Values of the exponential integral

edit · history · discussion · files · long url · special functions special values

Numbers

normalisation

$x$

value

$\operatorname{Ei}(x)$

1/6:

-1.040667234618592847325744968939192295538273711065393906449354131920757712760435236879128959752572774

$\operatorname{Ei}(x)$

1/5:

-0.8217605879024003156533108698999777247627413368316407871683302039878211995487151234211072930264778364

$\operatorname{Ei}(x)$

1/4:

-0.5425432646619137295335318517343131618605951501281112277469533634852988850539986918389027927614667532

$\operatorname{Ei}(x)$

1/3:

-0.1580921089711557103135773062301122859227892527545194175342259120266969176086114508383033262467941875

$\operatorname{Ei}(x)$

2/5:

0.1047652186193247932289297638834602626515833855115913381485977932685094240143112757359295621787046358

$\operatorname{Ei}(x)$

1/2:

0.4542199048631735799205238126628023652814055543526420451628177866881528969266622561361796585714373140

$\operatorname{Ei}(x)$

3/5:

0.7698812899373594370931373302578512478875976104689771065282126014589750521383655041018023196018801972

$\operatorname{Ei}(x)$

2/3:

0.9682884423157133042101457444789017671037064662107604376387832804311736501143960990875198202969568122

$\operatorname{Ei}(x)$

3/4:

1.207332816001221882759442381995318774542137580441758163088083974254569787851673323087301565607974692

$\operatorname{Ei}(x)$

4/5:

1.347396548212325938118996383303582934630495325758518775519483271167369602133005829750205762721107794

$\operatorname{Ei}(x)$

5/6:

1.439768242201348706990525188012230846757665674994078613252313198430539284169354387520579361434750191

$\operatorname{Ei}(x)$

1.895117816355936755466520934331634269017060581732707591646228431882513834533804153548900710126138957

$\operatorname{Ei}(x)$

7/6:

2.350110544072333042157646413421692994759592471570669332315663999790380970887963949043390115152947587

$\operatorname{Ei}(x)$

6/5:

2.442092285192651639729097264306492853237246453178423605394562091722303492144570163281261747038652983

$\operatorname{Ei}(x)$

5/4:

2.581047974355475819711615574122249801778564132902404726325373153330880782281620308397976217433809812

$\operatorname{Ei}(x)$

4/3:

2.815856442572886724020750042761566523842616243491269170660843787278328145047230504565401526450452631

$\operatorname{Ei}(x)$

7/5:

3.007207464150645540102576179422194438122257221733073198679595293715591226930322659319807763254112234

$\operatorname{Ei}(x)$

3/2:

3.301285449129797837957411316134742787656606985453422100762913970145362682901597708797238406961684389

$\operatorname{Ei}(x)$

8/5:

3.605319949019469208722853214270112487314764751473054217205496124262778325703735350857402775129391942

$\operatorname{Ei}(x)$

5/3:

3.814357260171930642954474186100901285256983702822866390236501388792933987222993760775587119280119230

$\operatorname{Ei}(x)$

7/4:

4.083653659076957361824701625291281135525586811060024146962754338946949306521869969285165326621961275

$\operatorname{Ei}(x)$

9/5:

4.249867557487933504149965706150155694086465905980879617745361735317011051648157390137242453969413960

$\operatorname{Ei}(x)$

11/6:

4.362738450749532811730797310053727530451636741433490137304802727879236201356912982105328658377984347

$\operatorname{Ei}(x)$

4.954234356001890163379505130227035275518053562420042054527095254570707913298407190883096971852850163

$\operatorname{Ei}(x)$

13/6:

5.597102485195556072396406031294363637775537113464104742292577999928551520341908443574453129848378328

$\operatorname{Ei}(x)$

11/5:

5.732614699814380576980363854534572539917014680722591976210111309703831018210512480653532155538592599

$\operatorname{Ei}(x)$

9/4:

5.940569075037337797084579775112962978207937764888001038124441299088683346596768966388652645001323495

$\operatorname{Ei}(x)$

7/3:

6.300309582898208019168094900364009956323489149425413182473212262939661353397522799225605869379827100

$\operatorname{Ei}(x)$

12/5:

6.600670276350539313730718424831592331808768673061268783600573039706636148291290668547264201417014212

$\operatorname{Ei}(x)$

5/2:

7.073765894578600711923551962451012546996320105690375846236171645219738638560098256676337517388231307

$\operatorname{Ei}(x)$

13/5:

7.576114769787598649886368530157901009187577103110602760043377192208687653966511712668030365644306693

$\operatorname{Ei}(x)$

8/3:

7.928556705796363052605611167118187498028645854061230304188061818306982793011857554840704721759685797

$\operatorname{Ei}(x)$

11/4:

8.390297228026056961111417283333293227074394313208146826085832202733080273058754087462260645961893146

$\operatorname{Ei}(x)$

14/5:

8.679297723809165259747361751316525862699097315849760322129014344069841123563244288108803237668421435

$\operatorname{Ei}(x)$

17/6:

8.877184594120468479839806574743946469552447561238507954240129663262953038737713760089184624764549419

$\operatorname{Ei}(x)$

9.933832570625416558008336019216765262990653022987582119693034065242496228332697443187187398902087851

$\operatorname{Ei}(x)$

19/6:

11.11465680549653697863559813796571664585153736232771872933460998680731083215477807734487940879003727

$\operatorname{Ei}(x)$

16/5:

11.36730265696872958807711657229459496694843478318960109313691427439742914055205553422924763873821086

$\operatorname{Ei}(x)$

13/4:

11.75730379330492525617224003784464733903272471861995450768948379099396003897006728017352777060426665

$\operatorname{Ei}(x)$

10/3:

12.43811716881943521144301143370491870530410035276772009016760516230384325210623960188484424599139157

$\operatorname{Ei}(x)$

17/5:

13.01207530414044598680191879564984864218213254860285440551417700847994277966886746381755868686421129

$\operatorname{Ei}(x)$

7/2:

13.92535399515233529160016417048754973665770724392382825121295089110551015248531061213610689104381518

$\operatorname{Ei}(x)$

18/5:

14.90625409953709047344822304786726393022910851057753432944748566727268673064596387353276693650643313

$\operatorname{Ei}(x)$

11/3:

15.60062010954062129804388470051705701633938821764589505661193728270605795402199819521125008082743368

$\operatorname{Ei}(x)$

15/4:

16.51731509818927556327428538997756018825951442451730067672468259168805840242343183524150361487355251

$\operatorname{Ei}(x)$

19/5:

17.09480226516090065609430326742872591455176118181366540475272521695157160281243002767980379324571402

$\operatorname{Ei}(x)$

23/6:

17.49177814196503278272797764135061434111779412408702547696024372353264480022531294061810441064878180

$\operatorname{Ei}(x)$

19.63087447005622002264572027972383885313207975115315216815404620859056206761114761767084729515828299

$\operatorname{Ei}(x)$

25/6:

22.05480276218490925641552412187989228898497121893112230717798007243182570348897192769579265013774555

$\operatorname{Ei}(x)$

21/5:

22.57740064778208318824211788227501493660758442753913506552358915394315965537551169363783962915482670

$\operatorname{Ei}(x)$

17/4:

23.38662020364075265230481454096015270494481611690287982921556836744401233009797712712991767527618055

$\operatorname{Ei}(x)$

13/3:

24.80607206231062326722990776372353479364694228372311118969532752827428174361116514939327346615077079

$\operatorname{Ei}(x)$

22/5:

26.00897327160514597762730743975318706342599686348251748204070544744083096808859374243454921368385708

$\operatorname{Ei}(x)$

9/2:

27.93369669793958062906070837769252157365620949255040070616613226136577519968499491479968652257911631

$\operatorname{Ei}(x)$

23/5:

30.01409929647569657084332362526120906941542402833407938080851832876340471615854765813391635187409379

$\operatorname{Ei}(x)$

14/3:

31.49422805055113823762454924342038230622022497748035366465031635963360136547468017628081198744136251

$\operatorname{Ei}(x)$

19/4:

33.45685770742831433620710506197991791930281439882930237402302968347485268780572600790024823967978825

$\operatorname{Ei}(x)$

24/5:

34.69788987377532762712291256001489636576223294276461258196679558824071298506847728283227830645288940

$\operatorname{Ei}(x)$

29/6:

35.55295146415912064786761346572461382246971071484904321191609651451531500498310238925578918043769475

$\operatorname{Ei}(x)$

40.18527535580317745509142179379586709541908739919593043418287525164264481392822057865779970191749916

$E_1(x)$

1/6:

1.374515400116086405173403990428475502466211893652895417185260218923365618514032810575548055647902266

$E_1(x)$

1/5:

1.222650544183893088334772577574422774059585999309928859561380908302429227919949588667361760687033596

$E_1(x)$

1/4:

1.044282634443738194536438161232282251891528374744802718635140467927968348132202998387154036496492311

$E_1(x)$

1/3:

0.8288877453485866361202069811634756826450010649093967332272735913382854921305272444761315598181019478

$E_1(x)$

2/5:

0.7023801188656624785830289940116447984912132305209679527149413288785283401891984157265013080500848170

$E_1(x)$

1/2:

0.5597735947761608117467959393150852352268468903163535152482932191073398988302742688603417524733251125

$E_1(x)$

3/5:

0.4543795031894021081559882389372316785929806124825842499805612887927490279013543235885993203745771694

$E_1(x)$

2/3:

0.3984089930702799837865538683381159040446381399255835262863276511282432123498107182077165192236234580

$E_1(x)$

3/4:

0.3403408129112300078065357686796446251898098614786599945045006640043517388914640392771578691313995455

$E_1(x)$

4/5:

0.3105965785455430346548245550756189053846989607723556858915710228033415902468211787478058185852060315

$E_1(x)$

5/6:

0.2925543681493770795332397611184512581851329688553432562481825469187755599805155657502407767665946495

$E_1(x)$

0.2193839343955202736771637754601216490310472934069082075779786130735686985591415447222102510351372500

$E_1(x)$

7/6:

0.1670370166378481662248336980215673956232320046034175414720757700871896126414326117151082300491247834

$E_1(x)$

6/5:

0.1584084368514625614249559707108617385341579768405789633785744207451827739586452935818843915065118853

$E_1(x)$

5/4:

0.1464133725259101766932585022903932235747701160803928539977552558331833055079293141966258117073507692

$E_1(x)$

4/3:

0.1286643042677474212644726321202210437715884812841720770164155412345428901867882762136441975629958814

$E_1(x)$

7/5:

0.1162193125713578892748793556211437382708618823656590244835764321289550308167445990735432562603428175

$E_1(x)$

3/2:

0.1000195824066326519019093399116669782617300061403505850505670603647199164241734387421982935560856726

$E_1(x)$

8/5:

0.08630833369753978558451624125276549238754625766003561065540227705174936121847640381111932926043626600

$E_1(x)$

5/3:

0.07833342960047432009208512987137040477246262250029714839439896485486242737322468121512415988366056211

$E_1(x)$

7/4:

0.06948868460463879263837412067261358609690518081619424747682377118796911839046950799299804963896272258

$E_1(x)$

9/5:

0.06471312936386885956357451048683198202694809018692908571758417144212279861690714814214372136993296146

$E_1(x)$

11/6:

0.06172987959772765111730454668858884419989315668040481615584668253065424962272815563184956122420855519

$E_1(x)$

0.04890051070806111956723983522804952231449218496302311632732287371169292871415219127926896100745164177

$E_1(x)$

13/6:

0.03891140477518787729408393913587117366308577982272507910487562534127454898166799716199073423797141247

$E_1(x)$

11/5:

0.03719113705193257438360263447178853302082891188289430881377803570202829390990220703511512687947170481

$E_1(x)$

9/4:

0.03476207311944459707617859810088544864118256789007691061300521528447772463753180749306688658337402030

