Search.setIndex({desctypes:{"0":"function","1":"attribute","2":"method","3":"class"},terms:{"9525945894834273060175651j":27,lerchphi:27,orthogon:[7,15,9,4,1],poorli:23,four:[0,27],"1e7j":10,prefix:28,totient:0,whose:29,accur:[0,8,22,23,15,25,16,26,9,27,2,10,28,11,29,20,6,21],"29j":19,"29e":20,illustr:[16,6,30],swap:1,under:[0,7,31,9],digit:[0,8,22,30,23,15,25,26,9,2,28,3,13,29,1,11,20,31,6],everi:[25,27,2,0,19,20],"19739830969552e":10,rise:[18,15,10,4,1,13],srivastava:3,trigonometr:[7,25,16,17,10,19,4,26],affect:22,primepi2:8,"272756997166375050700388j":18,"128379167095512573896159e":10,factori:[7,8,0,18,4,1,13],vector:[0,2,11,1,30,20,21],matric:[7,22,20,2],verif:[25,0,19],mp2:22,absmax:20,"434284j":10,"10j":[1,19,27,13,2],wronskian:19,repres:[0,8,22,15,25,16,17,30,2,28,11,1,19,20,26],naiv:[10,28],direct:[24,25,2,0,11,6,29,1,13,21],j_n:19,horn:1,second:[8,15,25,17,30,2,11,29,1,19,6,21],"325717j":10,dobinskisformula:8,even:[0,8,22,30,15,25,17,24,27,2,10,28,11,13,29,1,19,20,31,6],asin:[16,26],hide:[25,22],"3697207890154e":13,euclidean:[0,20],neg:[0,8,22,15,25,27,10,28,19,6,20,13],"2115710938304086076055298j":15,"7455257875808100868984496j":1,"391492658196305j":0,"82752982813540993502j":1,roundoff:2,"new":[22,20,6,21],symmetr:[15,17,27,2,4,30,20],told:30,"2453596101351438273844725j":19,elimin:1,"58775110440458732748279j":10,lerr:8,"09645e":6,never:[30,2],here:[23,15,17,9,27,11,29,3,26,6],path:[17,9,3],"630743639715893346570743e":19,interpret:[7,25,31,9,2,10],"085042824915332562522402j":19,intead:31,hankel:[19,4],"8057117299251e":8,"906755278982137052053974e":13,discontinu:[9,17,1,2],precis:[0,2,3,4,1,6,7,8,9,10,11,13,14,15,17,19,20,21,22,23,24,25,26,27,28,29,30,31],permit:[18,15,25,31,2,28,11,6,29,1,19,13,21],ainsworth:27,ellipfun:17,studi:[17,3,27,6],fourier:[17,2,27,23,5],"962288353j":8,golden:[26,8,11,4],fsub:2,"25e6":22,fsum:[28,0,8,6,2],linearli:[19,11,6],hypercub:6,"997921055881167659267063e":27,txt:27,unit:[31,23,15,0,17,9,27,2,1,26,13],highli:[28,30,9],plot:[7,23,15,28,17,2,0,19,12,21],describ:[22,9,19,6,21],would:[28,0,8,17],call:[0,8,22,23,25,17,9,18,27,2,10,28,11,13,29,1,31,30,20,26,6],"3326361850321887899e":27,recommend:[28,30,31,11],factorial2:13,type:[7,8,22,15,25,31,2,10,28,11,19,6],until:[28,9,6,2],matlab:20,"01085758014568876321379j":27,laguerr:[4,15],relat:[7,8,22,15,25,17,13,18,27,10,0,3,4,1,11,19,26,6],notic:27,splot:[12,15],exce:13,discard_known_zero:1,gammainc:[10,29],hold:[8,22,25,26,18,2,28,19,6,20,13],must:[22,14,23,25,17,27,2,0,11,6,29,30,31,13,21],springer:[3,27],"101014101401543575679639e":8,join:0,err:[14,23],"60408207642219e":13,setup:31,work:[2,3,1,6,28,8,9,10,11,15,17,18,21,22,23,0,31,27,25,29,30,26],"97746326050642263725661j":19,erf:[10,31,13],wors:[8,9],"55065599872579097221j":8,introduc:[25,8,21],root:[7,8,14,15,25,16,17,9,30,18,27,10,0,11,4,1,5,19,20,26,13],dhbailei:[9,3],overrid:22,i_n:19,polylogarithm:[10,7,27,4,13],give:[0,8,23,15,25,16,26,18,27,2,10,28,19,13,29,1,30,20,6,21],"61351758081e":23,indic:[8,22,25,27,28,1,6],liter:[28,2],caution:28,fibonacci:[8,20,4],want:[28,22,21],"01952474576025952984418217j":18,keep:[25,30,22],"157j":19,vanish:9,end:[7,14],straightforward:[25,20],thing:[25,20,26,19,13],"68e":8,cylind:[19,4],ordinari:[7,18,15,28,16,17,5,27,10,1,13,21],how:[23,25,31,27,2,10,30,20,13],answer:[25,6],verifi:[31,8,15,25,17,30,27,10,28,11,1,19,20,26,13],wolfram:[8,26,9,27,10,3,13,1,19,6],recogn:[9,11,29],earlier:[16,31,9],oscillatori:[9,19,5],befor:[9,2,11,1,20,13,21],wrong:[8,25,9,28,6,13],beauti:13,"55e":[8,20],law:2,demonstr:[11,1],handl:[25,9,6,1],attempt:[25,20,11,6,1],third:[15,17,2,19,29,6],"3619719871407024816e":8,exclud:12,alias:30,"314488899926921597203j":19,environ:31,enter:25,spoug:3,lambda:[8,30,26,23,15,0,17,9,18,27,2,10,11,13,29,1,19,12,6,21],order:[8,23,15,28,9,27,10,0,11,13,29,1,19,6,21],g11:27,g10:27,oper:[7,22,25,26,2,28,11,29,20,6],acsch:24,composit:[29,5],xlim:12,over:[8,28,16,26,9,27,2,19,1,12,6],fall:[6,25,0,4,1,13],"950440066552087387515653e":15,becaus:[25,30,23,2],"43429448190317j":10,veri:[8,22,23,25,17,9,27,2,10,0,13,29,1,30,20,6,21],pascal:13,"691023585965793j":0,"2607738054097702293308689j":19,fit:2,chebyfit:[14,23,5],fix:[15,25,17,27,10,28,11,1,19,26],inadvert:22,better:[8,16,9,11,1,30,6,21],"237220401382240876065501e":1,complex:[0,2,3,4,1,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24,25,27,28,29,30],condition:6,fadd:[0,2],comprehens:[3,1],easier:20,"1238733442561786843557315j":17,them:[28,30,9,20,27],"00222938275488344j":27,thei:[0,8,30,15,25,16,17,9,27,2,28,11,1,31,19,26,13],lambert:[0,30,3,4],safe:[25,22,11],overestim:9,rectangl:9,"break":9,"6313342832413452438845091j":17,"45626803423487934264j":27,silver:11,choic:[0,9,23,21,1],regula:30,"40j":27,each:[22,14,0,16,17,9,27,2,11,6,29,1,30,31,13,21],debug:[7,2],higher:[25,27,2,6,1,12,13,21],randmatrix:20,side:29,cl2:27,mean:[13,25,17,9,27,10,0,11,4,20,26,6,21],"449410587611035724984376e":10,clausen:[8,4,27],symmetri:[23,17,27,10,19,13],"24575026015369e":22,"1173597j":13,runtest:31,"575172198981096218823481e":27,logm:20,extract:31,matematica:27,emul:[25,7,22],dhbpaper:[9,3],"10000j":19,rewrit:[6,1],"283185307179586476925287j":13,gov:[9,3,27],"07895435170559j":0,diffs_loggamma:29,linear:[7,22,25,9,11,1,20],"01609578360403649340169406j":19,"838224957069370695926416e":8,abramowitz:[10,19,15,3,1],ntr:27,infin:[22,26,27,2,10,20,6],free:3,ineffici:[10,2],nth:[0,8,4],findroot:[8,30,26,9,18,27,2,10,11,5,19,13],crandal:27,"0195927472575932j":20,muller:30,"0001305757570366084557648482j":1,convent:[15,25,16,17,2,0,19],angl:[16,26,20,2],mpmath_nogmpi:31,traceback:[28,17,27,2,10,0,1,30,20,13],att:13,"07e":20,erfc:[10,28,19,1],unabl:6,isc:11,subtl:25,onto:2,erfi:10,rand:[10,2],rang:[8,23,15,25,16,26,9,18,27,2,10,0,11,13,29,19,12,6],expint:[10,6,1],"4469098397654815837307006j":19,independ:[22,19,21],wast:8,thereof:[11,1],system:[25,31,28,1,30,20,21],restrict:[25,22,11],unlik:[15,0,27,2,6,13],alreadi:[31,9,11],"06e":20,robust:[30,9,21],rel_ep:2,agre:[25,8,13],airi:[19,4],top:7,sometim:[31,23,15,25,17,9,2,28,11,13,1,30,20,26,6],diffs_exp:[29,5],necessarili:[20,11,2],"041242537363167632587245e":19,too:[8,25,17,9,27,2,28,11,1,6],tol:[2,11,30,20,6,21],similarli:10,recent:[28,17,27,2,10,0,1,30,20,31,13],"9409227v1":3,"009884375038322493720314034j":24,namespac:[28,22],tool:[31,13],"146045605088568607055454e":8,inexact:25,task:9,vonstaudt:8,somewhat:[25,0,30],conserv:9,bernpoli:8,"713602407841590234804143j":17,"17706465801374033261383j":27,"59960492420134e":16,target:[29,6],keyword:[22,23,15,9,27,2,10,11,13,29,1,30,20,6],provid:[7,8,22,14,23,28,17,9,27,2,10,0,11,1,19,31,6,21],zero:[0,13,23,15,25,17,9,27,2,28,11,4,1,19,20,6],project:[7,2],matter:[25,7,9,23],expm:20,entri:[20,6],minut:31,"2674610150305334025962j":27,entail:16,modern:[17,3,13],spectrum:25,rai:19,"7th":0,pessimist:28,manner:22,increment:[25,9],further:[7,22,25,31,11,29],seen:[28,6],seem:9,incompat:[25,20],lbl:[9,3],recreat:22,djeffrei:3,latter:[10,20,11,28],ryzhik:3,"5707963267949j":0,"95885418079535738828j":1,"5707963267949e":0,simplifi:[17,9,11],though:[25,9,20],object:[28,17,22,26,21],monomi:29,regular:[18,25,26,10,0,19,1,31,13],specifi:[22,17,9,27,2,11,1,30,20,12,6,21],letter:17,phase:[12,27,2],airybizero:19,everyth:[31,22],prematur:11,"2400236j":27,don:[26,20,2],kleinj:17,"0702051v1":19,doc:31,doe:[0,22,25,9,27,2,28,11,29,1,30,20,6],dummi:6,"1e100000000000000000000":13,"2602j":19,section:[8,22,25,17,27,10,19,1],dot:[12,2],radix:25,"62e":20,meijerg:1,random:[10,20,6],sage:[28,7,31],maxdegre:9,radiu:[23,0,19,29,1,6,21],syntax:[30,1,20,2],"6769247817778742392j":10,noninteg:[19,29,27,1],cospi:[0,16,19],pcfd:19,involv:[25,26,9,10,28,19,20,21],despit:11,exactli:[0,8,22,25,16,26,27,2,10,28,29,1,6],steffensen:30,"3283224323946162083579656j":1,explain:22,"1e300j":13,"83675545731519e":19,choleski:20,"327764910194842158583189e":15