Values of the imaginary error function $\operatorname{erfi}(x)$

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Numbers

$x$

$\operatorname{erfi}(x)$

1/6:

0.1898191279333074723371369415550906641491596668752633092528531236134869677455649041534570687728849370

1/5:

0.2287212992439712955669345156319766926411211513298020137408336078117034061595606670092621466416550643

1/4:

0.2880836197949719840347000657269359169306589416965274307337803815003726176198296766013926160256975889

1/3:

0.3905339043933965739486161842155485206462840095637153317221253992367843326644053005258338783866404816

2/5:

0.4766246396513396689069782806312721893551455693797608454583308292909188483090108442692425677785292951

1/2:

0.6149520946965109808396811856236413930513456178954035723266386934995887895613341957110356633853898032

3/5:

0.7678530692501766590785277058494796415910460336056608605958979896541775727415257400720213946879871434

2/3:

0.8802757880498597072637771198144497817064511991037866228491268698687420395245597192407780155076914969

3/4:

1.035757284411962967860153121657800186386445498888186031841503536934390384212110564948339413704166876

4/5:

1.138670789947370671452523895314842900335723332230242909851741314811401757578077364948658541712455617

5/6:

1.211966398005783061657366883116322017458228095472120585694263879325027820847574881079251707466880353

1.650425758797542876025337729561362443895679874874022877600257996316111304925645922533850168585456181

7/6:

2.263290123997997930030267153943250876716398622220644595626876781269736075399658906335455968716882849

6/5:

2.415912970899116496543887652385776851097068900340290260041504034622501753324242425265567356106706975

5/4:

2.669134288682375042514337772244237758133921758971865768319207159210644175983076451733006861348469832

4/3:

3.169102066485088035676858237983098293947087113087168335070947856304042577920194751478500536661806457

7/5:

3.656957483162534816793508832548842899678854793564308614868622240167085395857349094087647845088445609

3/2:

4.584733257284426942221381032382701256662242548351999108876627023744176949019770931419772312767308061

8/5:

5.837725374364932167710290904598892010893006702419424367854339367614678762557020157076069188448095748

5/3:

6.924110923070113018522143922695204521032194898169013115432676007556142413145258130538166513633300008

7/4:

8.671694929603424624528939507121413419276337679034314872444279081110210636019380482396659973506976466

9/5:

9.991119815744892511143284722686532606100485454283194569703543171630979390458222402528956486665781630

11/6:

11.01189912360984073600761047460911944989528712962923044751834621633651533546473503384409008781183527

18.56480241457555259870429191324101719885800172608340681930081872293634170145762056806755792436841688

13/6:

33.32107210366776667913990038184702195045718240803062778477898381688171606014340644638400564377891515

11/5:

37.74710898061624567738666957007976873368233372859761098940231552301501176825334121607608160634754546

9/4:

45.73515014065393064927848135636707229225727696812321162572860955852965232925736688365444583191492957

7/3:

63.80535366027345391778446847550554836433579116902597017990935723947696127088842877907369285870856804

12/5:

84.26307398561919995580887539566956586917377360658680929230288266859194834606444142221293157675759201

5/2:

130.3957550132469268137315308322899098275353223889881621708796595862923260866928305530408882960407974

13/5:

206.5188761365409682474898883723021726045996378902512659284039950379172128427848944420943146929468126

8/3:

284.2052429523492849071270949269713702999704585435003163637750611461366229969931880179645480039781721

11/4:

429.6800722766928978578778044318818072404027795547221931226519006742567645750785271093010510023810206

14/5:

554.7771910530312576762898138521654489300510484413085278555579250931026354767598852782472514496693812

17/6:

659.8609175513045106783724053676941827544301696413085463649582914029890738166060334866784142572525764

1629.994622601565651061647952076274162778889910915259988647982150695656826775680269243153186081103257

19/6:

4279.150398861044321772837668125229474965500439910063978813324393602514470262860273287438593383134154

16/5:

5227.918675842428441659105193331406991083848800979077334025465984242778804666535037571263206049836063

13/4:

7091.422701590346430101943651771310553441426392525812826029312161983913059329300085137405040892643058

10/3:

