Polylogarithm

Author

Eric W. Weisstein
December 2, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Polylogarithm.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Polylogarithm.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plots

Plot[Evaluate[Table[PolyLog[n,x],{n,-2,2}]],{x,-2,1},PlotStyle->{RGBColor[1,0,0],RGBColor[1,1,0],RGBColor[0,1,0],RGBColor[0,0,1],RGBColor[1,0,1]},PlotRange->{-1,1}]

Special Values

Table[PolyLog[n,x],{n,-2,2}]

FullSimplify[Table[PolyLog[n,1/2],{n,4}]]

(Subscript[Li, 0])(1)

PolyLog[0,1]

Limit[PolyLog[0,n],n->1,Direction->#]&/@{-1,1}

PolyLog[0,.99999]

(Subscript[Li, 2])(1/2)

v={1,Pi^2,Log[2],Log[2]^2,Log[2]^3,Zeta[3]};
c=PolyLog[2,1/2];
LatticeReduce[Rationalize[N[Transpose[Join[IdentityMatrix[Length[v]+1],
{10^15Append[v,c]}]],30],10^-30]]//N

soln=Drop[Round[First[%]],-1];
exact=Drop[soln,-1].v/Last[soln]

N[exact+c,20]

(Subscript[Li, 3])(1/2)

(1/24) (4 ((Log[2])^3)-(π^2) Log[4]+21 Zeta[3])

v={1,Pi,Pi^2,Pi^3,Pi^4,
Log[2],Log[2]^2,Log[2]^3,Log[2]^4,
Pi Log[2],Pi Log[2]^2,Pi Log[2]^3,Pi Log[2]^4,
Pi^2Log[2],Pi^2Log[2]^2,Pi^2Log[2]^3,Pi^2Log[2]^4,
Pi^3Log[2],Pi^3Log[2]^2,Pi^3Log[2]^3,Pi^3Log[2]^4,
Zeta[3],Zeta[3]^2,Zeta[5],Zeta[5]^2};
v={Log[2]^3,Pi^2 Log[2],Zeta[3]};
c=PolyLog[3,1/2];
red=LatticeReduce[Rationalize[N[Transpose[Join[IdentityMatrix[Length[v]+1],
{10^15Append[v,c]}]],30],10^-30]];
Last@Transpose[N[red]]

soln=Drop[Round[First[%]],-1];
exact=Drop[soln,-1].v/Last[soln]

N[exact+c,20]

(Subscript[Li, 4])(1/2)

Does there exist an expansion of this?

FunctionExpand[PolyLog[4,(1/2)]]

(1/24) (4 ((Log[2])^3)-(π^2) Log[4]+21 Zeta[3])

v={1,Pi,Pi^2,Pi^3,Pi^4,
Log[2],Log[2]^2,Log[2]^3,Log[2]^4,
Pi Log[2],Pi^2 Log[2],
Log[2] Zeta[3],Log[2]Zeta[5],
Zeta[3],Zeta[3]^2,Zeta[5],Zeta[5]^2};
c=PolyLog[4,1/2];
LatticeReduce[Rationalize[N[Transpose[Join[IdentityMatrix[Length[v]+1],
{10^20Append[v,c]}]],100],10^-100]]//N

Integrals

Bose-Einstein

Integrate[((k^s)/(Exp[k-μ]-1)),{k,0,∞}]

Fermi-Dirac

Integrate[((k^s)/(Exp[k-μ]+1)),{k,0,∞}]

Sums

(∑_k^∞)((z^k)/(k^n))==(PolyLog[n, z])

V7

Sum[((z^k)/(k^n)),{k,∞}]

(∑_k^∞)(k^n) (z^k)==(PolyLog[-n, z])

V7

Sum[(z^k) (k^n),{k,∞}]

(∑_n^∞)(z^n) (HarmonicNumber2[n, r])==((PolyLog[r, z])/(1-z))

V6

Sum[HarmonicNumber[n,r]z^n,{n,∞}]

V7

Sum[HarmonicNumber[n,r]z^n,{n,∞}]

Eulerian

Table[{
PolyLog[-n,r],
Sum[Eulerian[n,i]r^(n-i),{i,0,n}]1/(1-r)^(n+1)
},{n,6}]

Subtract@@@%

Analytic Expressions

v=Table[n,{n,-10,10,1/8}];
Select[{#,PolyLog[2,#]}&/@v,!MatchQ[#[[2]],PolyLog[_,_]]&]

35/2Zeta[3]-Pi^2Log[2]-(36PolyLog[3,1/2]-18PolyLog[3,1/4]-4PolyLog[3,1/8]+PolyLog[3,1/64])//N

2Log[2]^3-7Zeta[3]-(-24PolyLog[3,1/2]+18PolyLog[3,1/4]+4PolyLog[3,1/8]-PolyLog[3,1/64])//N

10Log[2]^3-2Pi^2Log[2]-(-48PolyLog[3,1/2]+54PolyLog[3,1/4]+12PolyLog[3,1/8]-3PolyLog[3,1/64])//N

