Values of the cosine integral $\operatorname{Ci}(x)$

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Numbers

$x$

$\operatorname{Ci}(x)$

1/6:

-1.221480216179132439385958569588648331375312698400800090560441027179204143791971848875436102398836924

1/5:

-1.042205595672781975362916906675889523453125168105938907771508473753606092361988028437555390508844830

1/4:

-0.8246630625809456530858650345190850733960949224566059347682517446693093913051280769434625217991471300

1/3:

-0.5490461177821529164924525479297681167338624436976178196272705743073352352134085700006670106860662555

2/5:

-0.3788093464252443320846617572294418403950775789258904420265940717903741069282477550573352960658268818

1/2:

-0.1777840788066129013358102710705690780905194748126219686668253575951272650159020967709876446027217324

3/5:

-0.02227070695927976252660274390206256054895346166617113439621444450803963484177001894338898449986846908

2/3:

0.06267685723397644057329364723403093093104615776151953816369804711369987814635697351174732629136011457

3/4:

0.1521636009803302365530954317485839924767229650719516624930530247867910532813859875420284530786757556

4/5:

0.1982786159524671770158335822338409704801842574813290740397061148629087412161926160967486766078121600

5/6:

0.2262296605427398996618457282580153621911623409121384276936472724168765591990769390943389364859775594

0.3374039229009681346626462038891507699975780325857318948013185424361303300250560528968481830973229946

7/6:

0.4098135310560801156028285733775913334436732439419077075999602335812994507539622482807967713742840000

6/5:

0.4204591828942405027153707851337276721807543441757304423455769594781043437964822626861826793628011721

5/4:

0.4343007240335523888279514437200584916159835414361156093464599403554915727648755159135475756845809715

4/3:

0.4521049719736518855881234164123107650468070336187719894386414200355613692895523761137027657360403484

7/5:

0.4620065850946772763120557463026489629630494641177488271301110704370648943502400716883795946894987948

3/2:

0.4703563171953998866750821522365605516152327005730752953941674282481411205940176830407817186047829299

8/5:

0.4717325169318778033675123539297478882295603587462680466161836462867049714898140969432623259209851220

5/3:

0.4691910063392783002072332753397421856166611157462984476032628436623587776491456763897298701297542365

7/4:

0.4625199967269357432971344114889846689255726845059131391505893545818140543728575727934651023830488796

9/5:

0.4568111294183368931235789000704366605653288452187655569632348777173667433307135141398143957624385113

11/6:

0.4523460522522531657981392844370593987343114621168000143430502311675980807729462488943190674916368973

0.4229808287748649956985651531982558941357377563061876881101624557656023036996416214224698023378797897

13/6:

0.3838516206912247741246536006073385596814142570245296664079873217981765924155504327800953720464073495

11/5:

0.3750745990498321540120568641971008688530051819136208894181145314484824693719462687907868431608940910

9/4:

0.3614023419226268753480678790435491427101097351554286827942005389332983083059601883415850754072002240

7/3:

0.3374121317318208309970155509183090076285539924257871506584142410142507843327346654868727289877813896

12/5:

0.3172916174366979837296400010360354759304354058206386106981075721771724759243346566569734940210491212

5/2:

0.2858711963653834953891006479252360719692328766710792544459513853856266304601511301150912020083634105

13/5:

0.2533366160625841922157200717955879502643798641651834895768285962794055204774087846942600266704927995

8/3:

0.2312249861974898808832518088220141898538565118355301717876639769501605464006793912597849829205535575

11/4:

0.2033074019240613845603322984982753703311378112996081387995001682389668876179597669365343259348615915

14/5:

0.1864883896431757677481902066665125006395327773541410825794896921532710941334197904007326756511506878

17/6:

0.1752738210172123320405363789196919846100316352747221086751558270319698731255075756372281744640712234

0.1196297860080003276264722811766778505468365249870718971038261126757762890096713114279671247503315067

19/6:

0.06571911927293680546460458441083871366134970817202460751242575647400863209469423944978950150292375718

16/5:

0.05525741171994249171889678313037534803808339376372777808200996182071119305710926727709542116872525964

13/4:

0.03980864951956497319292479702418717588343218647217496223386002661965467883385992102955237634535404573

10/3:

0.01478199101356882212149352963777926835564409729017595793009075401263810571912548291095479945332216581

17/5:

-0.004518077930741953583450480495694832860724068520240886082360185836292346167306147179156347554894746493

7/2:

-0.03212854851248111561669442340485852995868667997796086274206818496531288383922100670683297598104806284

18/5:

-0.05797435185980087898774604915004012034681944291824348330590989057020072820664414961043904013149252190

11/3:

-0.07414723134207814403128572428409776645777627399729653839291795284414830378163191326644128720305245959

15/4:

-0.09310297301222044326248329339495120416692294340896456997955860439410241889165780186058462202797423763

19/5:

-0.1037781503568977059602932564075068926453652042947928508646133321986509830476419380313695362618430205

23/6:

-0.1105960078261121827771106900103977906475124754523449144002843965495359073746568546689642890386483052

-0.1409816978869304116391448986940359267129683554616938615858568244712435627860090592938105570169452208

25/6:

-0.1649915094939960198107769331419170521642148050631438483726158427441440362142858369366791597972570215

21/5:

-0.1690131567671567386896276606729614620840664166996508143791788298450158096021830342143787199744510243

17/4:

-0.1745554141316271504239775237186211540930453130869891106590538600941422795752286316870980363019492327

13/3:

-0.1824862415699467114897912095006939119738858586786274523273885667692836248645976729553102606006590059

22/5:

-0.1876602868004406847519059286176014297623388081251598569424395369371432814019814321918178060178399235

9/2:

-0.1934911221017387574161305949079949989553488553962802588686158932986290137495238738154692230166140743

23/5:

-0.1970470797223561954373327012114427787125416662262357723438012526889560255677547900630128550953998343

14/3:

-0.1981843422265760850729919747652605006106432326412085797929757146177988353674397288160614085472891200

19/4:

-0.1982582796123961685308794335942390502402224591794189609455372063871722436115739049762795129980185487

24/5:

-0.1976036133099352363986917796081882921938905340334112962985699542706710012468236462656874244584291112

29/6:

-0.1968834383348622136325311046406702101193865690605653201921654795500129901337629010946361020072017060

-0.1900297496566438786184589001163008064967391561018566289128122162558916359306146468940248636027555073

Definition

The cosine integral is $\operatorname{Ci}(x)=\gamma+\log x+\int_0^x(\cos t-1)/t\,\mathrm{d}t$ [4]. This table holds it at rational $x>0$.

Parameters

$x$

— argument ($x>0$)

Formulas

(1)

For $x>0$, $\operatorname{Si}(x)=\int_0^x\sin(t)/t\,\mathrm{d}t$ and $\operatorname{Ci}(x)=\gamma+\log x+\int_0^x(\cos(t)-1)/t\,\mathrm{d}t$ [4].

(2)

The functions $\operatorname{Si}$ and $\operatorname{Shi}$ are odd [4]. With the principal branch of the logarithm, for $x>0$, $\operatorname{Ci}(-x)=\operatorname{Ci}(x)+i\pi$ and $\operatorname{Chi}(-x)=\operatorname{Chi}(x)+i\pi$ [5].

Comments

(3)

The constant $\gamma$ in the definition is the Euler-Mascheroni constant.

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))
RBF(QQ(1)).Ci()

(P2)

Python

import mpmath

mpmath.mp.dps = 100
mpmath.ci(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 A099282: decimal expansion of $\operatorname{Ci}(1)$

Similar tables

Values of the sine integral $\operatorname{Si}(x)$ — the companion trigonometric integral

Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$ — $\operatorname{Chi}$ is the same integral with $\cosh$ in place of $\cos$

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 $\operatorname{Ci}(x)$ 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 $\operatorname{Ci}(1)$ [7]. The widest returned ball had radius less than $10^{-117}$ for 100 requested digits.