,klein:17,mascheroni:[26,27],rerr:8,stop:[29,6],ceil:[12,13,2],watson:[3,13],aand:[10,19,3,1],reconstruct:8,bar:17,twice:[20,19],bad:[22,9,2,10,30,12],rung:21,angerj:19,hyper2d:1,pade:[20,23,6,5],subintegr:9,doublefactori:13,full:[25,31,9,2,11,29,1,6],result:[0,8,22,23,25,16,9,2,10,28,11,13,29,1,30,20,6],bihyp:1,"000159154943091895334973109j":19,"57095384420474e":27,"561034538142413947789221e":19,best:[25,9,21,6,2],"27e":20,hopefulli:1,differintegr:[19,29,5],wikipedia:[14,15,25,27,3,1,30,20,13],"04942396360948e":8,discoveri:[9,3],figur:19,elementwis:20,simplest:11,awai:[25,19],approach:[8,17,27,10,19,6],"003183098861710582761688475j":19,attribut:28,accord:[22,19],"21133054865546e":13,extend:[8,25,17,27,11,6],extens:[7,8,15,25,17,2,1,13],"5990701173677961037199612j":17,add:[25,31,27,2,20,6],"69919063017432362805663e":8,extent:11,eulerpoli:8,toler:[7,25,2,11,1,30,20,6],"7007434154172148e":2,sanchi:3,protect:2,accident:25,cot:[16,4,27,12],irregular:[9,19],howev:[8,22,25,16,9,2,10,11,29,30,19,6],against:2,howel:27,"1574325240413029954685366j":13,logic:28,uncertainti:[25,28],com:[7,8,26,9,27,10,3,13,1,19,31,6],col:20,"16j":0,"634292858119162417301683j":17,"521291677592745527851168e":17,"2nd":19,"13805976125208e":13,permut:[0,8],shortcut:[26,9,13,1],struveh:19,diff:[8,23,15,0,17,5,27,2,10,19,13,29,1,30,26,6],summat:[18,0,26,9,27,1,5,6],assum:[8,22,15,25,16,17,1,12],gurante:25,code:[7,22,25,26,9,2,28,3,31],degener:[17,27],union:27,"94463017605008e":13,numpi:20,three:[8,14,25,17,27,0,3,6,12,13],been:[8,22,9,2,11,29,6,21],"10361006263698e":13,much:[0,8,25,31,9,2,28,6,29,13],basic:[2,1,6,7,8,9,10,11,13,15,16,17,18,19,28,20,21,22,25,0,29,30],"294418620024357476985535j":10,pro:30,tini:[0,29,27,1],quickli:[8,0,17,11,13,21],"25j":[8,17,18,27,19,1,20,6],chug:25,life:20,acoth:24,sawtooth:2,suppress:22,"27461836811016e":13,ani:[0,8,14,23,25,17,9,30,27,28,11,13,29,1,31,19,20,26,6,21],"141290706038744575749139j":19,gravesmorri:[3,6],"593897639029319267398803e":10,page_228:10,emploi:28,exception:9,ident:[8,15,28,17,18,27,10,0,3,30,19,20,26],tanh:[9,3,6,24],properti:[7,8,22,28,17,9,2,10,0,19,13,1,20,6],"0912451798066779j":27,"322076267321435669290334e":19,counterclockwis:[0,9],loggamma:[29,19,27,13],calcul:[0,8,22,14,15,25,26,9,2,28,11,13,29,30,19,20,6],unsolv:9,aid:2,"455191523e":27,carlson:[17,3,4],a000178:13,tabl:[25,11,1,3,19,6],clausentheorem:8,pidata:27,h_n:15,conj:[15,19,2],sever:[23,31,9,27,2,19,30,11,20,6],"6846779342253e":10,"002235061547756185403349091j":10,perform:[7,22,14,25,9,2,0,11,29,30,20,6,21],make:[25,17,9,27,2,0,19,1,30,20,31,21],"75744874290013526417059e":10,dhb:[9,3],drawback:28,split:9,complet:[15,28,17,9,10,31,13],elli:3,hand:[9,21],"3619719871407e":[10,8],fairli:[9,11],rais:[0,22,25,17,2,28,30,20,12,6],"318704469415522036482914e":8,"115945364792025420300208e":1,"314700021761367j":20,"6f5":1,tune:9,"0129721075905630604445669j":1,lommelfunct:19,bewar:10,thu:[8,15,0,16,9,10,19,29,1],hyp1f1:[19,1],hyp1f2:1,ellipticintegr:1,wherebi:[21,15],thi:[0,2,3,1,6,8,9,10,11,12,13,15,16,17,19,20,21,22,23,25,26,27,28,29,30,31],"5327107430640678181200491j":1,quadgl:9,"95105651629515357212j":14,hyp3f2:1,left:29,hurwitzzetafunct:27,identifi:[10,17,11,13,1],just:[22,14,25,31,9,27,0,11,29,1,20,21],newton:[0,30,14],horwood:3,psyco:31,farther:25,yet:[28,22],"53e":20,easi:[25,29,15],borweinbailei:3,conjug:[19,14,27,13,2],ideal:9,"8930312845682213562172j":1,expon:[25,22],applic:[0,30,13,6,15],"540725411970948860173735e":19,"04081912627366673332369512j":17,preserv:[25,16,26,6],measur:[25,22,9,2],specif:[23,15,25,0,11,6],arbitrari:[7,8,14,23,15,25,17,13,0,18,27,10,28,19,4,29,1,26,6],"485168143266784862783596e":13,manual:[25,31,2,6,29,1,30,13],pslq:[7,11],"1918250181801757416908224j":19,underli:[28,1],www:[8,27,10,11,1,3,19,13],right:[25,17,29],old:31,deal:9,negat:[10,2],interv:[7,8,22,23,15,28,16,9,27,2,10,0,13,30,20,12,6],"3463514534j":27,maxim:20,airybi:19,intern:[14,25,31,28,19,29,20],hypot:0,superfac:[8,13],guess:[30,11],total:2,cholesky_decomposit:20,"710055114624614989262335e":19,subclass:9,track:[25,28,22,20,30],tracker:31,condit:[19,17,14,23,21],sfu:[10,11,3,27],confluent:[19,4,1],"948359035324558320217678j":19,"84901771719130765656316j":19,"46944695195361e":20,repositori:31,"3504j":1,hadamard:26,"super":[4,13],get_nod:9,chapter:[17,22,19,1],"97272691416282e":13,slightli:[22,15,0,9,11,29,6],"22532e":6,simul:21,done:[8,25,9,27,1,20,13],acot:[16,11],"1020485197430267719746479j":13,mnorm:20,"053285248188621896314384e":8,"float":[7,8,22,23,25,16,26,0,2,28,11,1,20,12,13],contriv:25,workdp:[22,2],karlsson:3,down:2,hermitian:20,slight:9,wrap:[28,2],abel:6,precomput:11,"0133048601383712j":20,accordingli:[25,31,9,29,23],wai:[8,22,23,15,25,26,9,18,27,10,0,19,6,20,13],segment:[16,23],support:[7,8,23,28,16,17,9,27,10,0,19,13,29,1,31,30,20,26,6],transform:[17,9,11,6,1,13,21],guess_degre:9,"25318497558335j":0,avail:[8,22,23,28,31,9,10,0,11,4,1,20,12,6],width:[25,28],anderson:30,fraction:[7,8,22,25,26,5,0,2,28,11,29,1,19],"540520869144928548730397j":13,amazingli:8,fresnel:[10,9,4,12],lowest:8,p117:3,form:[0,8,22,23,25,17,27,2,10,28,11,13,1,19,6,21],forc:[30,31,6],epsilon:[26,9,2,3,20,6],"0999007139290278787734903j":19,"1071073099422j":15,"1737173j":19,"906948208j":27,"591553486811969380442171e":8,"true":[2,1,6,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,28,26,27,0,29,30],reset:22,"138269638197858094311122e":19,coeff:14,maximum:[18,14,23,25,9,2,11,6],srivastavakarlsson:[3,1],"08254224233701381482228e":19,inaccur:[9,27,1],cosec:[16,24],"4402528821177305173599j":1,fundament:[28,13],opensus:31,"5217735339984807829755815j":17,"02058050372068958069241j":15,maxterm:[6,1],classic:27,twinprimesconst:26,cyclotom:[8,4],textbook:25,"17920694574172j":0,"1e1000000j":1,"1256256609322773150118907j":19,check:[8,25,31,27,2,10,28,11,13,21],"08e":20,"041223996670201399005265e":19,viduna:[3,1],excel:[25,6],"912249367458445419235083j":19,"366197716545672122983857e":19,"05e":20,test:[7,8,25,31,28,19,29,1,20,21],tie:2,presum:29,node:9,"663270217461022307645625j":15,intend:28,determin:[8,22,15,2,1,20,21],"337048736538615712436929e":19,nonrectangular:9,substract:20,scale:[28,0,17,30],consid:[15,17,9,2,11,1,30,13],"254528549217097862167015j":19,"934099289460529j":30,"15j":19,equidist:0,"291911213325184901239155e":1,merten:[26,4,27],faster:[0,8,25,26,9,2,28,11,29,30,31,6,21],bullet:11,abscissa:9,anywher:[10,1],"100600143302797880647194j":19,"48478e":6,"23378441241546j":27,cmath:[28,16,4],fact:[28,9,11],time:[8,23,25,31,9,27,2,28,11,13,29,20,12,6],backward:8,mpfr:3,"9272952180016122324285125j":0,reproduc:19,skip:[22,23,28,10,20,12],global:[28,22,24],signific:[25,28,8,2],isnorm:2,"6419643823412927142424049j":19,computation:[8,13,15],row:[20,6],depend:[23,25,9,27,2,10,28,11,1,30],"06100816202125885450319632j":10,graph:12,decim:[7,22,23,25,26,2,28,11,6],intermedi:28,abs_ep:2,cach:[8,28,9,2,19,21],quasi:17,"47255216721987688224357j":17,"285447104177860806j":15,lommels1:19,multivalu:[0,16],sourc:[10,25,26,3,31],string:[22,25,17,2,28,11,30,20],asymptot:[8,31,9,27,10,19,1,30,13],feasibl:[8,11],broadli:17,octob:3,word:[0,11,2],exact:[0,8,13,25,17,9,2,28,11,4,29,1,19,26,6,21],riemann:[8,26,27,19,4,29,30,3],"645432555362369624818525j":10,solver:[30,27,21],die:9,gui:31,restor:[22,2],iter:[14,0,9,2,19,6,29,1,30,20,13],googolplex:22,item:[22,13],"742664438034657928194889j":[27,13],team:3,quick:[7,2],appellf3:1,round:[0,8,22,25,26,27,2,28,1],gradshteynryzhik:[19,3],upper:[10,17,19],slower:[28,8,9,6],version:[22,23,15,25,16,31,2,28,1,13],arccosin:16,maclaurin:[24,0,26,9,27,6],htm:[10,19,1],"28669174871j":27,cosh:[25,23,6,24],cost:[23,6,2],spectral:20,appear:[8,14,26,19,1,13],"14484006887480537268266j":27,"536943175722j":20,current:[22,25,26,27,2,28,1,20,31,6],nsum:[8,28,26,9,18,27,2,10,0,13,1,6],deriv:[8,14,25,17,9,27,0,19,13,29,1,5,30,26,6,21],"347525544923600