\right)\, {\left(\frac{\left(2916270054609093779698514\, \sqrt[3]{2}\, \sqrt{69}+7558771862066415470773832\, \sqrt[3]{2}\right)\, {\sqrt[3]{3292165368405969\, \sqrt{69}+3223144068915206797209803}}^{2}+\left(-341909352350280891965532227260595\, {\sqrt[3]{2}}^{2}\, \sqrt{69}+886205600633308441288164541143965\, {\sqrt[3]{2}}^{2}\right)\, \sqrt[3]{3292165368405969\, \sqrt{69}+3223144068915206797209803}-415602083435429457554072014414190772251900}{29708830686115443991170412591201094558520862500}\right)}^{n}}\right)
\]
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