Wrong Identities

$$(\ln 2)^2=\Li_2(\h)-6\Li_2(\q)-2\Li_2(\fu{8})+\Li_2(\fu{64})\eqno\eqnum$$

PolyLog[2,1/2]-6PolyLog[2,1/4]-2PolyLog[2,1/8]+PolyLog[2,1/64]-Log[2]^2//N

$$\displaylines{
{\Li_m(\fu{64})\over 6^{m-1}}-{\Li_m(\fu{8})\over 3^{m-1}}-{2\Li_m(\q)\over 2^{m-1}}+{4\Li_m(\h)\over 9}
-{5(-\ln 2)^m\over 9m!}\hfill\cr
\hfill +{\pi^2(-\ln 2)^{m-2}\over 54(m-2)!}-{\pi^4(-\ln 2)^{m-4}\over 486(m-4)!}
-{403\zeta(5)(-\ln 2)^{m-5}\over 1296(m-5)!}\hfill\cr
\hfill =0.\quad\eqnum\cr}$$

Polylog Derivatives

PolyLog[-2,x]

PolyLog[-1,x]

PolyLog[0,x]

PolyLog[1,x]

PolyLog[2,x]

PolyLog[n,1]

PolyLog[n,2]

D[PolyLog[n,x],x]

D[PolyLog[n,x],n]

PolyLog[n,-1]

zeta[n_]:=eta[n]/(1-2^(1-n))
eta[n_]:=Zeta[n](1-2^(1-n))

Simplify[D[PolyLog[n,-1],n] /. Zeta->zeta]

PolyLog[n,-1] /. Zeta->zeta

Limit[D[eta[x],x],x->0]

Limit[D[PolyLog[n,-1],n],n->0]