321838281e":19,gener:[0,2,3,4,1,6,7,8,9,10,11,13,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],coeffici:[8,14,23,15,0,17,13,27,2,11,4,1,6],dougal:1,satisfi:[8,23,15,0,17,18,27,2,10,19,1,30,20,13],slow:[23,25,27,0,1,30,20,12],tangent:[16,9,24],address:25,"42477796076937971538793j":13,locat:[18,14,15,31,27,19,30],along:[8,22,23,0,16,17,9,27,2,19,6,13],do_someth:22,nois:8,shift:13,step:[0,31,9,2,29,30,6,21],superfactori:13,"7053886880648105068343509j":1,a002109:13,"302j":[27,13],extrem:[8,25,17,9,27,0,6,1,13],"22007971483662e":0,commonli:[10,4,13],novic:25,regardless:[30,6],"65024064048415e":20,extra:[22,25,31,9,2,29],circumv:9,tweak:30,modul:[25,28,31,20,4],prefer:11,"807708544441684513934213e":1,bjoerk:30,instal:[7,31],kei_n:19,"84955592153875943077586j":0,prove:28,univers:[3,13],airyai:19,prec:[22,25,31,9,2,28],subvers:31,"6474077j":19,bender:3,checkout:31,afford:20,peopl:27,finit:[18,15,25,16,9,27,2,28,6,29,1,13],appel:[3,1],easiest:23,"3167287567024e":10,"299602851624027900335391e":19,valle:26,uniqu:[25,0,20],descriptor:9,can:[0,2,3,1,6,7,8,9,10,11,12,13,14,15,17,19,20,21,22,23,25,26,27,28,29,30,31],"9th":3,primecountingfunct:10,purpos:25,eps1:1,"117867654j":29,complic:6,"5876309929580408028816365j":1,j0zero:9,"514272j":20,critic:27,nint_dist:2,unfortun:[11,6],"14j":[30,27],occur:[25,2,10,11,1,6],alwai:[8,25,16,26,28,11,6],differenti:[7,8,23,15,0,26,9,27,19,13,29,1,5,6,21],subnorm:25,multipl:[22,25,26,9,27,2,3,6,30,19,20,13],winter:27,write:[28,18],pure:[29,9,27,6,15],parameter:17,"643437226836021723818215j":27,map:[20,1],product:[7,8,25,26,5,18,2,3,13,1,11,19,6],atan:[16,26,21],"01144471378225e":13,max:[25,30,8,23],clone:22,legendr:[17,9,4,15],meromorph:27,"4th":[3,13],mag:2,inhomogen:19,mai:[2,1,6,28,8,9,10,11,12,13,14,15,17,19,20,21,23,0,26,27,25,29,30,31],"1347251417346937904572519836j":30,data:[28,9,27],grow:[8,23,0,10,19,11,13],"9967577965693583104609688j":16,harmon:[15,26,27,4,13,21],hyperu:[19,1],corless:[0,3],practic:[8,9,27],stdin:20,explicit:[17,21,20,15],doubleseri:6,produc:[23,25,2,28,11,1,12,6],quadrant:16,inform:[7,15,17,9,27,10,19,1,30,31,6],"switch":31,combin:[14,17,30,11,6,1,19,13],gamma:[7,8,26,25,17,30,18,27,10,0,3,4,29,1,19,12,13],callabl:30,michel:19,irrat:[28,8,11,26],atanh:[27,24],increas:[0,8,22,25,16,9,27,2,28,11,29,1,12,6,21],"916117957092593868321778e":27,still:[28,9,1],ieee:25,group:0,monitor:2,"82e":20,rosetti:3,mpcontext:28,eulernum:8,window:31,main:[25,31,9,27],"114976146679907015775102j":1,prime:[8,22,0,26,27,10,4],non:[0,17,9,2,11,3,20],"806659285336231052679025e":19,initi:[8,28,9,19,5,30,21],bound:[28,8,11,2],"08405337962602933636096103j":27,zwilling:3,half:[31,8,28,17,9,27,0,19,29,20,26,13],now:[25,6],introduct:[7,22],"3789978119185981665403j":27,term:[0,8,22,30,23,15,25,17,9,27,2,10,28,11,13,29,1,19,26,6],name:[26,9,20,6],revers:[10,11,23,13],"887561841253070615257336e":8,"93e":20,linspac:[9,2],separ:[8,25,31,2,11,29,1],"7817789262839266148j":27,"52359877559829887308j":27,diagram:3,"143324917492256448770769e":19,"43114569750291e":13,compil:[25,7,31],domain:[10,12,15],"41567059811949e":19,replac:6,individu:[7,31,22],gaussian:[9,2,10,3,1,13],ensur:[25,30,8,22],significantli:[25,8,11,26],operand:[25,28,26],"3096175398562811896208j":19,happen:[0,11,27,21],"99999998328279e":29,"164633059729968297147448e":19,accomplish:9,"100j":[19,27,1],space:[27,6,2],kummer:[19,1],barnesg:[26,13],"3619619242700451992411724j":13,formula:[8,22,23,15,25,26,9,27,0,11,13,29,1,19,6],correct:[7,22,25,16,26,9,0,10,28,3,1,11,6],integr:[1,3,4,5,6,7,8,9,10,13,15,16,17,19,28,21,23,25,26,27,0,29],after:[15,31,11,1,30,6],pseudocod:11,"201985j":27,million:[8,11],seventh:3,theori:[17,9,27,10,3,26,13],carl:27,org:[8,14,31,30,27,3,1,19,20,13],care:[22,23,15,25,10,30],nthroot:[0,13],expm1:[0,2],suffici:[23,25,9,28,11,6,21],frequenc:9,"822462726981357180929024e":19,recov:[0,8,15,1],turn:[10,19,23,1],"79e":20,place:[22,14,23,16,26,9,6],dpi:12,addprec:29,principl:27,imposs:[25,9,21,2],frequent:[25,17,27],first:[8,14,23,15,25,17,9,30,27,2,11,13,29,1,19,6,21],origin:[0,17,19,11,6,21],directli:[8,25,17,27,10,11,1,19,20,26,6],carri:22,onc:[9,6,21],arrai:20,"106169926814270418999315e":1,"long":[25,11,6],"585621643835618941044855j":19,oppos:1,ring:31,open:31,angular:9,size:[8,25,2,11,29,21],given:[2,1,6,8,9,10,11,12,13,14,15,16,17,19,20,21,23,25,26,27,0,29,30],varianc:10,cotang:[16,11,24],"27287187546220705604627e":1,workaround:6,whitw:[19,1],"0000005769149277148425774499857j":1,adjac:14,nonintegr:[27,13,15],"290402449901174752972486e":10,whitm:[19,1],cumul:[10,13],circl:[0,26,9,27,2,1],kinkelinconst:26,arithmet:[7,8,22,14,25,16,26,0,2,28,11,4,20,31,6],conveni:[8,22,0,17,27,2,1,20,26,6,21],cite:3,cabral:3,hue:12,especi:[25,30,31,22,6],"162235978298741779953693e":19,copi:[20,3,2],nonsmooth:[29,23],artifact:[25,22],mathworld:[8,26,9,27,10,3,13,1,19,6],f_diff:19,hypergeometr:[7,18,15,25,17,27,10,3,4,1,19,26],mostli:28,borwein:[9,3,27],"11072073453959156175397j":17,quad:[23,15,28,16,17,9,27,10,0,19,29,1,5,26,13,21],than:[0,8,22,14,25,16,9,30,27,2,10,28,11,13,29,1,19,20,6],wide:19,"28735528717884j":20,nstr:[8,22,20,2],nzero:27,"939717517906339193598719j":19,sech:[8,6,24],posit:[18,22,30,25,16,26,9,27,2,10,0,11,29,1,19,20,13],l_n:15,"237132026969293954162816e":8,seri:[0,1,2,3,4,5,6,7,8,9,10,13,15,17,18,19,20,21,23,24,25,26,27,28],"4249117735058037649915723j":1,"03347835361657e":19,sai:13,ann:27,radial:19,argument:[2,4,1,6,8,9,10,11,12,13,15,16,17,18,19,21,22,23,24,0,26,27,29,30],dash:12,anger:[19,4],notimplementederror:27,"20j":[8,27,1],"8952500412050989295774e":19,engin:[11,3],squar:[0,25,16,17,9,2,28,11,1,20],bailei:[11,9,3],"715826296033590204557054j":27,note:[2,1,6,28,8,9,10,11,12,13,15,16,17,19,21,22,23,0,31,27,25,29,30],tanhsinh:9,"1e1000000":1,denomin:[11,13],take:[22,15,28,16,31,9,0,1,12],kerner:14,"107795j":20,begin:[13,15],sure:31,normal:[15,25,16,27,2,10,19,4],multipli:[23,25,9,27,2,19,29,11,20],nprod:[26,19,6,2],beta:[27,4,13,1],borweinborwein:3,pair:[22,23,25,9,2,28,11,1,30,12],"45285242480185e":13,"1998650072605976600724502j":15,subinterv:[9,2],later:[31,2],quantiti:17,feynman:3,"100000j":[10,19,27],runtim:31,hmag:1,axi:[8,25,16,27,2,10,0,19,13],sigma:10,wynn:6,"00000000005e":[0,2],show:[22,23,28,26,2,0,11,1,12],leibniz:6,logarithmicintegr:10,cheap:[9,20],bright:12,gegenbau:[4,15],"06561206166091757973112783j":27,help:[28,7,31],quadraturerul:9,onli:[0,8,22,30,25,16,17,9,27,2,10,28,11,13,29,1,19,20,6],explicitli:[8,22,25,11,1,12,6],ratio:[26,8,11,4,17],proportion:25,state:[10,25,22],"973341637511505389128708e":19,parametr:[0,12],dict:[11,1],thompson:19,analyt:[8,30,23,25,9,27,3,13,1,19,6,21],"11j":1,nearli:[25,28,23],variou:[8,15,0,17,9,19,1,20,26,13],get:[8,22,25,31,9,10,11,30,21],pegasu:30,repr:[25,28,22],secondari:27,cannot:[25,16,31,27,10,28,1],progress:[9,12,6],infiniteproduct:6,requir:[25,16,9,2,10,28,11,13,29,1,12,6,21],"855468179979844983174628j":19,ramanujan:[8,13],"08848758376254e":10,yield:[0,30,29,6],paper:[11,3,27],aris:19,"60784095465201e":13,where:[2,1,6,28,8,9,10,11,12,13,14,15,16,17,18,19,23,0,26,27,25,29],wiki:[14,27,1,30,20,13],"163588003770709748720107e":19,meijer_g:1,"822879187542699127327782j":19,assumpt:9,appellhypergeometricfunct:1,"49052338550226e":8,"07595948379084571927228948j":17,nonstandard:23,concern:[20,11],infinit:[0,8,26,25,17,9,2,10,28,11,13,19,12,6],"078258353474186729184421e":10,nonreal:30,detect:[31,11,6],kfrom:17,anewton:30,"49795891462086e":13,enumer:[8,1],lemnisc:17,"680424678918e":27,enough:[8,25,28,11,6,1,30,13],"31102877714605990523242j":17,between:[25,16,17,9,27,2,10,0,11,30,3,19,6,21],"import":[0,2,1,6,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],"78j":27,spars:20,parent:2,comp:[19,3],amplif:9,ln2:[27,13,6,1],come:25,noconverg:[1,2],pertain:27,"780848544363503984261041j":27,improv:[30,8,9,23,6],"189190365227034385898282e":1,"437j":19