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

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Numbers

$x$

$\operatorname{Si}(x)$

1/6:

0.1664096792540574716599058562927961810870834565763752223843732727409831885570998033913193994699363213

1/5:

0.1995560885262338214004569447641659509866038788055132198626167003976643976469523521694243846567117434

1/4:

0.2491335703197571640954025664870223741307663926517214583611496950623570857846851751656936622594611292

1/3:

0.3312825659301884225135046991300795291449458929061728671816055806988514753290583497706352337143285106

2/5:

0.3964614647513728830203342631351744541442199635092304671195286795950023354716800536552732641566170850

1/2:

0.4931074180430666891616267075727646536413371384287211316602426140302374769327562878716402493512687074

3/5:

0.5881288096080800668996479040068368191964074189278061513244112603158240919708007617995235896858585336

2/3:

0.6504235890459940064064993202167472497745744134235330418961672422612440086989405906725190143007145579

3/4:

0.7269542471500869790123856014645862052001017234670621898329382420137309287856167098133839020000084327

4/5:

0.7720957854819965602538897124798954918992817055288749928957661624285290578585723977589080745050667875

5/6:

0.8018450720846252413114995957716959670914241849119349486204793550875526676874530494958411607251032040

0.9460830703671830149413533138231796578123379547381117904714547735666870365407979180887021330817407112

7/6:

1.081966652079617383478008463762681814304297291090507094226349756139137571318440682454386252809985821

6/5:

1.108047199013718590042783172702787641451604079596307320528573628403908966266730609846348505021423115

5/4:

1.146446415673234426032111651201808436737704236407957467491945059053570292885389334301809647171549238

4/3:

1.208461089440520896431996644238316852096772267979999511255709780944851164078745984004467447597794884

7/5:

1.256226732779217943175975117490768170905499120479816437186588921709549376538070609413805867280661205

3/2:

1.324683531172119680370472846875214042814140454625112248480722201262320219553895386407207332512598140

8/5:

1.389180485870438428069328873025725984142178462200072102317420810216231997004963020331541082786850161

5/3:

1.429915714737521143248762223517670489867417533941209579016562765774258624980976956816244512814813740

7/4:

1.478233418855844811362147280921221245315651888164274330916050859838745455189772825148478850415739826

9/5:

1.505816780255578571630507268103528549027427191705672878607100400756398584250580460991020719984928508

11/6:

1.523613487872797371005273687670774756246481941505290107024408863884724995780404457067473165556651491

1.605412976802694848576720148198588940848583422328499660289006306564529357172666149845344797162787032

13/6:

1.675133309546340480601888540010125517712216618153351896617479247047601368805579228163404722527432996

11/5:

1.687624827241098520373882470966760098309237284586543331295352540319087205145954307936541694365878615

9/4:

1.705457197538423584490398533463756401499361488953180663403352521776588266289548330748874976999748994

7/3:

1.732775875303600009980463948850573534769433053649382029446581154130311055485275883326921419576275752

12/5:

1.752485500761767386061361670606119572811240834086889274985978526176301172855193321342591242178430040

5/2:

1.778520173443826642100311981736229478709573849477781989880603283096823470654203933549809140993681158

13/5:

1.800394450526770159887698672674272289441718561079972880257364096260130960911198451351001664062979723

8/3:

1.812716376484998736607483384903689491015818590751198317806172092040769593963721633993974541725511511

11/4:

1.825637507599783275320626274186462808358773435311143303014665295410546786092418272610238299243072328

14/5:

1.832096589081322326875284208226359485943572879317655551274210047330715056559018341893284282421110651

17/6:

1.835874799086595585202765323748867394079960645605758475747831797761939380180615569454044356059978017

1.848652527999468256397730251111973245164512730309465694979723072435605912718515020049181788126173506

19/6:

1.851837524751437858499832169132898002034233788213939168874110868858701933609025074929683555230431263

16/5:

1.851400897018440279339813882325421857783306739017117928387743602963342132804471363851435926928397374

13/4:

1.850110365396640970674536484006683746295855051042389862344076773175928999664706245650208267704966178

10/3:

1.846330562376076670796985931785464073340836278368846782528668751903377332800681322418672623352981601

17/5:

1.841913983326143035922639385021581582964266161324227699516718176770338881587372403447934051658797747

7/2:

1.833125398665997047939606433285520701503876096619300690232904821698961486403665002006665807724069145

18/5:

1.821948115649503541269811487430347970284366787288597581909608735664286974511169540502638627257392895

11/3:

1.813287620400429575055862351177089833161930081120293370287257095438494212737514112760882698288865691

15/4:

1.801228726765879156794112125472244434657427737387570319294900432546507053574024073920449843892398007

19/5:

1.793390354849570170709716720930992560151727415371812961780746875968950092791004564600728420749590348

23/6:

1.787932615125460934213834229512376895178692926604709235008092992764572067028916690957683387954942075

1.758203138949053058105559303358501617209579460962609792592209216499274979704482197242669622646187422

25/6:

1.725246950284295282089798522800353690177873384636116958429524073174103296040042024533942835914500734

21/5:

1.718368563690868598554360239050667847456743986819695132070869872491974917045484060918129049183136196

17/4:

1.707913488578430771695909632614293882029618302429457190861919716609620759447674076368156936024986513

13/3:

1.690194970605318455510610867163669877006611843460027695840949318289048211729183721724150370834247631

22/5:

1.675833959408374161021180721161152324700252363101894641200394459974540008228848669068834221434117391

9/2:

1.654140414379243983503922486851540015538781833091970145431412011245857766574035878741187645288615622

23/5:

1.632460352500349890247159032195217894905291206969186564358382128671274611962169232865394319195870533

14/3:

1.618119301940070990889472728931588937312515668823190734010864650582034880703673117812434726877837962

19/4:

1.600425000251444619590685811599525746577545690670149686501378440982785267954574547728127766440672286

24/5:

1.589975278172365569408329803677180342157353760546922471340664354165346780162807070482488058976459552

29/6:

1.583092731925181562561139015127723679285910088573536574749247458769587069174447874292254603794906472

1.549931244944674137274408400730639012183184893966372210477969710681487208951511074986007223927691325

Definition

The sine integral is $\operatorname{Si}(x)=\int_0^x\sin(t)/t\,\mathrm{d}t$ [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)

$\operatorname{Si}(\pi)=1.8519\ldots$ is the Wilbraham-Gibbs 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)).Si()

(P2)

Python

import mpmath

mpmath.mp.dps = 100
mpmath.si(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 A099281: decimal expansion of $\operatorname{Si}(1)$

Similar tables

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

Values of the hyperbolic sine integral $\operatorname{Shi}(x)$ — $\operatorname{Shi}$ is the same integral with $\sinh$ in place of $\sin$

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