Order-16 Ladder

l={{1,657517430619563136979927560311907031673621211923757556039680},{2,359886000860172792447593380451304667509342919316395916742210},{3,-141684894142517938914234443949080358856724456088731117486080},{4,196781701168242866806602275650062000534697577654107935879825},{5,745935348767332443069712397147123742304948147537862045466624},{6,405488816708650056442253795413843043306948102060187723235830},{7,-416406976403628958702584514595218236213894104883105761525760},{8,-547505077081618409188348198016100365604103632542087633890050},{9,-168975342608695257899656804048229345162975246342759199539200},{10,-230249343614447384417237255880683163036311216601594850723110},{11,-686361499027198737843324163158516540202013109745360933355520},{12,-555211669354532402241841641332915138353295668921519456494790},{13,192815301162191732499187837598868261035304857047105259765760},{14,679734532959241358849438888514522682758692394636208781152890},{15,542262843988600709919208184100477463027334034487684580835328},{16,424235600610741544849445172143286300309734125137633061204035},{17,523715028246745518068354598319413022191932310750064994549760},{18,391765228725624091642051568538398618159246632370196009838220},{19,-198765352117361763621609618366336349702431673161386991943680},{20,-681771795414951811771620988305943452975785965232369963176182},{21,-711962908052232232505015799802945837905471681104673666826240},{22,-474711713652489086239653502137393573002849538293395396856580},{23,-305563488793803341514102820998121597667241109957343639961600},{24,-72481617150888165437226250731733584917169373028647243993490},{25,277441845925552929689261105309806909812357891559963458994176},{26,560813990488672759149774541727869507575569280403768300321140},{27,529382955893422715273603248526440343418480284327032762204160},{28,320081042366233059912738009855303427011191818027491322896270},{29,83494538290613239874707297912762677581023131627678971658240},{30,-98012487241465589885244822793749074739104108244040188504014},{31,-259850303587986534656435580762491706946152146552951829626880},{32,-312381473762254154637700526528599595535218857011814459284405},{33,-246779089811622911261098199061852288809784802679547056947200},{34,-101370164345933351431594273523846287706927174737460926584690},{35,10117727926311381644405806386440429351025303665206665019392},{36,92813012460581905456644049699805057100444523762661712331760},{37,126324305118118269024517948173064327264644326236420977131520},{38,115549700605836091570299617434612530310676419462990155960060},{39,55524385597088576211029994800590286128713854655226225950720},{40,8681384820081409297328583655855503616180543689183078659790},{41,-19635305068579015455281545202962471423318427323980082053120},{42,-25853290618480167202972487269611444701702175335712826764200},{43,-31825441142023707248687536800399542265330919953456915087360},{44,-21238990403232997150237193584099678135190886191312613084840},{45,-5941171482587742832139996778309010890215227722504605532160},{46,6867449991711733361905911701040369140850427873891452167820},{47,2683941033568052348250476195016210444676454909493450178560},{48,750276335951027139956797045951168533453325950288012793755},{49,902953609381753090186332205561649133432166531686386565120},{50,2845791617893774854256316723847280908911507508173997476472},{52,-211518744564361609748949526157272452992328540189820187580},{54,1586305520716919558891270499363536151171957733664553920190},{55,317399573539556012199105353632753466414027589073604640768},{56,-411955065839184813654206085597826201289284276111276432380},{57,-633034566415505822568457398619791655535556228581668945920},{59,-422235192545960711449176250527328955777038182764768133120},{60,-100775809060677875668353632672084730591567385836002358340},{62,103161260941635325589940465107263862028544860193511288430},{63,-71239824525936132532239108327787334379823490760552611840},{64,-758961502664172364046541735177189723413489520451845090},{65,-31614329666390498308045970054858836918855503060815904768},{66,82590308807941016435157163298430216384039435573691923590},{68,-6111893964221453989555021260745192324274991457297917375},{69,36723207357049059339848864763934042997356103557409013760},{70,56587501615035807034718738004427311071764030581924717666},{72,6246571495623829969272301654360557650197045971497886770},{74,4662327569727635575068647456890293743838455770712101950},{75,14887216546252872098059151093987262212120215877194153984},{76,4933420845476441755128409798981902904608462970609785585},{77,-3718375935394449285719242013677644130212998760206499840},{78,797543027544694301981107935492458580524804545433769960},{80,-756655048589380952520516473920663602783235627959554055},{82,599221956438568586892136999602126203104200052611696840},{84,1711070904277206791256558392136135387624754115179524010},{86,971235386414297706563950708020005562296475828657742770},{90,712743581327703411348920012655781994420044560878114988},{91,581533337739120206692138569579040767870857035407360000},{92,281171208042281370705030709443218746311661931924480940},{93,-123474844527238243730367824366624184577805521172234240},{96,45358670699766680263040092382358986091316360899268450},{98,-27555957317558382879221563890431186933354691518749590},{102,-13957477138823786670980509990907214192819210555392000},{105,-51592349989081948844625156379637404692389576604581888},{106,-49757317433194357135489925493889989029913561751388160},{108,4104159613575523414703444177171856613076465917080545},{110,436503492818404043948452074062000221710038770175550},{118,12885595475645773664830818192362333855500432823631840},{120,-4458271535866469116508257194072443804522804186654722},{123,684209085353296089217162854388925631175894697902080},{124,-2568170165780877908485719228826518196430780458722240},{126,-1238638691691188975878629542803876608020141601562500},{130,1661804350411188506429822540247968366095406030963502},{132,-1633602118796242535519471886453758168320481063983580},{135,1847684862191853167535965349249779010200990442520576},{138,-1120703349519319437861598656125916839518924058758820},{143,174515941261835218319905200778076418833749415362560},{144,-317934100357607293764664092299747258866505243251140},{154,113475828106520058768287414968189823309722862555130},{160,-10275921042899706253486332862758642766530474426501},{165,-23116946816383260126447835448067026267507493044224},{175,-32347514260620927078585304389860102844649182658560},{180,-4479106615252570847623887867187960984130837217280},{182,-17746989066745611776493486620454124996058869488750},{186,3768153214332221793529291515094732195367600133430},{195,-2675778372359720766614632517822289310362850099200},{204,425948399011956380339981383999853948755469072125},{210,-291194657074029253989297333815629327367478738414},{212,1518472822057933262191465011410216950375780082745},{240,76199283284999932540837893030287171626166802389},{246,-20880404216104006628941737499662037084225302060},{270,-56386867132319737778807536293023041082793897782},{286,-5325803871515967355954138207338757898979169170},{315,-664989084046735447844305843955260271683239936},{360,136691486061174647449459468603148223392664710},{630,20293856324668440180795466429298714345802}};

TableForm[l,TableAlignments->Right]

Length/@IntegerDigits/@Last/@l

a0=2^3 11 1770708910425291120521033962427 23216857398851664164043691705297

α=Last[Reduce[x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1==0&&x>1,x,Reals]]

N[α,20]

N[{Total[#[[2]]PolyLog[16,(α^(-#[[1]]))]&/@l],a0(( PolyLog[16,1])/3617)},10]//Timing

N[((Total[#[[2]]PolyLog[16,(α^(-#[[1]]))]&/@l])/(( PolyLog[16,1])/3617)),100]

Borwein Integral Indentity

((2^n)/(n!))Integrate[(((Log[x+Sqrt[(x^2)+1]])^n)/x),{x,0,1/2}]

Do[Print[FullSimplify[((2^n)/(n!))Integrate[(((Log[x+Sqrt[(x^2)+1]])^n)/x),{x,0,1/2}]]//Timing],
{n,3}
]

L[n_]:=With[{g=GoldenRatio-1},Zeta[n+1]-((n(((-Log[(g^2)]))^(n+1)))/(2(n+1)!))-Sum[((((-Log[(g^2)]))^(n+1-j))/((n+1-j)!))PolyLog[j,g^2],{j,2,n+1}]
]

Table[L[n],{n,5}]//N

Table[((2^n)/(n!))NIntegrate[(((Log[x+Sqrt[(x^2)+1]])^n)/x),{x,0,1/2}],{n,5}]

FullSimplify[L[1]]//TraditionalForm

FullSimplify[L[2]]//TraditionalForm

FullSimplify[L[3]]//TraditionalForm

FullSimplify[L[4]]//TraditionalForm