,among:[14,11,13,19],acceler:[30,6],rescal:[9,19,1],overview:25,inspir:11,period:[16,17,9,27,19,26],lacim:27,pochhamm:[18,4,13],twin:[26,4],featur:[28,7,26,22,23],cancel:[0,23,25,9,27,2,10,28,1,30,6],"351577584254143j":20,poli:23,fib:[8,20,26],invert:1,invers:[24,25,16,17,9,10,0,11,4,30,20],addit:[31,23,25,17,9,27,2,10,19,6,30,20,26,13],valueerror:[28,17,2,10,6,30,20,13],schroding:19,arcsin:16,thousand:9,catastroph:[0,2],"625115880451325002151688j":10,weierstrass:[17,13],those:[0,22,25,16,27,10,28,11],"case":[0,18,22,30,15,25,26,9,27,2,10,28,11,13,1,19,20,31,6,21],"50j":[10,19],borweinzeta:3,add_package_repositories_to_yast:31,casi:27,whittak:[19,3,4,13,1],"003238627353609801446359j":16,"158j":19,"4949812409615803110051e":19,qr_solv:20,exist:[0,20,11,6,1],region:6,advantag:[28,9,13,6],stdout:2,henc:9,equal:[0,8,15,25,26,27,2,28,31,13],"07595948379084571927228946j":17,ivp:21,fdiff:29,"04120098628857412646300981j":16,ber_n:19,develop:[25,11,31,3],"887705114889527446891274e":19,author:10,maxcal:2,same:[0,23,15,25,16,17,27,2,10,28,19,29,6],binari:[0,22,25,31,2,28],html:[8,26,9,27,10,11,13,1,19,31,6],"417977827655098025684246j":19,document:[7,22,25,16,26,9,27,2,28,3,1,30,20,31,6],hermit:[19,4,15],kutta:21,finish:[31,11],"4342944819025j":19,closest:[0,30],"5019859055341699223453257j":19,kelvin:[19,4],nest:[29,9,20,13,6],siegel:27,"04914370344703j":10,companion:2,jeffrei:3,two:[2,4,1,6,28,8,9,10,11,14,16,17,18,19,20,21,23,0,31,27,25,30],mani:[0,8,13,30,26,23,15,25,17,9,2,10,28,11,4,1,19,12,6],extern:29,solvabl:9,appropri:[31,9,13,6,1],"950930337531768015892987e":1,h2_n:19,floor:[12,2],without:[10,22,8,9,2],promis:10,dimension:[30,6,21],"0317017309957721j":20,heegner:17,execut:2,"3907724215792121244867j":19,stegun:[10,19,15,3,1],gausslegendr:9,"30566371247811e":19,polish:30,speed:[23,25,28,30,20,6,21],"684053288183046528703611j":19,"434290j":19,miscellan:11,"4f1":1,except:[0,8,22,30,25,16,17,9,27,2,10,28,19,13,1,26,12,6],param:1,estimate_error:9,identif:[7,11],color:12,exercis:25,notori:6,"33333333333333e":13,"29047999988194114177943e":8,real:[0,2,3,4,1,6,8,9,10,11,12,13,14,15,16,17,18,19,20,22,25,27,28,29,30],"80192435203j":27,psi:[29,26,27,13],around:[0,9,19,27,23],hypothesi:8,read:[25,7,6],"3197663987712e":13,"374906442631438038871515e":19,"21j":27,"71130938718796e":10,exploit:9,integ:[7,8,22,15,25,17,13,18,27,2,10,0,11,4,29,1,19,31,6],"943133917910336090459748e":19,mpmath:[0,2,3,4,1,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],updat:[22,6,2],output:[8,22,25,9,2,11,12],meijerg1:1,manag:[7,31,2],polylog:[10,27,6,1],"02203963877155499262848j":27,constitut:1,nonzero:[0,14,15,25,27,2,28,29],qbarfrom:17,ellip:[17,1],"109227893755336597979197j":16,definit:[0,8,15,25,16,17,27,2,10,28,3,13,1,20,26,6],exit:[22,11],cylindr:19,borel:1,bessel:[7,9,19,4,1],refer:[7,8,14,15,25,17,9,27,10,0,3,13,29,1,19,20,26,6,21],besselk:[19,1],besselj:[1,19,2],compris:17,power:[7,8,25,0,27,2,28,11,4,19,20,6],"49952562843408e":2,garbag:6,ration:[8,23,15,25,27,10,0,11,13,1,19,6],orszag:3,fulli:[25,21],appli:[24,25,26,2,28,11,3,20,6],comparison:[7,8,28,9,2,1],mdnewton:30,central:[26,29],"117960409887505906848456e":19,opensuse_11:31,aco:[16,17],"606j":1,chop:[8,24,23,15,0,16,17,9,27,2,10,19,13,29,1,20,6],degre:[23,15,16,26,9,11,4,1,21],"166227124177353j":0,act:2,"24851410674842e":13,luck:11,hyp2f3:1,hyp2f2:1,hyp2f1:[17,1],hyp2f0:1,effici:[8,15,25,17,9,27,2,0,3,29,1,30,20,26,6],scorer:[19,4],elementari:[25,17,27,10,28,19,1,13],"008222855781077j":20,ucla:[19,3,1],"75e":20,strip:27,"75j":[18,19,2],your:31,hep:3,log:[0,30,25,16,17,9,27,2,10,28,11,13,1,19,26,6],area:[26,9],omega:[10,9],start:[0,30,9,24],"005753848733883879742993122j":1,interfac:[28,7,31,6],low:[7,8,23,15,28,2,11,1,30,21],strong:[9,6],strictli:[9,27,1],"102j":15,monoton:[16,19,6],newtonian:30,mantissa:[25,2],tupl:[8,14,25,9,27,2,29,1,12],diffun:[18,19,27],oscil:[9,19,23,6,21],qfrom:17,"001224335676457532180747917j":19,diffs_prod:[29,5],conclus:28,longer:[21,11,15],notat:[17,19,1],tripl:[28,9],coulomb:[19,4],possibl:[0,23,25,17,30,2,28,11,6,1,19,20,26,13,21],"default":[22,25,31,9,2,28,11,29,1,30,12,6,21],mpmathifi:[15,1,2],"71678149306807j":27,benderorszag:[3,6],k_n:19,"34814309345871e":0,connect:19,taylor:[8,14,23,15,0,16,17,5,18,27,10,19,6,24,1,20,13,21],creat:[28,7,22,20],certain:[11,27,1],pole:[15,16,27,10,1,13],cbm:[10,19,3,1],littlewood:26,"925320392280406229e":8,decreas:[16,9,19,1,6,21],file:[31,20,12],secondzeta:27,girgensohn:[9,3],"699193712834466537941283e":8,legendrep:15,incorrect:25,denot:[8,22,15,0,17,9,18,27,10,1,26],googl:[7,31],mathematica:[16,9,15,1,2],field:25,binomi:[8,4,13,15],cleanup:14,immens:13,geometr:[0,4,6,1],you:[31,22,25,17,9,2,11,30,20,12,6,21],"1e3000j":13,d2f:30,poor:6,polar:[15,2],isint:2,"03194209021885226400892963j":1,sequenc:[15,25,27,28,6,29,13],symbol:[18,13,11,4,6],"1e1000":[10,2],"0000000001e":2,multidimension:[30,20,6],polynomi:[7,8,9,14,23,15,5,27,10,11,4,1,19,13,6,21],"628120031122876847764566e":8,isinf:2,peak:9,"3003j":13,"0005200017201681152909000961j":19,reduc:[8,15,0,17,27,10,11,6,1,19,13],taufrom:17,expjpi:0,directori:[25,31],"07595200133501806872408048j":13,descript:[22,28,9,2],"169178177065714963568963e":1,laplac:19,potenti:[0,19,1],acsc:16,extraprec:[22,14,17,27,2,20],represent:[7,8,22,25,17,27,2,10,19,13,1,26,6],all:[0,8,22,14,23,15,25,16,31,9,30,27,2,10,28,11,1,19,13,21],messi:1,lack:[25,20],scalar:[20,3,21],"858877821790010714163487j":19,tricomi:19,abil:[25,22],acosh:[17,24],"728805278735270407053139e":10,follow:[2,3,1,6,28,18,9,10,11,12,13,14,15,16,17,19,22,0,26,27,25,29,30,31],"8011892476145118551756j":19,disk:[17,6],"9294486669502295512503127j":19,grampoint:27,"1454441043328607980769742j":1,sympi:[20,7,31,11],articl:[25,15],"3095598756531121984439128j":0,"16470269330255923143453723949e":0,program:[8,2],present:[8,22,9,2,11,6,13],"41421j":13,norm:[0,30,20],mpf:[0,8,22,23,25,17,18,27,2,10,28,11,13,29,1,31,19,20,26,6,21],"04081912627366673332369511j":17,mpc:[28,26,22,2],bitcount:25,fals:[0,22,14,23,25,17,9,27,2,10,28,11,13,29,30,19,20,12,6,21],faq:25,mpi:[28,8,2],util:[7,2],worst:[25,11],failur:31,fab:2,fac:[8,15,0,26,27,10,19,6,1,13],abramowitzstegun:[17,3],orthonorm:15,"693919841408089681743j":15,list:[7,8,14,23,25,17,9,30,27,2,0,3,29,1,31,11,20,12,6],hyp:1,arithmeticerror:6,d_n:19,cosin:[23,24,16,9,27,10,19,1,20,21],small:[8,25,27,2,0,11,29,1],node3:11,"462130727102185701817349e":27,dimens:[9,20,6],maxprec:[1,2],unnorm:16,fresnelc:[10,12],rate:[30,9,13,6],design:[25,28,9],pass:[8,23,0,17,9,27,2,30,31,6],"500j":10,trick:9,what:[25,28,11,1],nonlinear:[30,21],richard:27,diag:20,sum:[7,8,9,25,26,5,0,18,27,2,10,28,13,1,20,6],advanc:[7,25,28,3,29,20,6],abl:[25,31,9,11,6],delet:9,sinpi:[0,16,26,19],p155:27,intersect:30,"16966398138119450043195899746j":8,consecut:[9,27,13,6],"22568799288165021578758j":10,method:[8,22,14,28,9,27,2,0,11,29,30,3,20,6,21],contrast:6,maxstep:[14,11,30],"369385787943539818688433e":19,"031911949062334538202567e":27,legenq:15,legenp:15,modular:[17,4],"_fast_abort":6,"19j":19,"434291j":19,standard:[23,15,25,16,17,9,10,11,4,29,1,5,20,31,13,21],modifi:[16,19,4,29,30],ignor:6,valu:[0,5,2,1,6,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],search:11,ctx_mp:28,indetermin:[28,22],twinprim:[26,4],autoprec:2,polyexp:[8,27],amount:[8,25,9,18,2,28,13],doctest:[22,23,28,31,9,10,0,11,29,30,20,12],action:2,"333333j":1,"002472407104546216117161499j":1,unrecogn:2,"293142504363642687204865e":13,magnitud:[8,22,25,9,27,2,0,11,29,1,19,20,12,6],quotient:[13,6,2],"700j":19,primit:[0,8],int_0:19,put:[28,21],bernfrac:8,famili:[10,30,22],heurist:[25,2],hyperfac:13,select:[15,0,9,6,29,1,13],wirefram:12,distinct:[0,1],"303985255686378497193063e":19,regist:25,rightmost:13,taken:[25,17,9,27,19,6,1,13,21],stieltjesgamma:27,minor:25,more:[2,1,6,8,9,10,11,13,15,16,17,18,20,21,22,23,24,25,26,27,0,29,30,31],diagnost