11928.36924528281921453237842038637918480509706308917990103404900345416042061457719139862793073448681

17/5:

18276.59861630598942981971302882902750865556323730291379724872360194251395945160301164894609523573724

7/2:

35282.28771517168531015799721641242794918212620500451316165244609857143540600864126939235727164856078

18/5:

69567.21939096751129226188259073020332759417958689599811157753738653787189184993700498131113122355147

11/3:

110675.9582422770200182315587397292779777452264069834235105955610370474180231145212212909621883206043

15/4:

200361.4093850143371089487534959998007051184294002251496175881247882688775369158530078438522172829722

19/5:

288073.4994598208617182296759776938076271368248760914007002166642804773811086067941249072043714876814

23/6:

368036.4129156390036082924585409218232144654317515818204910632852115531397813143805398026415369831363

1296959.730717639231527940950621767303818858159091052932776495827461878100973663174250475225833312313

25/6:

4842221.718156472051717183410215319921234270427362419308978773372536473567417632117088575304592984632

21/5:

6345553.158504535059217683638828228412379102138001520810399054792621373260406460613558189256544253128

17/4:

9560467.515846173411849930213529016033425102391876792492598390265026228406094321672699335294203969665

13/3:

19149153.27231795492605132496009151165419193293591567011055585790026610077549742484535677568860756267

22/5:

33726349.82286847296772382568862908584858266420952016823704883061663946580308229967918466269827625653

9/2:

80197458.90121747817731640016987353558785471784411429121341237039012310308574616788604421506619753952

23/5:

194667358.4700888469339192090863520005288050237166443905690227423727514450681430858128254626913988982

14/3:

355638666.5011564760969867605979215031712281080183910983737955529807334877241515419913208627463597569

19/4:

765117754.5300527295138451248561439737701670732784147915838993167282915171185889818061212509352641994

24/5:

1219917228.934533797717324843211854267565342260635949773454974454817334940908705854462545868020128917

29/6:

1669687818.286824647717766001939849061976673422755016206078254773051588327203360197932222293651726540

8298273880.676803516146223190746919951872735737541398445643912895480973194160213047611291366792059648

31/6:

43653690419.79043016796588491605951397471957684006567913015581986838654621711247543178787470914558128

26/5:

61261223838.31724798340152552337550512137804165527847081608603936513669754754649033029396447813864258

21/4:

102277495356.0307600098360615507619079313742265556341032336099624183854744413513124697039283151987756

16/3:

243049463353.3756317581026031790147813534143187766717223698071563671624675380036672977662827093142679

27/5:

490744386833.1318457415426717698543770556462226176390053401650172099960423307369721728401430888025021

11/2:

1432099172039.832821476869289855699288225956985038177549255896001948978650820505900723972120774311724

28/5:

4265176077690.003661243953855897901878120255798709916728420635685926803768744448042698889273727922091

17/3:

8929440413491.333411728197821794197570678761331816449460986574338664630863603736131990745749048393433

23/4:

22774848015911.85607629307448387235151915560744566790308265892088035049130139312134138863891620817024

29/5:

40214076398794.75585709948981798460158704247560230353331550089882903532161382976512603487293366522842

35/6:

58914114486117.16926656308566924338770138244547237673709203041156358876655810008660216296532647054547

411275145582823.8709717312025942799233643828595576520625777592471798892862178261086327727104544359520

37/6:

3037666201531482.903044838828684431422292556676321261545198404230715041643626011654717150978065737645

31/5:

4562038914606190.850084346721051697566145008011516549869131443287263400465851106953784919910039226619

25/4:

8431842912883763.041256213110105322360485591713400937855280540883174275818429239456326528966474089894

19/3:

23736720269727757.80590408936319392026326885415499965650881098534943911871267148262783383683418509957

32/5:

54880872608904296.50625838779060519015023282169802482113817564715544912772263858958566908407504184309

13/2:

196225267754784050.0494187831613913812130145394433100711902355147853467988079761107428342421900977265

33/5:

715954910374046471.0315853633179602115481639984714594464114583552426682995661257387668578166377329736

20/3:

1716029843524596956.159833589635342826470991972714121175572135920304939001538002285392477318786204294