$E_1(x)$

7/3:

0.03108339065063154960156018170321238994661088878835033893879431333994799175689582745845324040720266510

$E_1(x)$

12/5:

0.02844026093579567172653371455738800778144232758075766791655677686591264055280833199961389045401069262

$E_1(x)$

5/2:

0.02491491787026973549562801227460963594584838471142737701193454449656612251978548469146160042946169026

$E_1(x)$

13/5:

0.02185022180408219361058235015951009616982349166788812811466496937051412404455763774176354741817633380

$E_1(x)$

8/3:

0.02003083321130836333943796316139571171394042311611889468043755647799979856400217548422482373660448789

$E_1(x)$

11/4:

0.01797893821545440659344439154546103850433039338503197871856553574552179696973691960352984591146315213

$E_1(x)$

14/5:

0.01685529244521605051657093889817972781306505727034169724209206443558797099665235573969419985270872887

$E_1(x)$

17/6:

0.01614747713626066547221951908781707025870378125436273670563142053139325435557049782343253949392912947

$E_1(x)$

0.01304838109419703741250074582864502294847763408022360117699903114085916543429526306181307900231552789

$E_1(x)$

19/6:

0.01056703197885409736834101513680838158419840872511187017906133436982553872542147941192137001947977523

$E_1(x)$

16/5:

0.01013299249934910993755023111077662505188151651426081466279739677691477336459411122065264805452377557

$E_1(x)$

13/4:

0.009516507415488778197975644280843255256116117277972316439725546196008431775498484688232787645801562996

$E_1(x)$

10/3:

0.008574454203539531305053406521411740718599831007760863254219105234360488618079894741402796538755518820

$E_1(x)$

17/5:

0.007890973505565841691109931284879642398762073147392155386439586914596961192313417869229040993075086958

$E_1(x)$

7/2:

0.006970139857548392919272924566140513707547369068072265421013952416259589940706646399802382423292057777

$E_1(x)$

18/5:

0.006160414331797309424943311915448213974785195658590354968836623460970629495116175153356635178166169131

$E_1(x)$

11/3:

0.005675345069884025175296989855236098204347951124985700165887437850752532059060886921651399775169266709

$E_1(x)$

15/4:

0.005124102198586809787413117875176032515654218836788087929012607990808296245976608005566532477215735454

$E_1(x)$

19/5:

0.004820246816497640609699173707892741636487216418843804896443032710360015371164399481537164283353313441

$E_1(x)$

23/6:

0.004628083430238760538256291268547048206235308846449156271049455435521482212051155382854352987649965220

$E_1(x)$

0.003779352409848906478874860132466414851716547042489580360734020313584606172362404699077971896569385492

$E_1(x)$

25/6:

0.003090265939196316240019864594984151113574450680307162217020004002100403800609950497326262629705577556

$E_1(x)$

21/5:

0.002968762181941947791146798724087576518501464679429153170668452197726250805159370807740589110795813934

$E_1(x)$

17/4:

0.002795652712273958452670962693078440365877014256493160474847105414189938846161434580803143856988498556

$E_1(x)$

13/3:

0.002529859749726570573114368768751280129260515075010128043815347056585501038763090088682759600303449847

$E_1(x)$

22/5:

0.002336010040655828994103804068128677695837434475249174764311805784406450579721404968111814834349754920

$E_1(x)$

9/2:

0.002073400754714614432885593869579788488931972570144300385874044088006080024432630985788844925503724030

$E_1(x)$

23/5:

0.001841005817241503071312829720984602408000887312354881531945537006276130380396555974113830284134973906

$E_1(x)$

14/3:

0.001701077141506636935568216936082758022029896096505920576962375074175246954327043291347121368895219309

$E_1(x)$

19/4:

0.001541364705263895250505442242605119387265678851445896482452582098702541689282044040362223305907184294

$E_1(x)$

24/5:

0.001452993936878324220193684837645472955391084372490581355294835111614194641293374119244329767848714814

$E_1(x)$

29/6:

0.001396978131282777305588115116874929097213108298636484717329464639269589738335682840304684319571065456

$E_1(x)$

0.001148295591275325797330561969819722076266095470697866764845494043583439236859430913119905624993329877

Definition

For $x>0$ the exponential integral is the principal value $\operatorname{Ei}(x)=\operatorname{PV}\!\int_{-\infty}^x e^t\,\mathrm{d}t/t$, and $E_1(x)=\int_x^{\infty} e^{-t}\,\mathrm{d}t/t$ [4]. The two are one function in two normalisations, $E_1(x)=-\operatorname{Ei}(-x)$, and this table holds both at rational $x>0$.

Parameters

normalisation

— normalisation

$x$

— argument ($x>0$)

Formulas

(1)

For $x>0$, $\operatorname{Ei}(x)=-\operatorname{PV}\!\int_{-x}^{\infty} e^{-t}\,\mathrm{d}t/t=\operatorname{PV}\!\int_{-\infty}^x e^t\,\mathrm{d}t/t$ and $\operatorname{Ei}(-x)=-\int_x^{\infty} e^{-t}\,\mathrm{d}t/t$ [4].

(2)

For $x>0$, $\operatorname{Ei}(x)=\gamma+\log x+\sum_{k=1}^{\infty}x^k/(k\,k!)$, where $\gamma$ is the Euler-Mascheroni constant [4].

(3)

For $x>0$, $E_1(x)=\int_x^{\infty} e^{-t}\,\mathrm{d}t/t=-\operatorname{Ei}(-x)$ [4].

(4)

For $x>0$, $\operatorname{Ei}(x)=\operatorname{Shi}(x)+\operatorname{Chi}(x)$ [4].

Comments

(5)

For positive real $x$, $E_1$ is the case $n=1$ of the generalized exponential integral $E_n(x)=\int_1^{\infty} e^{-xt}t^{-n}\,\mathrm{d}t$ [6].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

RBF = RealBallField(numberdb.bits(100, losing=64))
x = RBF(QQ(1))
x.Ei()               # Ei(1)
-RBF(-QQ(1)).Ei()    # E_1(1) = -Ei(-1)

(P2)

Python

import mpmath

mpmath.mp.dps = 100
mpmath.ei(1)
mpmath.e1(1)

Links

[1]

Wikipedia: Exponential integral

[2]

Wikipedia: Logarithmic integral function

[3]

Wikipedia: Trigonometric integral

[4]

DLMF 6.2: Definitions and Basic Properties

[5]

DLMF 6.4: Analytic Continuation

[6]

DLMF 8.19: Generalized Exponential Integral

[7]

OEIS A091725: decimal expansion of $\operatorname{Ei}(1)$

[8]

OEIS A099285: decimal expansion of $E_1(1)$

Similar tables

Values of the logarithmic integral $\operatorname{li}(x)$ — $\operatorname{li}(x)=\operatorname{Ei}(\log x)$, so that table is this one after a change of variable

Values of the hyperbolic sine integral $\operatorname{Shi}(x)$ — $\operatorname{Ei}(x)=\operatorname{Shi}(x)+\operatorname{Chi}(x)$ (4)

Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$ — the other half of $\operatorname{Ei}=\operatorname{Shi}+\operatorname{Chi}$

Wilbraham-Gibbs constant — stores $\operatorname{Si}(\pi)$ and its normalisations

Stieltjes constants — contains $\gamma_0$, the Euler-Mascheroni constant appearing in $\operatorname{Ei}$, $\operatorname{Ci}$ and $\operatorname{Chi}$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds $\operatorname{Ei}(x)$ and $E_1(x)$ at every rational $x=a/b$ in lowest terms with $b\leq6$ and $0<x\leq5$)

How they were obtained:

Each value was computed as a Sage real ball with arb at numberdb.bits(digits, losing=64) bits, and checked against mpmath at 150 decimal digits and against the OEIS value for $\operatorname{Ei}(1)$ [7] and $E_1(1)$ [8].

The widest returned ball had radius less than $10^{-117}$ for 100 requested digits.