:31,flat:20,diamond:9,desir:[25,9,6],canon:17,hundr:9,"01497616165031102661476978j":1,"02074960467222823237055351j":19,function_nam:7,flag:[31,2],"399512j":20,particular:[14,25,26,27,11,6,1,19,13],known:[0,8,25,26,18,27,10,28,19,6,29,1,30,13,21],s_method:30,elliptice2:1,none:[0,18,25,17,9,2,28,11,29,1,30,20,12,6,21],endpoint:[28,9,2,10,29,6],nint:2,eta:[18,19,27],deq:15,det:8,erfinv:10,remain:[14,13],nine:27,nonsingular:19,"890569218254486878220752e":1,def:[22,23,15,0,17,29,1,30],transcend:[4,27],prompt:7,mangoldt:[8,27],coulombc:19,dobinski:8,coulombg:19,coulombf:19,accept:[22,1,20,21,2],sphere:15,"003327897536813558807438089j":27,minimum:[25,30,24],"46458120882585e":13,pointwis:23,sharp:[9,12],"547394054060638j":0,"3619719871407032404e":8,concept:25,huge:[25,17,27,0,19,1,13],cours:[25,9,10,3,20,13,21],goal:25,divid:[9,11,23,2],rather:[9,27,19,6,29,13],anoth:[28,9,19,29,30,6],comfort:20,csc:[16,4,27],divis:[25,1,11,6,2],lozano:3,reject:6,jborwein:9,simpl:[8,14,23,25,16,9,27,2,10,0,11,13,30,19,20,12,6],"897755278982137052053974e":13,wave:[9,19,4,2],resourc:25,primezeta:27,algebra:[7,22,25,17,11,20,13],variant:[30,4,27],uniti:[0,16,8,14],reflect:27,plane:[16,17,27,2,10,19,1,12],plana:6,"691337610097577791134j":15,wilei:3,associ:[27,15],circumst:0,"short":[25,22],confus:[10,27,13,1],caus:[9,20],atan2:16,zerodivisionerror:[0,30,1,20,2],spheric:[4,15],rotat:[20,19,24],soon:[25,29,6],pseudorandom:6,scientist:[25,3],through:[25,27,6],barnes_g:13,gammaprod:[13,1],suffer:0,qfac:18,paramet:[15,28,17,27,10,19,29,1,20,13],style:22,altzeta:[27,6],"7647282253046207836769297j":1,brows:31,perturb:[8,1,29,2],rapidli:[8,17,9,27,10,19,6,1,13],liouvil:[19,29],"634482152088184489550553e":19,jacobi:[8,15,17,18,4,6],might:[25,26,9,2,0,30,20,31,6],fool:2,"634963914784736j":20,good:[22,23,25,9,27,2,10,28,30,12,6],"return":[2,1,6,28,8,9,11,12,13,14,15,16,17,19,20,21,22,23,0,27,25,29,30],"49j":1,"59947894041409j":14,qgamma:18,siegelz:27,complain:25,"4f3":1,ldexp:2,hornfunct:1,gamow:19,level:[25,28,9],"0001j":1,instruct:[31,23],easili:[15,11,2],achiev:6,radian:[16,26,4],fmod:2,"12th":0,found:[14,25,16,17,11,30,3],truncat:[14,23,25,9,11,1],"0447701735209082j":9,weight:[15,9,23,6,1],hard:6,idea:[25,11,6],procedur:6,slowdown:2,expect:[30,20],quantum:19,beyond:[23,6],todo:[6,21],reduct:[6,1],"46771150871022j":30,meijer:[4,1],robert:3,"45677330576j":27,"578535j":27,research:[3,13],titchmarsh:27,print:[2,3,1,6,7,8,9,11,13,14,15,16,19,28,20,21,22,25,31,27,0,29,30],"596619701076529803290973e":19,belief:25,"1084662449961914580276007j":15,dealt:9,suspici:25,reason:[25,26,9,21],base:[0,8,25,26,9,27,28,11,4,1,30],ask:[25,27],"5f4":1,rect:2,"36950073615897616781j":27,khinchin:[26,4],exponenti:[7,8,13,25,16,17,9,27,10,0,11,4,29,1,19,20,26,6],perhap:25,"202966666673054671364215j":27,"06154836525312066938147793j":1,interact:[25,7,31,22],integrand:[10,17,9],singleton:9,"556691003883483531962095e":10,"22044604925031e":[20,14,27],prevent:11,famou:[17,11,27],mersenn:[0,22],number:[0,2,3,4,1,6,7,8,9,10,11,12,13,14,15,17,18,19,20,22,23,24,25,26,27,28,29,31],smaller:[14,25,27,2,30,20],"1904534692378346862841089j":10,least:[8,23,25,9,2,28,11,20,6],spherharm:15,"58810701527j":13,"6141851464j":1,"97262736954392e":13,differ:[0,8,22,15,25,16,17,9,27,2,28,19,13,29,1,30,26,6],quadt:9,nome:17,"5th":14,construct:[22,27],"783578125755972803440364e":1,contra:30,transcendent:[7,28,17,10,25,11,26],"5863603889018621585533j":19,"33262283196162e":13,scheme:9,"86e":20,cosm:20,store:[28,22,20,2],"792526803190927323077905j":1,stirl:[13,6],statement:22,relationship:8,blindli:25,besselyzero:19,mnewton:30,part:[7,8,22,14,15,25,16,9,27,2,0,19,29,1,12,13],"32579597597705e":13,moder:8,sign:[25,16,9,27,2,10,28,29,30,12,6],kind:[15,0,17,9,19,1],"2f0":1,contrari:25,doubli:[17,9,6],cyclic:0,"18j":2,remov:[29,31,11,12,6],reus:[9,2],str:[8,22,25,26,2,0,20],toward:30,openopt:30,randomli:2,comput:[0,2,3,1,6,8,9,10,11,13,14,15,16,17,18,19,20,23,24,25,26,27,28,29,30],"008156449464604044948738263j":13,"570796326794896619231322e":13,packag:31,"47925220414035e":20,aperi:[26,4,13],option:[7,31,22,23,0,17,9,27,2,19,29,1,20,12,6,21],built:[25,30,22],lib:31,"013370403397787023602j":27,also:[0,2,4,1,6,8,9,10,11,12,13,14,15,17,18,19,20,21,22,23,25,26,27,28,30,31],sum_next:9,build:31,jacobian:30,techniqu:25,npdf:10,"17e":20,distribut:[10,9,4,13],"875060251748226537857298e":10,zeroth:27,interpol:21,reach:[11,23],most:[0,8,22,15,25,16,17,27,2,10,28,11,13,1,30,20,31,6],madelung:6,alpha:[19,1],powm:20,clear:[9,1],cover:25,"707106781186547j":[20,2],roughli:[8,22,25,26,9,2,10,0,11,29,19,6,21],lommels2:19,prettier:22,exp:[0,22,30,26,23,15,25,17,9,27,2,10,28,11,13,29,1,19,12,6,21],azimuth:15,usual:[22,15,25,17,9,27,28,11,1,30,20,31,6],"410032405664160838288752j":13,"232856260909107391513108e":13,"4000000j":19,chebyshev:[15,4,23,5],unari:26,hyper:[10,27,1],frexp:2,"3561052815014558335412538j":17,"33262154439442e":13,particularli:6,fine:19,find:[7,14,25,27,0,11,6,5,30,13],"4985390447553234305e":27,trap_complex:22,pretti:[2,1,6,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,24,28,26,27,0,29,30,31],solut:[30,23,15,0,17,9,27,10,11,1,19,20,26,6,21],technic:[25,27],quadosc:[10,16,9,19,5],factor:[18,25,17,27,2,10,19,29,1,30,20,26],"107759j":20,express:[8,0,16,17,27,10,11,1,19,13],"096692038332790615j":17,"674397j":20,"624622876523j":27,coth:24,qhyper:18,whenev:[26,6],"2773099198055652246992479j":27,hankel2:19,hankel1:19,common:[7,25,17,9,28,19,4,1,26],crd:[9,3],set:[7,8,22,14,23,15,25,31,9,0,2,28,11,13,29,1,20,6,21],seq:6,"905169561300910030267132e":1,mutabl:20,fneg:2,see:[2,1,6,7,8,9,10,13,14,15,16,17,18,19,20,22,23,28,26,27,0,30,31],arc:17,sec:[16,17,4],arg:[10,16,17,19,2],transpar:[28,31],close:[0,8,23,15,25,17,30,27,2,10,28,11,13,1,19,20,6],contour:[29,1],analog:[22,26,27,2,10,1,20,13],barn:[26,13],"499471750566824369520223e":27,"711508683721355528322567e":19,feriet:1,"399196580646054789459839j":10,won:22,nontrivi:[30,27],"131705322874965310390701e":1,altern:[15,0,17,9,27,2,10,19,4,29,1,30,31,6],signatur:22,unitroot:[0,8],numer:[0,1,3,5,6,7,8,9,10,11,12,13,14,15,17,19,20,21,22,23,25,26,27,28,29,30],"56637061435917295385057j":0,"31j":13,"75527e":6,len:[0,8,2],complementari:[10,8,27],solv:[15,25,27,0,19,5,30,20,21],froberg:27,"17563978290601j":27,both:[0,13,23,25,16,26,9,27,2,10,28,11,4,1,19,31,6,21],last:[0,22,25,17,27,2,10,28,6,1,30,20,13],nonrigor:8,"05658474496e":13,kamp:1,context:[7,22,25,2,28,13],"236047850352252236372551e":19,"915965594177219j":27,whole:0,"7432082105196321803869551j":6,simpli:[0,25,2,28,11,1,6],point:[0,2,3,1,6,7,8,9,11,12,13,14,15,16,17,19,20,21,23,25,26,27,28,29,30],residu:[9,20,27],suppli:[9,11,12,27],written:[8,21],zeta:[7,8,26,13,27,3,4,30,12,6],coincid:[19,27,13],"0121089837381789j":20,backend:31,"48071721019185e":19,nogmpi:31,vertic:[0,12],due:[8,25,9,27,28,20,6],calculu:[7,9,30,5],sinc:[8,22,15,28,16,17,9,27,2,10,0,11,4,1,12,13],"066421087689197632388731e":19,asinh:24,hurwitz:[4,27],imag:[22,28,27,2,19,1,12,13],great:27,coordin:[15,19,2],understand:25,func:[28,27],elliprc:17,elliprg:17,elliprf:17,elliprd:17,scorergi:19,look:[25,31,11],"2503670480325997418156j":17,"168906959966564350754813j":16,erron:[16,11,12],kwarg:[8,30,24,0,16,17,9,18,27,2,10,11,13,1,19,6],histor:17,zeroprec:[15,1],pace:13,"while":[8,22,25,28,19,29,1,30,13,21],"2609739839216039203708921j":18,"867245506962312577848166e":19,abov:[10,31,11,27,15],error:[7,8,13,14,23,24,25,9,0,27,2,10,28,4,1,30,20,6],"82441278771834794493484e":15,loos:8,loop:3,propag:25,"10000000j":27,"0215277159194482j":20,vol:[19,3],von:[8,27],"526683999721399103332021j":17,"89e":20,quadrat:[14,9,11],tripli:29,illinoi:30,"93147180559945e":0,decor:[22,2],minim:[20,11],u_n:15,"46072418957528e":10,behav:[23,25,10,19,29,21],tower:0,"149364285543783412210773e":8,optim:[7,8,23,28,5,2,1,30,6,21],wherea:[10,28,2],temporari:25,user:[25,28