Complete elliptic integral of the first kind $K(m_r)$ at the singular values

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Numbers

$r$

$K(m_r)$

1.854074677301371918433850347195260046217598823521766905585928045056021776838119978357271861650371897

comment: $K(m_1)=\Gamma(1/4)^2/(4\sqrt\pi)$ [4].
equals: Complete_elliptic_integral_of_the_first_kind_K#1/2

1.645568395293458039866051685287072715999557026055401037265292137149578863729330871593184129832048067

comment: $K(m_2)=\sqrt{1+\sqrt2}\,\Gamma(1/8)\Gamma(3/8)/(2^{13/4}\sqrt\pi)$ [4].

1.598142002112540144460965105390112888900611097136742843395001850717051182376299048153673076745023371

comment: $K(m_3)=3^{1/4}\Gamma(1/3)^3/(2^{7/3}\pi)$ [4].

1.582551727223715911833135071070409876529488149618789243497169448478208535186663551736209814065543222

comment: $K(m_4)=(1+\sqrt2)\Gamma(1/4)^2/(2^{7/2}\sqrt\pi)$ [4].

1.576390394802592476323757555260863247741937034744074005958674897925807012418125362905820159225379507

comment: $K(m_5)=(2+\sqrt5)^{1/4}\sqrt{\Gamma(1/20)\Gamma(3/20)\Gamma(7/20)\Gamma(9/20)/(160\pi)}$ [4].

1.573656231264378266467658770764262615447771495128054395083197322369080422472509636815994782912924097

comment: $K(m_6)=\sqrt{(\sqrt2-1)(\sqrt3+\sqrt2)(2+\sqrt3)}\sqrt{\Gamma(1/24)\Gamma(5/24)\Gamma(7/24)\Gamma(11/24)/(384\pi)}$ [4].

1.572339753406137527416490497176982668311869155892281594080158998982190209092426217240368536444070701

comment: $K(m_7)=\Gamma(1/7)\Gamma(2/7)\Gamma(4/7)/(4\cdot7^{1/4}\pi)$ [4].

1.571665689206437042031861058437591105397806921254527952296192373178656438735452990723968471906921488

comment: $K(m_8)=\sqrt{(2\sqrt2+\sqrt{1+5\sqrt2})/(4\sqrt2)}(1+\sqrt2)^{1/4}\Gamma(1/8)\Gamma(3/8)/(8\sqrt\pi)$ [4].

1.571303417736683352208481497338796809391174821642956915852621765840171701094284147643588839960207742

comment: $K(m_9)=3^{1/4}\sqrt{2+\sqrt3}\,\Gamma(1/4)^2/(12\sqrt\pi)$ [4].

10:

1.571100880616723881003002373510769622686969652665615951839918153838543584392595612006151433598168530

comment: $K(m_{10})=\sqrt{2+3\sqrt2+\sqrt5}\,\sqrt{\Gamma(1/40)\Gamma(7/40)\Gamma(9/40)\Gamma(11/40)\Gamma(13/40)\Gamma(19/40)\Gamma(23/40)\Gamma(37/40)/(2560\pi^3)}$ [4].

11:

1.570983855774829120451912490395857958850098334483956144147357301154551652926723296219209863525485519

12:

1.570914318015180699495529303260880207317602201733998524304216557139476320543522033579062257357751935

13:

1.570871985028043952749736101704886627387040866366313479008915381835799340337021811653869373998330152

14:

1.570845661713460841142136866693181381505564807352363146293953030731769301777275789346509408539752770

15:

1.570828983715412189014263369690341331157381974519987484413652593357045521096949475513596117316161571

16:

1.570818238489563571061465624767418199316574829280794041540263412897062335464087574121035010812223044

17:

1.570811210569246350315833542812695219975250411517325149941147076638890250732735903365533100133375389

18:

1.570806550820860128459264105073204076999713941053078149614108321027265943125168968231381696017455202

19:

1.570803422600539339147263427534529867936127278497503291871877834131983532904300361899650830890676902

20:

1.570801298480576273076828176038966953042066924863155606033675531419662408308177857085577143458815867

21:

1.570799840960255689540380725484294688344479267147672113759890502212856069457723602326453805515470460

22:

1.570798831102597942427213451217345101849901250799098176934411312864488160504963907894365093172581116

23:

1.570798125089600450799332700055066736548088666998897666213026876012652006331037877296446458392403757

24:

1.570797627348300815818900626198938104378607935735114561699332315966250852544718915512886255240617429

25:

1.570797273681919556386615751574864629728925148595052292776397459868624710874083700183914110963972972

26:

1.570797020537337764053708658796850307385974779105990598542309007845676718351355037185094871251269745

27:

1.570796838090910814596668635681037842448978965908923686069254149298276239434313240178931931554353790

28:

1.570796705742217178181219147748659480170667852551030876164894549915542986435953618912551506432903829

29:

1.570796609145268110282182838284465469881425041312944924020828792335080181415016126280160088452812315

30:

1.570796538232790280232806657195617357197921542094411298211495154951328009768928198021243731820477144

31:

1.570796485888910458598789198774993251614549402905163405907619206620479693157577054685702438508116629

32:

1.570796447049497577532156058067250102727897517833724064845089153948064628447915975000133900284642459

33:

1.570796418087257122332626717435851867343338766147455037886837880521753812315040159735257125338292067

34:

1.570796396388125620985711791278147484787574018874063464624413914941264668495776234179959009499430632

35:

1.570796380057269235182508858811971930284382969918975778544082609148992318405954650177873588292896135

36:

1.570796367713589133425967663504495952361254556562457966678576148419977019506905670672466806961346081

37:

1.570796358345125063315498446322001862480128531372327059851580279082244644619749771677800498210060710

38:

1.570796351206660974881532554934077800790177542057333085236058251926358969784877490154375875396489122

39:

1.570796345746775441747181762488336866775003103075419776590719581423951150390054475749298837228778201

40:

1.570796341555569025468133169800120084822583718698986782886668900698743426141466990278278017187763039

41:

1.570796338327001112000963077512918173340827991439382615232911426682879650215776356160301170915690632

42:

1.570796335831609933409288142922122968880189815494206490715088079065062772319998350306254121907299254

43:

1.570796333896653562065973299445682779664220715163555980410292586319444916781796703385108857788982454

44:

1.570796332391583263999439853378107213382015870957778323053630285389736495547661068777419260325927121

45:

1.570796331217368155607128543702039856513385232080723089966092013586524525642349117286798113882478019

46:

1.570796330298615194968675939580017721588218755972871268025013975199586536128731698236910544837372961

47:

1.570796329577727677017453655111453457919477232681593758841232755494410391965863795751906740001763129

48:

1.570796329010557074299465757945398938745038629633055761758503478753433204016544909644382240561200160

49:

1.570796328563153641982175045818781613738580489175103029773123352665366582032112395046835074583171319

50:

1.570796328209328867760252289734130056976554395772655151614014344157092592209855635214913417367896351

51:

1.570796327928820202661057632062664994553637303920307903102284418989090686509892691484142795275715116

52:

1.570796327705904350870999563016529830797881410452962852502451574954797387938215203244480795327557466

53:

1.570796327528345891671854276095271432952110059828770032771197183337919261018281283823226178181192190

54:

1.570796327386597500767858918246605151719873969287567516030082520743153114720569195128161361146950437

55:

1.570796327273189610802651294049672113922265944368436364735080204155400791487637459278962036758111730

56:

1.570796327182263193612086568906575281720110458528837220812310375038310663527849948283182784201997330

57:

1.570796327109211027362264766185313660841576153661045791394538341940289405259917311620941680942571337

58:

1.570796327050401456389318691007637636796443162741001760219310644493396988723492018515053783429761842

59:

1.570796327002965072895787821455539429599641145308235581628066508837204011589773319980271533387698920

60:

1.570796326964629536465272276703809838586320261738242198747275666052751130934628195633560983622544365

61:

1.570796326933591327997818043699545540269043641733030522390396625246324207414823981110252750357948281

62:

1.570796326908415922016381475453612057836123538264064109392633694807096814655691372872475728630275565

63:

1.570796326887959884739628623996513660278533715256081042663547127768677027843224555293573566849794055

64:

1.570796326871309933661242882791513724375785371421285278066318895920333193230564426073477992806358674

65:

1.570796326857735152283731787797640657882199880526378349773573844016370580362822236052348473108708269

66:

1.570796326846649433509842184335468629714547978422197314231936247312215292264337634874847732827746822

67:

1.570796326837581895691312290722580344957945411348901468360477473204709970497921208476717150477641165

68:

1.570796326830153527718698094966616035414685325787489293665534949136166426478373130879783089529048314

69:

1.570796326824058712158263393523788102649561404083225069562036905179559007410364881246979650550155285

70:

1.570796326819050577091604612875907450257592755460382544954248117105811613597614034031617689838050762

71:

1.570796326814929359667093337158871057596661836524182435899304484942867285693349389423904923557456781

72:

1.570796326811533143797786850271791028917611502943417518809561565619563434562702641286794617016259825

73:

1.570796326808730473333587261485782433775827440127047349635388565400406323365880606628538780342147101

74:

1.570796326806414449808826336043547232306465736820084642035602177494487123560107249699848390492157869

75:

1.570796326804498007687101443199687263760089409812101305357333170473637226093378661493614461092190456

76:

1.570796326802910125364474647747338739442668541899416755659805030295661563849160341585949585394879647

77:

1.570796326801592781237702505520103774855012980201428052109678363005994174379110227369827902791295882

78:

1.570796326800498504109071238199152582134633989418281611543263770494639401927944357399401457079868535

79:

1.570796326799588398555005321292989282634267010502953370698474335686958452100650019136127429746211179

80:

1.570796326798830550537234664578836515255547789174648259155095780188005804763878648197065055382038824

81:

1.570796326798198737955321811120023544485697790678742230470936729526816079631954891118694298056168967

82:

1.570796326797671386168141272447900035645771082539218385486322139622439782482674372125757157037029375

83:

1.570796326797230720634659510906809710949910300330522222006654790126054490142565387018923507312009279

84:

1.570796326796862078429682280637314009024262810833092762837962618861007596315077042711400663699309148

85:

1.570796326796553348014911813914125450004354819699068496171333879221455533884107552679789366154905921

86:

1.570796326796294512709014299690530347524805883650353581394352105468132934680206002551534682431177839

87:

1.570796326796077278130661050250714423098371790069926158188282634153787296361063738814453487453507802

88:

1.570796326795894767743096500853535637715482289368010393188468856527615866100610417453739683264069958

89:

1.570796326795741273709986873140382226553177694211587024352358819023650724035950962916062295065656217

90:

1.570796326795612052739372688698784420833749912357852960008298200576266330188427464962157836692181291

91:

1.570796326795503158571102169636490585209677676371645369518879992895966875997782720606037116710312368

92:

1.570796326795411304352421480873304292219148626626580504496526840507582239804391286693404956807795529

93:

1.570796326795333749425016284867691947081587178162483996887221786120892666058688540126497249462080825

94:

1.570796326795268206077045211345772072117255195308225206840463956470426886884348943203308365763336693

95:

1.570796326795212762645073096061943994534299413948768739522129395265701206927570909103183966755824571

96:

1.570796326795165820022680608884222547366708671157230267044357232066801746101198765020057403085966102

97:

1.570796326795126039176282413961384830904839431784674194275419003263559593092854078479513759812565093

98:

1.570796326795092297709381919663102885502202451038653916471504605510594077608281569401075926094580239

99:

1.570796326795063653874157987741729334932631721723888315814408848604466337203982037165395175002306963

100:

1.570796326795039316719957832966164435280270179702545499007551308367269481618751968918809444736644588

Definition

For $r\in\mathbb Z_{>0}$, set $m_r=k_r^2=\lambda(i\sqrt r)$, where $\lambda$ is the elliptic modular function [3]. The entry is $K(m_r)$, classically written $K(k_r)$.

Parameters

$r$

— index, with $\tau=i\sqrt r$ (positive integer)

Formulas

(1)

$K(m_r)=\frac{\pi}{2}\theta_3(0,q)^2$, where $q=e^{-\pi\sqrt r}$ and $\theta_3(0,q)=\sum_{n\in\mathbb Z}q^{n^2}$ [4].

(2)

$K(1-m_r)=\sqrt r\,K(m_r)$.

(3)

$K(m_r)=\frac{\pi}{2}\,{}_2F_1(1/2,1/2;1;m_r)$ [2].

Comments

(4)

This table writes $K(m_r)$ throughout: $K$ takes the parameter $m_r=k_r^2$, the convention of arb's elliptic_k, of the complete elliptic integral $K(m)$, and of the elliptic alpha function in its table. The classical literature on singular values, and DLMF [1], write the same number $K(k_r)$ with the elliptic modulus $k_r$. The two notations name one number, but read in the wrong convention $K(k_r)$ is a different one: $K$ at parameter $k_r$.

(5)

The lattice $\mathbb Z+\mathbb Z\,i\sqrt r$ has complex multiplication by the order $\mathbb Z[\sqrt{-r}]$, of discriminant $-4r$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField

r = 2
CBF = ComplexBallField(numberdb.bits(100, losing=256))
tau = CBF.gen(0) * CBF(r).sqrt()
m = tau.modular_lambda().real()
CBF(m).elliptic_k().real()

Links

[1]

DLMF 19.2: Definitions of complete elliptic integrals

[2]

DLMF 19.5: Hypergeometric function representations

[3]

DLMF 23.15: modular functions and the elliptic modular function $\lambda$

[4]

MathWorld: Elliptic Integral Singular Value

Similar tables

Singular values $k_r$ of the elliptic modulus — holds $k_r$ and $m_r=k_r^2$; the entries here are $K$ at those $m_r$

Complete elliptic integral of the first kind $K(m)$ — that table stores the same function at rational parameters; the entry $m=1/2$ equals the $r=1$ entry here

Values of Ramanujan's signature hypergeometric functions $F_r(x)$ — signature $2$ satisfies $F_2(m)=2K(m)/\pi$ by (3)

Values of the Gamma function at rational numbers — the low-index singular values have Gamma-product closed forms

Pólya's random walk constants — the return probability is $p(3)=1-1/u(3)$, where Watson's integral is $u(3)=\frac{12}{\pi^2}(18+12\sqrt2-10\sqrt3-7\sqrt6)K(m_6)^2$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds $K(m_r)$ for every integer $1\leq r\leq100$, matching the singular moduli $k_r$; over this range the order $\mathbb Z[\sqrt{-r}]$ has class number at most $12$)

How they were obtained:

Each value is computed in Sage's arb-backed ComplexBallField at numberdb.bits(digits, losing=256) bits. The generator evaluates $m_r=\lambda(i\sqrt r)$ as a complex ball, checks that the imaginary part contains zero, and evaluates arb's elliptic_k at $m_r$.

Before the values were sent, the generator checked (1), (2), and (3) on every computed row.