27/4:

5182842294781432575.668638051059822916748943347424659365055510942091381068900420588947942021441404426

34/5:

10127992935839409783.59277420977587150189549776718869504591687232947096076878907169251256224691561749

41/6:

15875010787354864795.66376616829831325446578365251079020137746680476694257238386975195239935503490450

155348625346050399388.8981032171520673074441522622257394463914918963803029281004646550750056215714500

43/6:

1608018599353625853283.083939979329980636663893254010114521400951362834495267673582079168290353350319

36/5:

2583529828475891343363.142822859265500323293585369369651218235802395739844983772094654053341340199742

29/4:

5283542226056481870811.380549302851291624644420853420190032782177086210355677626371541131958624767492

22/3:

17605651933989299919541.97375132562625512915368995014586758401119566681524492988666745472464089406996

37/5:

46582146379980214867353.04879303314214361131106255583361979993279889979117281597516588671032737592312

15/2:

203881871917862113077072.8335863434632816480313397994316258675990515218942699113210081603677775243097

38/5:

910553124938862116550304.7661978006953516912518877527516431650650388972026861599606424011837393919416

23/3:

2497235431519977107466498.867364848485409777144020869821576228464915260580219414426757663651747055321

31/4:

8925438593955495394023975.960513442191479009338257792722907147561592282341872175730767183391030387263

39/5:

19295337787512492619549183.14363267616030389063011137585540823156753550490300623439129279888587230515

47/6:

32350755545947454711507204.71748777514615886002538952425194739391787802904451025595799295448247388329

443244974600233463199411342.4773495106566246866324164061330101393887528093502346021253756055501749787

49/6:

6422863475739157406126712717.041604681793675573446272394348971364057669635063724835666066050580771735

41/5:

11037328637261982607999544304.18599961405720846935506958443958750937289139301915139341949852338167951

33/4:

24968389801841129504783650162.40851939447424802194661867269602911049815599553548651050439838132998705

25/3:

98430716238880286171993916852.14091459654638144514219935882942017725224825714129627830947344169596668

42/5:

297919158498441445206261382790.2849678043810778972973472556698434863821715811909161887526996953797991

17/2:

1595298001394745672777335107074.809316531949637651485592394324661850315875542552506536382606686707324

43/5:

8716334032468621712978367322239.765635687657722630839991636026116921081983688037207229588360915320624

26/3:

27343546447825399458120063663631.50779693700181496643288034964461918314444718509164091091440721474414

35/4:

115603867029421510025623128195241.6310868403285899341456092203993466197176015157718459544723490668377

44/5:

276412650551227168310196837857332.7363032415799519238211719984139523454345336912948166536139094788120

53/6:

495635924463147128728525372659492.4929522406335097593739168867839859051941815738996228164962349114832

9500776643665995567475170723057540.497704097510899010139071851189708078130548629502407629412244057367

55/6:

192591756533041223635838522157613719.1879964543920757607330566989207797627826269557217630531140923255

46/5:

353937652776407876766721154392425515.9067725997078528522746435543387532346568564261731919806273058473

37/4:

885484594745060938214451468725798286.2405716357819582332457562140587570120374895899534717110683039119

28/3:

4128504320594411145112104217961063334.721251464699012375071304975798093674925774280391140558059118596

47/5:

14290615608741503159759524352470727986.99578008069858918388581576615503498432756842148349245707408405

19/2:

93587702885323339422598733389389903601.63898226686121765952889678156969631896245887990024950279771702

48/5:

625348812526802053660169128634106143787.7589935754557887906147273584558551508167071468884877288754344

29/3:

2243420362536612388626729260446496054033.422502712824256127775080465619440709729274427251347202828962

39/4:

11216432264373567439480227418578644808313.21801581587105172870118703871342509093224896542967419368797

49/5:

29657343795175631099955706206351807886348.94757701535244105418926632089362276992637962290927843727516

59/6:

56867411588452399878069946748019023598930.84159527843162104975028781790425558496428644707650609188732

10:

1524307422708669699360546614726544062463812.074698857310557201311229700714271763759939487330097258173

Definition

The imaginary error function is $\operatorname{erfi}(x)=-i\operatorname{erf}(ix)=\frac{2}{\sqrt{\pi}}\int_0^x e^{t^2}\,\mathrm{d}t$ [4]. This table gives its values at rational $x$ with $0\leq x\leq10$.