,31,9,11],"__add__":20,"7018j":13,"807755278982137052053974e":13,riemannr:8,"963625832768189140995j":27,"1000j":[0,19,27,13,15],"__new__":9,subpackag:31,lower:[25,10,11,1,19,20,6],digamma:[3,13],gauss:[15,9,1],els:30,chebyu:15,chebyt:15,noncommut:20,parenthes:8,whittakerwatson:[17,3],person:[3,27],expens:[8,23,0,9,29,21],"491808047480981808903986j":27,academ:23,know:[25,20,6],italiana:27,cuboid:9,pylab:12,bilater:[4,1],"0770637306312989892451977j":10,zetazero:27,shape:9,"67j":19,"67e":20,lerch:[4,27],"8414709848078965066525023j":0,cut:[8,0,16,17,10,1,31,13],complexnumb:31,"78669862804508537384j":27,theoret:[7,8,23,17,27,4],recurr:[8,15,9,27,10,13],rgb:12,quadratur:[7,23,15,9,10,3,29,5,6],theorem:[10,0,8],input:[7,22,25,16,31,9,2,28,11,1,20,12,6,21],coffei:27,euler:[8,13,0,26,9,18,27,10,11,4,6],fire:31,format:[20,9,14,23],big:9,judgement:25,catalan:[26,27,11,4,13],bit:[22,25,9,27,2,28,11],"1e3000":13,"160893281017115450519948e":19,"_method":30,"2454393389657273841385582j":1,success:[25,19,9,11,6],"9328495143101e":13,signal:[9,2],resolv:9,elaps:2,collect:2,princip:[0,16,17,10,28,13],popular:25,interestingli:9,encount:[17,22,13],doubleintegr:9,often:[0,22,25,16,9,27,2,28,19,6,1,13],visibl:28,some:[0,8,22,30,15,25,16,17,9,27,10,28,11,13,1,31,19,20,12,6,21],back:[25,0,31,20],understood:[19,15],unspecifi:0,sampl:[25,27,23,6,2],cabralrosetti:[3,1],brent:30,riel:27,bilateral_hypergeometric_seri:1,t_n:15,stieltj:[26,4,27],"850502209186044j":13,"510544741058517621110615j":1,per:[30,9,21],recognit:[7,11],mathemat:[7,8,25,16,26,9,0,2,20,28,3,4,11,19,13],larg:[0,8,13,15,25,9,27,2,10,28,22,11,4,29,1,19,20,6,21],appellf4:1,frobeniu:20,appellf1:[17,1],appellf2:1,either:[22,25,17,9,2,11,13,29,1,6],machin:[25,20,11,23],previou:[9,6],run:[7,25,31,9,27,11],"675159751729877682920301e":15,oei:13,dirichlet_eta_funct:27,"983617742029605093121118j":10,from:[0,2,4,1,6,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],"1752011936438j":20,subtract:[25,23,2],hilbert:20,stieltjesconst:27,constraint:11,morri:3,"46923988977204e":8,memori:[25,2],"125j":1,"365499358305579597618785e":1,asech:24,cecm:[11,3,27],block:[22,27,2],"012757178975667428359374e":15,"__future__":22,primarili:9,"365981529122220091353633e":1,p_n:15,within:[25,31,9,27,11,29,30,6,21],nja:13,"302396543466867626153708j":13,ellipsi:[22,0,9,11,29,30,20],sqrtm:20,equival:[0,8,24,25,16,17,27,2,10,28,19,1,6],chang:[22,25,31,9,27,2,28,30,20,6],institut:27,"434287j":10,riemannzetafunct:27,scorerhi:19,combinator:13,fast:[7,8,23,28,31,9,27,0,19,29,30,6,21],custom:[22,9,27,2,11,12],"935753855797342532571597e":1,handler:2,handbook:3,grave:3,forward:[11,1],"4059040020276937085081127j":1,properli:[28,9,13],arbitrarili:[8,23,0,27,10,19,1,13],repeatedli:2,"1e30":13,"7024814731040726393156375j":17,decomposit:20,asymp_tol:1,reed:27,translat:[25,11],expintegralei:10,atom:11,line:[15,16,9,27,19,30,20,12,6],mitig:11,"61349954245e":23,info:27,"78309e":6,infj:17,consist:1,confusingli:1,scipi:30,fdiv:2,"0906111371962375965296612j":16,backlund:27,complex_numb:31,similar:[30,18,24,1],curv:[7,12],findpoli:11,constant:[7,13,15,25,17,9,27,11,4,19,26,6],wavelik:19,thereabout:12,"8382249570693706959264156336481764738284680928013e":8,lectur:27,incomplet:[10,17,4,13,1],poussin:26,behavior:[25,19,27],nonsymmetr:23,parabol:[19,4],peter:[9,3],polygon:0,sequenti:8,nan:[22,25,16,26,2,19,13],"1e100":1,nice:[30,9,6],"559955931096196875845858e":19,mertensconst:26,"765587078525922j":27,"100000000j":27,eval:[25,28,22,11,2],asec:16,formal:[27,1],agm:[0,3],"972260393822275434196053j":15,"54668863216891e":1,algorithm:[8,14,25,9,30,27,10,28,3,1,11,6,21],inaccuraci:20,confirm:27,evenli:2,tradeoff:[28,21],delta:[28,8],nasa:27,transform_nod:9,far:[25,28,8,6,1],extrapol:[7,9,6,5],jtheta:[8,18,17,6],ellipf:17,"536643565778565j":10,scratch:25,ellipk:[17,26],schoenfeld:8,airyaizero:19,ellips:17,ellipt:[7,17,3,4,1],edu:[19,3,27,1],"71e":20,entrywis:20,"776445785304252879026585e":15,compact:17,cython:28,"1415926535897932385j":27,sensit:[29,2],elsewher:27,send:31,nonposit:[27,13,1],"14322868174828e":13,sens:[9,15,1,19,2],logarithm:[7,8,15,25,26,27,2,10,0,19,4,29,1,20,13],"01e":20,sympifi:11,pcfu:19,"34294481903252e":10,speedup:[28,9],electron:3,volum:3,pcfw:19,"999977909503001j":10,implicitli:23,pcfv:19,notabl:[9,6],"921627163136j":27,fewer:[9,21],"try":[22,25,9,11,1,30],"09136296657021e":8,"755146243738946045873491e":10,"23936276195592e":16,pleas:[28,30,31,1],impli:[16,19,27,2],nonconverg:27,natur:[0,26,20,4,1],"325697j":27,fold:19,investig:[11,27,6],wolframfunct:[19,3,1],download:[7,31],odd:[8,23,24,25,27,10,0,19,6,13],"007180821099823942702249e":27,append:[6,2],compat:[28,20],index:[7,8,0,31,19,6,20,13],compar:[8,25,17,18,2,10,6,20,13],accurate_smal:1,access:[7,20],experiment:[28,30,9,3],secant:[16,30,24],"021799872770764149793693e":1,imaginari:[8,14,24,25,26,27,2,10,0,19,29,1,12,13],absolut:[8,23,25,2,28,29,1,20],matplotlib:12,lambertw:[0,30],sinm:20,"59e":20,gmpy:[7,31],sinh:[25,9,3,6,24],ubuntu:31,slater:[3,1],sine:[23,24,16,9,27,10,19,1,20,21],implicit:1,"07269254613282301012735797j":1,convert:[0,25,16,17,2,28,11,20,26],convers:[7,25,16,26,2,28,4,31],larger:[8,25,10,11,29,13],ctx:[14,23,28,9,30,2,11,29,1,19,20,12,6,21],"1f1":1,converg:[8,14,23,0,17,9,27,2,10,1,30,26,6,21],typic:[14,25,9,2,28,19,6,29,1,13],chanc:25,hardi:26,nearest:[25,8,26,2],cbrt:[0,17,12],approxim:[7,8,22,14,23,15,25,26,27,2,0,11,13,5,30,20,31,6],inequ:[28,8,19],apt:31,"58778525229247312917j":14,"3448325764012e":8,underflow:25,"684731303500835514850962e":19,nonrat:27,"163368465314934589818543e":1,zip:[31,29,2],chi:[10,27],doubl:[7,8,25,17,9,0,27,28,6,20,13],upgrad:31,particl:19,next:[14,11,29,6],websit:[7,31],few:[8,22,15,28,31,9,27,0,11,6,29,1,19,13],nprint:[8,22,14,23,15,0,16,17,24,27,2,10,11,13,29,1,19,20,6,21],"28318530717959j":9,remaind:6,sort:14,kinkelin:[26,13],normalize_output:2,"346791453661149089338186e":8,"33j":1,"33e":20,"66577118985331e":2,uwo:3,topic:3,account:[16,29],retriev:[17,9,1],alia:[0,8],arang:2,when:[0,22,30,23,15,25,17,9,27,2,10,28,11,13,1,19,26,6],cambridg:[3,13],salam:3,donut:12,obvious:25,calcbn:8,squarefre:27,"19850014994_1985014994":27,proof:3,"849835186175096516193e":1,tau:17,"554898911356806978892748j":13,tar:31,"05494e":2,process:11,fourierv:[23,5],sudo:31,high:[0,8,22,14,23,25,17,9,28,11,13,29,30,31,3,19,26,6,21],"11353553941214591260783j":0,onlin:[31,9],tan:[16,17,24],everywher:16,surfac:[7,12],"883714220959876246415651e":1,polyexponenti:27,uqam:27,"9911485751285526692385j":0,six:1,ellippi:17,interfer:31,memoiz:2,subdirectori:31,instead:[31,8,22,23,15,25,17,9,27,2,28,11,13,20,26,6,21],sin:[2,1,6,28,9,10,12,13,15,16,17,19,20,21,22,23,24,0,26,27,25,29,30],inher:25,frac:2,singular:[23,0,16,17,9,27,10,19,29,1,20,12,6],"17192140544213j":20,"1347251417347j":27,gcc:25,notebook:31,intercept:2,"905250692824046162462058e":19,redund:25,physic:19,alloc:25,"083660630910611272288592e":10,essenti:[8,23,15,16,17,24,10,11,6],"875060250875106915277943e":10,correspond:[23,15,25,27,28,19,1,20,6],element:[8,14,15,2,1,20,6,21],issu:[7,31,23,25,17,0,11,3,26,6],encyclopedia:3,allow:[18,15,25,9,2,28,11,1,20,6,21],"052951783922142735255881j":1,sigmoid:24,setuptool:31,move:6,report:31,theta_q_lim:17,"66e":20,perfect:[9,6],"096964974460199200551738j":1,"87221167872304520599418j":27,remark:23,chosen:[0,17,27,2,19,29,1,13],gammabetaerf:[10,13],csch:[6,24],infrastructur:9,decai:[9,19,6],differint:[19,29,5],therefor:[15,25,9,2,28,6,1,13,21],weber:[19,4],greater:2,"13e":20,nonneg:[8,15,25,27,2,19,13],dal:9,bernoulli:[8,4],nsimplifi:11,"65531023921j":27,ferguson:11,"9683e":19,"159388148811981535941434e":1,ker_n:19,edit:[3,13],hypergeometricfunct:1,sincpi:[16,26],astronom:13,"1684e":30,mode:[31,2],truth:8,fprod:[8,18,6,2],subset:28,expans:[8,23,15,0,17,27,10,19,1,30,13,21],arxiv:[19,3,27],tolist:20,bump:9,barnett:19,combinatori:[7,8,4],eisenstein:6,hyperbol:[10,7,9,4,24],tenth:3,twodimension:1,"892316224968072424732898e":27,out:[22,23,31,9,2,1],variabl:[15,31,9,4,1,17],"25e30":2,matrix:[7,22,28,30,20,6],"54768762233649126154534j":1,mere:1,categori:[25,1],"23828e":1,suitabl:25,rel:[10,25,8,29,2],lattic:6,hardwar:[25,28],libari:16,kerner_method:14,clsin:27,respecto:1,math:[28,10,19,4,1,3],insid:1,parenthesi:27,standalon:1,control:[22,25,2,28,1,6,21],almosteq:2,"9038644233367868273362446j":27,dictionari:20,releas:31,"1454701525056682297614029j":1,"003538923749396546871j":17,afterward:[28,30,22,2],ultraspher:15,dirichlet:[18,4,27],transposit:20,"15779412229419e":13,via:[25,16,31,27,10,0,6,1,13],unwant:11,could:[30,11,23,15],david:[25,11],counterpart:6,length:[23,17,9,27,1,13,21],enforc:2,outsid:[25,17,27,0,1,6],south:15,softwar:[25,16,17,31],"72510874248161608216j":27,"34584913683906e":1,"1240770953126831093464055j":19,fmul:[19,2],besseljzero:19,"1e7":10,"1e6":19,facil:[26,11],"9069482075392389420041147142092708393819935j":27,sumap:6,"32718175j":27,"_may_rot":20,dlmf:[19,17,3,27],strict:[31,11],unknown:[11,2],licens:31,perfectli:28,"4763860949415867162066j":0,wrapper:[30,11],rosser:27,"14159265358979j":0,termin:[15,6,1],siam:3,man:25,behaved:25,"final":[22,17,27,30,20,31,6],"994957561486776072734601e":8,"213856669836495986430094j":17,"426j":27,arctang:16,"43433j":19,stupend:8,bei:19,mfrom:17,"65j":19,structur:[15,1],charact:[17,27],claim:25,num:1,becom:[25,17,9,27,19,13,21],ber:19,stricter:25,"757089288053346261497686e":19,"2118232380980201350495795j":27,exhibit:27,falsi:30,respons:27,clearli:28,fail:[30,31,9,11,2],goldberg:25,have:[0,22,14,23,15,25,16,31,9,30,27,2,10,28,11,13,29,24,19,6],bess:[19,1],need:[22,23,25,31,9,11,29,30,20,12,6,21],"86579517236206j":0,clco:27,border:6,ylim:12,"42236140124728e":13,"9253203909477020467e":8,bdiff:6,verbatim:11,"627291764358919410440499j":19,diverg:[10,13,6,1],"773874j":20,mid:[28,9],accuraci:[22,23,25,9,28,11,29,1,6,21],mix:22,builtin:[28,2],which:[2,1,6,28,8,9,10,11,13,14,15,17,19,20,21,22,23,0,26,27,25,29,30,31],fac2:[19,13],subject:11,singl:[0,18,25,16,17,9,2,28,13,29,1,20,12,6],analysi:[25,3,13,2],unless:[8,1],who:25,"824993j":20,rigor:[25,28,8,20],realnumb:31,gradshteyn:3,judici:2,"class":[22,9,2,10,30,20],hyp0f1:1,laurent:[17,27],"32e":20,url:31,request:6,inde:[17,11],"275758229007902299823821e":1,"400j":27,polygamma:[27,4,13],"92532039161405e":10,polyroot:[8,14,15,11,5,13],fdot:2,north:15,fibonaccinumb:8,spuriou:11,text:[17,31,23,21],"12j":[27,13],verbos:[9,27,2,11,30,12,6,21],"31790195707456404099817e":19,"2660553j":19,"5309210862485198052670401j":24,trivial:[30,26,15],"101124983878j":13,qbar:17,"20000000j":1,"989901335682408421480422e":8,"1026623j":13,"61398543581249e":13,losslessli:[25,2],correl:25,should:[0,31,22,14,23,15,25,17,9,27,2,10,28,11,13,1,19,12,6,21],y_n:19,durand:14,smallest:[23,2],suppos:9,"257913707524362408877881e":10,local:[30,22,27,21],pborwein:[3,27],hope:25,cube:0,"00000012791934e":0,ludicr:10,familiar:2,autom:11,overdetermin:[30,20],extradp:[22,17,1,27,2],workprec:[22,6,2],"141592653589793238462643j":[10,13],lucki:6,endless:11,triangl:13,enabl:[31,6,1],organ:11,bisect:[30,27],"577879007912758274533309j":17,"481821753552703090628793j":18,gram:[8,27],partit:[8,18,2],contain:[23,25,9,2,28,11,1,12],shi:10,nist:3,view:[17,14,1],modulo:[0,2],"8503366631752726568782447j":1,madelungconst:6,temporarili:[22,9],dyadic:27,modulu:17,isnan:2,conjectur:26,correctli:[25,11,23],pattern:27,powm1:0,tend:[9,13],"3504398860243e":10,force_seri:1,yast:31,theta:[8,15,17,18,27,4,6],neither:9,"1e300":13,"76821518787438186031973e":1,predefin:26,kei:[19,1],"13472514173469379045725j":27,"568864481005783j":20,moebiu:8,entir:[25,16,9,27,19,13],"32579620567308e":13,ker:19,thumb:23,hypercomb:1,staudt:8,"1e30j":13,"90210294051982j":27,lazi:[26,2],"302585092994045684007991e":13,etc:[8,22,23,25,13,2,28,11,4,29,1,6],admit:[17,29,1],instanc:[8,22,28,26,9,27,2,1,31],equat:[7,18,30,23,15,0,17,5,27,11,1,19,20,13,21],"0343v1":27,comment:25,distinguish:[25,28,1],"46j":13,respect:[23,15,25,17,9,27,19,1,26,20,12],page_504:[19,1],quit:[25,9,20],cauchi:[10,17],slowli:[23,17,9,19,1,30,26,6],why:22,insuffici:2,compon:[7,12,2],treat:28,"519839j":20,modif:8,immedi:6,date:31,partial:[22,0,9,29,1,6],apmath:3,presenc:2,"308937j":19,ridder:30,maxcoeff:11,togeth:21,bei_n:19,"53155328326114e":20,"806818225662921587261496e":0,replic:9,automag:26,hallei:30,"05804937947429712998395177j":27,"3109263360275413251313634j":27,rectangular:[28,22,12],defin:[2,1,6,8,9,10,11,12,13,15,16,17,18,19,20,21,23,25,26,27,0,29],rgamma:13,voro:27,ill:[25,17,23,21],erik:27,"988897705762865j":20,helper:21,almost:[28,20,8,11],demo:31,site:3,"93665300979176e":13,archiv:27,adiff:6,substanti:6,f_int:19,scienc:31,"35e":8,let:20,welcom:7,legendreq:15,lu_solv:20,cross:[25,17],sqrt:[8,14,26,23,15,28,17,9,27,2,10,0,11,13,29,1,19,20,12,6],fractionalpart:2,python:[7,22,25,16,31,0,2,28,11,4,20],cope:9,largest:[25,20,27,6,2],antideriv:[10,0,19],probabl:[8,25,9,2,10,28,20,6],difficult:[23,21],phi:[8,15,17,27,11,4,20,26],http:[7,8,14,20,26,9,30,27,10,3,13,1,11,19,31,6],cubic:30,innermost:6,futil:25,effect:[22,23,28,31,27,2,19,6],"866025j":0,phy:19,pdf:[9,3,27],h1_n:19,expand:[6,1],"866025403784438j":0,off:[9,27,6,2],center:[0,9],"551991234890371873502105j":17,"866025403784439j":0,well:[7,8,22,30,23,25,26,9,0,28,11,13,29,1,19,20,6,21],polyv:[14,11,23,5],"2360679774997898j":28,exampl:[0,2,3,1,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,27,28,29,30,31],choos:[10,30,9,29],undefin:[27,1],latest:31,"11604466157397267043655j":27,"7298060598018049369381857j":15,distanc:19,"5118225699873846088344638j":24,therebi:[11,1],"24570818422368e":13,less:[0,8,25,26,2,10,28,11,1,21],"135483865314736098540939e":0,"26j":[19,29],"765587078525921j":27,obtain:[8,22,23,25,17,18,27,10,0,29,12,6,21],"219637303741238195688575j":13,"2050415770437884900574923j":1,simultan:[14,27,21],keep_aspect:12,adv:3,web:25,rapid:23,"5756221777743631j":27,"114650753239580827144204e":19,elsevi:3,bell:[8,4,27],"68162300487989766727122e":27,script:31,trigamma:[3,13],valid:[23,0,17,27,2,11,13],"02107292320897e":19,densiti:10,match:[11,13],futur:[23,2],interlac:19,max_degre:9,"49895233563961e":10,"1e6j":19,"107491j":10,"568997623337943121769938j":18,press:[3,23,13],height:27,recurs:[8,9,15,2],"83493347225108e":13,expj:[0,15],loss:1,like:[0,22,14,25,16,17,9,27,2,10,28,11,13,30,19,20,12,6,21],lost:[6,2],betainc:13,"22464679914735e":[0,26],weisstein:[19,3,6,1],primepi:8,necessari:[25,17,9,11,1,30,13],"8565440833654995949e":27,lose:25,ncdf:10,page:[7,8,9,31,2],"198140234735592207439922j":1,"12954868316652e":2,glaisher:[26,4,13],"436102233553582711818743j":1,"539991603850462856j":1,growth:[26,19,13],with_stat:22,convex:24,proper:[16,9],guarante:[25,7,8,28,2],"00000008274037e":[0,2],librari:[7,22,25,31,3,20],win32:31,trust:[25,20],lead:[0,14],struvel:19,avoid:[22,0,17,2,10,29,1],thank:11,estim:[8,14,23,0,9,2,10,30,6],leav:30,summand:[13,6],"385040221241629041351886e":19,glacial:13,struve:[19,4],"92726697721369j":27,"656628229175902018939005e":19,quasiperiod:17,calc_nod:9,journal:3,usag:[28,7,22],docstr:[7,1],"8598304758094664213e":30,although:[14,23,25,26,9,27,10,28,1,30,6],offset:10,"70710678118654746j":28,simpler:[30,17,27,1],hyperfactori:[4,13],about:[7,8,25,26,9,27,2,11,31,6],actual:[22,25,9,2,28,11,6],column:[20,6],odefun:[0,21,5],sumem:6,includ:[7,8,22,23,15,25,16,17,24,18,27,2,10,0,19,4,1,31,30,26,13],discard:20,"478450291165243789749427e":19,disabl:[28,30,31,6,2],own:[0,16],fro:20,"96014e":6,easy_instal:31,automat:[22,25,31,9,2,10,28,19,29,1,20,6,21],"871672518271819j":0,diagon:20,guard:29,"23370055013617e":0,"2464022641351420167819697j":19,empti:[20,13],lune:27,"56e":20,ctx_mp_python:28,"5988311440454888436888028j":1,van:27,"131967049321725588398296e":19,renorm:27,lommel:[19,4],meissel:26,libmp:[25,31],inner:1,log10:0,overflowerror:[25,2],"function":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],offprint:3,"707106781186548j":22,"844952708937228j":27,shank:6,"519555750556794218804724e":27,"202762j":20,triangular:[20,6],"9219865355293e":8,continu:[8,25,17,27,19,6,29,1,26,13,21],"6141851416j":1,uninstal:31,"1000000j":[10,27,1],overflow:25,bug:[25,31,22],count:[8,26,2,19,4,20],made:[9,23],wise:20,whether:[25,9,2,28,11,20],wish:6,"455089860562227j":0,smooth:[8,9,23,6],displai:[25,0,12],troubl:9,"333332j":1,below:[22,9,27,11,1,6,21],limit:[7,18,24,15,25,16,17,9,10,0,11,13,1,5,19,26,6],otherwis:[8,16,9,27,2,11,30],problem:[25,17,9,28,11,5,30,6,21],"21728689675164j":20,reliabl:[25,30],"catch":2,reciproc:[0,26,8,18,13],evalu:[0,1,2,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,21,23,25,26,27,28,29,30],"int":[8,25,26,2,6,13],dure:[31,22,6],"968299831601008419949804e":19,implement:[0,8,13,14,15,25,16,26,9,27,10,28,22,11,4,1,19,20,6,21],evalf:11,run_doctest:31,"2774499251557093745297677j":27,inf:[0,8,22,24,15,25,16,17,9,18,27,2,10,28,19,13,29,1,20,26,6],dirichletetafunct:27,"480717210191852440616782e":19,inc:3,mutual:15,"47e":20,"243554598118700063281496e":19,detail:[23,25,26,9,27,2,10,31,6],virtual:[25,9],preinstal:31,other:[0,31,13,14,15,25,17,9,27,2,10,28,22,11,4,1,19,20,26,6,21],lookup:11,special:[8,22,15,25,17,18,27,2,10,0,19,4,1,20,26,13],branch:[8,15,0,16,17,10,1,13],"3937692045497259717762649j":15,cplot:[0,17,19,12],repeat:[9,14,23],"97336609573659199632j":1,"8660254037844386467637232j":8,june:[19,3],shown:[8,11,12,27],"14228140091932e":20,anal:3,siegeltheta:27,"03284785591801e":2,debian:31,"56964379417266e":13,sphinx:31,elliprj:17,baker:23,rule:[9,23,5],"22311232304258e":13,richardson:6,decemb:3,invari:[0,17]},titles:["Powers and logarithms","Hypergeometric functions","Utility functions","References","Mathematical functions","Numerical calculus","Sums, products, limits and extrapolation","Welcome to mpmath’s documentation!","Number-theoretical, combinatorial and integer functions","Numerical integration (quadrature)","Exponential integrals and error functions","Number identification","Plotting","Factorials and gamma functions","Polynomials","Orthogonal polynomials","Trigonometric functions","Elliptic functions","q-functions","Bessel functions and related functions","Matrices","Ordinary differential equations","Basic usage","Function approximation","Hyperbolic functions","Precision and representation issues","Mathematical constants","Zeta functions, L-series and polylogarithms","Contexts","Differentiation","Root-finding and optimization","Setting up mpmath"],modules:{"mpmath.functions.elliptic":17},descrefs:{"mpmath.mp":{phi:[26,1],e:[26,1],degree:[26,1],twinprime:[26,1],glaisher:[26,1],apery:[26,1],khinchin:[26,1],catalan:[26,1],pi:[26,1],euler:[26,1],mertens:[26,1]},"mpmath.calculus.quadrature":{TanhSinh:[9,3],QuadratureRule:[9,3],GaussLegendre:[9,3]},"mpmath.calculus.quadrature.GaussLegendre":{calc_nodes:[9,2]},mpmath:{polyroots:[14,0],hyper:[1,0],jtheta:[17,0],ellipe:[17,0],meijerg:[1,0],rand:[2,0],lommels2:[19,0],lommels1:[19,0],ellipk:[17,0],pcfu:[19,0],mnorm:[20,0],pcfw:[19,0],pcfv:[19,0],zetazero:[27,0],acsch:[24,0],identify:[11,0],csc:[16,0],gammainc:[10,0],gammaprod:[13,0],pcfd:[19,0],grampoint:[27,0],isnormal:[2,0],cholesky:[20,0],ellipf:[17,0],ln:[0,0],laguerre:[15,0],rf:[13,0],li:[10,0],frexp:[2,0],powm1:[0,0],degrees:[16,0],quadosc:[9,0],hermite:[15,0],jacobi:[15,0],fdiv:[2,0],norm:[20,0],psi:[13,0],legendre:[15,0],factorial:[13,0],maxcalls:[2,0],nzeros:[27,0],qgamma:[18,0],ceil:[2,0],workprec:[2,0],timing:[2,0],airyaizero:[19,0],siegelz:[27,0],eulernum:[8,0],coth:[24,0],qp:[18,0],qhyper:[18,0],primepi2:[8,0],betainc:[13,0],expjpi:[0,0],radians:[16,0],arange:[2,0],clcos:[27,0],hankel2:[19,0],diffs_prod:[29,0],hankel1:[19,0],fmod:[2,0],fprod:[2,0],shi:[10,0],pslq:[11,0],angerj:[19,0],cospi:[16,0],stieltjes:[27,0],hyper2d:[1,0],pade:[23,0],gegenbauer:[15,0],cyclotomic:[8,0],cosm:[20,0],polylog:[27,0],fresnelc:[10,0],j1:[19,0],cosh:[24,0],airybi:[19,0],hypot:[0,0],sec:[16,0],arg:[2,0],superfac:[13,0],logm:[20,0],fresnels:[10,0],sincpi:[16,0],kfrom:[17,0],fneg:[2,0],isnan:[2,0],qbarfrom:[17,0],isint:[2,0],loggamma:[13,0],ff:[13,0],struvel:[19,0],barnesg:[13,0],qfac:[18,0],re:[2,0],sinpi:[16,0],nsum:[6,0],appellf4:[1,0],hyp2f1:[1,0],clsin:[27,0],appellf1:[1,0],appellf2:[1,0],appellf3:[1,0],besseli:[19,0],besselk:[19,0],besselj:[19,0],csch:[24,0],power:[0,0],lerchphi:[27,0],eulerpoly:[8,0],legenp:[15,0],bessely:[19,0],nint_distance:[2,0],ei:[10,0],sinm:[20,0],almosteq:[2,0],findroot:[30,0],sinh:[24,0],altzeta:[27,0],e1:[10,0],sinc:[16,0],rect:[2,0],monitor:[2,0],hypercomb:[1,0],diffs:[29,0],cos:[16,0],dirichlet:[27,0],cot:[16,0],root:[0,0],odefun:[21,0],polyexp:[27,0],lambertw:[0,0],chop:[2,0],sumem:[6,0],binomial:[13,0],erfc:[10,0],scorerhi:[19,0],extradps:[2,0],acos:[16,0],fourier:[23,0],shanks:[6,0],acot:[16,0],gamma:[13,0],hyp2f3:[1,0],hyp2f2:[1,0],ci:[10,0],hyp2f0:[1,0],fsub:[2,0],mag:[2,0],bell:[8,0],cbrt:[0,0],fsum:[2,0],harmonic:[13,0],ldexp:[2,0],besseljzero:[19,0],webere:[19,0],bernfrac:[8,0],sqrtm:[20,0],primepi:[8,0],struveh:[19,0],diff:[29,0],zeta:[27,0],hyperfac:[13,0],kei:[19,0],plot:[12,0],log:[0,0],whitw:[19,0],chi:[10,0],airyai:[19,0],ker:[19,0],whitm:[19,0],fraction:[2,0],acosh:[24,0],nprint:[2,0],log10:[0,0],asinh:[24,0],acoth:[24,0],hyperu:[19,0],j0:[19,0],besselyzero:[19,0],fadd:[2,0],bernpoly:[8,0],splot:[12,0],qfrom:[17,0],elliprj:[17,0],linspace:[2,0],elliprc:[17,0],quad:[9,0],sign:[2,0],elliprg:[17,0],elliprf:[17,0],elliprd:[17,0],bei:[19,0],scorergi:[19,0],nint:[2,0],tanh:[24,0],mfrom:[17,0],ncdf:[10,0],nstr:[2,0],erfinv:[10,0],expj:[0,0],erf:[10,0],rgamma:[13,0],limit:[6,0],expm:[20,0],chebyt:[15,0],ber:[19,0],mpmathify:[2,0],fibonacci:[8,0],backlunds:[27,0],taylor:[23,0],coulombc:[19,0],spherharm:[15,0],coulombf:[19,0],unitroots:[0,0],findpoly:[11,0],im:[2,0],workdps:[2,0],mangoldt:[8,0],secondzeta:[27,0],conj:[2,0],tan:[16,0],fourierval:[23,0],polyval:[14,0],expint:[10,0],fabs:[2,0],floor:[2,0],atanh:[24,0],ellippi:[17,0],sqrt:[0,0],legenq:[15,0],bernoulli:[8,0],fac2:[13,0],memoize:[2,0],npdf:[10,0],sin:[16,0],fmul:[2,0],cplot:[12,0],polar:[2,0],frac:[2,0],atan:[16,0],asin:[16,0],primezeta:[27,0],powm:[20,0],bihyper:[1,0],nprod:[6,0],beta:[13,0],isinf:[2,0],siegeltheta:[27,0],ellipfun:[17,0],asec:[16,0],expm1:[0,0],airybizero:[19,0],riemannr:[8,0],diffs_exp:[29,0],differint:[29,0],hyp0f1:[1,0],taufrom:[17,0],agm:[0,0],digamma:[13,0],kleinj:[17,0],chebyfit:[23,0],chebyu:[15,0],sech:[24,0],asech:[24,0],hyp1f1:[1,0],hyp1f2:[1,0],atan2:[16,0],si:[10,0],erfi:[10,0],exp:[0,0],fdot:[2,0],autoprec:[2,0],richardson:[6,0],coulombg:[19,0],sumap:[6,0],hyp3f2:[1,0],acsc:[16,0],extraprec:[2,0]},"mpmath.calculus.quadrature.QuadratureRule":{guess_degree:[9,2],clear:[9,2],calc_nodes:[9,2],sum_next:[9,2],summation:[9,2],estimate_error:[9,2],get_nodes:[9,2],transform_nodes:[9,2]},"mpmath.calculus.quadrature.TanhSinh":{calc_nodes:[9,2],sum_next:[9,2]},"mpmath.calculus.optimization":{MDNewton:[30,3],Newton:[30,3],ANewton:[30,3],MNewton:[30,3],Ridder:[30,3],Illinois:[30,3],Anderson:[30,3],Pegasus:[30,3],Secant:[30,3],Halley:[30,3],Muller:[30,3],Bisection:[30,3]}},filenames:["functions/powers","functions/hypergeometric","general","references","functions/index","calculus/index","calculus/sums_limits","index","functions/numtheory","calculus/integration","functions/expintegrals","identification","plotting","functions/gamma","calculus/polynomials","functions/orthogonal","functions/trigonometric","functions/elliptic","functions/qfunctions","functions/bessel","matrices","calculus/odes","basics","calculus/approximation","functions/hyperbolic","technical","functions/constants","functions/zeta","contexts","calculus/differentiation","calculus/optimization","setup"]})