Parameters

$x$

— argument ($x\geq0$)

Formulas

(1)

For real $x$, $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm{d}t$, $\operatorname{erfc}(x)=1-\operatorname{erf}(x)$, and $\operatorname{erfi}(x)=-i\operatorname{erf}(ix)$ [4].

(2)

Dawson's integral is $F(x)=e^{-x^2}\int_0^x e^{t^2}\,\mathrm{d}t=(\sqrt{\pi}/2)e^{-x^2}\operatorname{erfi}(x)$ [2].

(3)

The functions $\operatorname{erf}$, $\operatorname{erfi}$, $F$, $S$ and $C$ are odd, and $\operatorname{erfc}(-x)=2-\operatorname{erfc}(x)$.

Comments

(4)

Negative arguments are not listed separately. Values at negative arguments follow from (3).

(5)

At $x=0$ the value is exact, $\operatorname{erfi}(0)=0$, and is stored as the integer rather than as a hundred places of one.

Programs

(P1)

Sage

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

field = RealBallField(numberdb.bits(100, losing=64))
x = field(QQ(1))
x.erfi()

(P2)

Python

import mpmath

mpmath.mp.dps = 100
mpmath.erfi(1)

Links

[1]

Wikipedia: Error function

[2]

Wikipedia: Dawson function

[3]

Wikipedia: Fresnel integral

[4]

DLMF 7.2: Definitions

[5]

OEIS A099288: decimal expansion of the imaginary error function at 1

[6]

MathWorld: Probability Integral

[7]

DLMF 7.3: Graphics

Similar tables

Values of the error function $\operatorname{erf}(x)$ — $\operatorname{erfi}(x)=-i\operatorname{erf}(ix)$

Values of the complementary error function $\operatorname{erfc}(x)$ — another member of the error-function family

Values of Dawson's integral $F(x)$ — $F(x)=(\sqrt{\pi}/2)e^{-x^2}\operatorname{erfi}(x)$

Values of the Fresnel sine integral $S(x)$ — another member of the error-function family

Values of the Fresnel cosine integral $C(x)$ — another member of the error-function family

Values of the exponential integral — stores $\operatorname{Ei}(x)$ and $E_1(x)$, with exponential kernels $e^t/t$ and $e^{-t}/t$ rather than the Gaussian kernel $e^{-t^2}$

Values of the sine integral $\operatorname{Si}(x)$ — stores $\operatorname{Si}(x)=\int_0^x\sin(t)/t\,\mathrm{d}t$, with linear phase in place of the Fresnel quadratic phase

Values of the cosine integral $\operatorname{Ci}(x)$ — stores $\operatorname{Ci}(x)=\gamma+\log x+\int_0^x(\cos(t)-1)/t\,\mathrm{d}t$, with linear phase in place of the Fresnel quadratic phase

Values of the hyperbolic sine integral $\operatorname{Shi}(x)$ — stores $\operatorname{Shi}(x)=\int_0^x\sinh(t)/t\,\mathrm{d}t$ at rational arguments

Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$ — stores $\operatorname{Chi}(x)=\gamma+\log x+\int_0^x(\cosh(t)-1)/t\,\mathrm{d}t$ at rational arguments

Values of the Gamma function at rational numbers — contains $\Gamma(1/2)=\sqrt{\pi}$, the normalising constant in $\operatorname{erf}$ and $\operatorname{erfc}$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds $\operatorname{erfi}(x)$ at every rational $x=a/b$ in lowest terms with $b\leq6$ and $0\leq x\leq10$, which are the arguments a reader is likely to have written down rather than the ones base ten makes short)

How they were obtained:

Each value was computed as a ball with arb at numberdb.bits(digits, losing=64) bits, and checked against mpmath at 150 decimal digits, against the identities in (3) and against the OEIS value at $x=1$, $\operatorname{erfi}(1)$ [5].

The value at $x=0$ is exact and is checked to lie in the computed ball rather than taken from it.