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

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Numbers

$x$

$\operatorname{Shi}(x)$

1/6:

0.1669240827487467789238295107446416034639690912937507553679530435013039528767987868482095479476647459

1/5:

0.2004449781407463863407308538372225246484223312391440361965253521573040141856172326231272338302778800

1/4:

0.2508696848909122325014531547489845450154666123083457454440935522213347315391021532741256218675127790

1/3:

0.3353978181887154629033148374666816983611059060774386578465238396557942872609578968189141167856538802

2/5:

0.4035726687424936359059793789475525305713983080162796454317695610735188821017548457312154351143947264

1/2:

0.5069967498196671958336598759889438002541262223344977802055555028977463978784682624982607055223812133

3/5:

0.6121303965633807726245627845975414632402891114757806782543869451258620400198599138452008199882286833

2/3:

0.6833487176929966439983498064085088355741723030681719819625554657797084312321034086476181697602901351

3/4:

0.7738368144562259452829890753374816998659737209602090787962923191294607633715686811822297173696871190

4/5:

0.8289965633789344863869104691896009200075971432654372307055271469853555961899135042490057906531569128

5/6:

0.8661613051753628932618824745653410524713993219247109347502478726746574220749349766354100691006724203

1.057250875375728514571842354895877959024053937569807899612103522478041266546472849135555480580638103

7/6:

1.258573780355090604191240055721630195191412238087043436893869884938785291764698280379249172601036185

6/5:

1.300250361022057100577026617508677295885702215009501284386568256233743133051607728431573069272582434

5/4:

1.363730673440692998202437038206321512676667124491398790161564204582032043894774811297301014570580291

4/3:

1.472260373420317072642611337440893783807102362387720623838629664256435517617009390389522862006724256

7/5:

1.561713388361001714688727767521669088196559552049366111581585862922273128873533629196675509757227526

3/2:

1.700652515768215244929660328023204882959168495796886342906740515255041299662885573769718350258885031

8/5:

1.845814141358504497153684727761438989851155504566544913930449200657263843461105877334261052194914104

5/3:

1.946345344886202481523279657986135845014723162661581769315450176823898207298109220995355639581889896

7/4:

2.076571171840798077231537872981947360811245995938109197219789055067459212456169738639081688130461999

9/5:

2.157290343425901181856770108318493838056706998083904351731472953379566925132532269139693087669673461

11/6:

2.212234165173630231424050928371158187325764949056947476730324705204945225489820568868589109801096451

2.501567433354975641473372482727542398916272873691532585427209064141200421006279691081182966430150902

13/6:

2.818006944985371974845244985215117405719311446643414910698726812634913034661788220368221932043174870

11/5:

2.884902918433156575681983244503180536468921796302743142511944672702929656060207343844323641209032152

9/4:

2.987665574078391197080379186606924213424560166389038974368723257186580535617150386940859765792348758

7/3:

3.165696486774419784384827541033611173135050019106881760706003288139804672577209313342029554893514882

12/5:

3.314555268643167492728626069694490169795105500321013225758564908286274394422049500273439045935512452

5/2:

3.549340406224435223709589987362811091471084245200901611624053094858152380539941870683899558908846499

13/5:

3.798982495795840421748475440158705552678700297389245444079021080789600889005534675204896956531241513

8/3:

3.974293769503835707972524565139791604871293138588674599434249687392491295787929865162464772748145142

11/4:

4.204138083120755683852430837439377132789362353296589402402198869239301035014245503532895245936678149

14/5:

4.348076508127190655131966345107352795256081186560051009685553204252714547279948321924248718760565082

17/6:

4.446666035628364572656013046915881769905575671246435345472880541897173146546642128956308582129239274

4.973440475859806797710418382522705142969565328533902860435016548191677696883496353124500238952201689

19/6:

5.562611918737695538001969576551262513717867885526415299756835660588568185440099778378400389404758524

16/5:

5.688717824734039349007333401702685796000158149851930953899855835587171956958324822724950143396367320

13/4:

5.883410150360207017185107841062745297144420417948963412064604668594984235372782882430880279125034106

10/3:

6.223345811511487371374032420113165223011350091887740476710912133769101870362159748313123521265073545

17/5:

6.509983138823005914246514363467364142290447310875123280450308297697269870430590440843393863928643188

7/2:

6.966162067504941842259718547526845125182627306495950258316982421760884871213008629267954636733553620

18/5:

7.456207256934443891436583179891356072101946853118062342208161145366828680070540024343061785842299649

11/3:

7.803147727305252661609590845186146557271868084385440378388912360278405243040529541066450740301301473

15/4:

8.261219600193931186530849253926368110387584321677044382326847599839433349334704221623535073675384125

19/5:

8.549811255988699148352001220568309328094124199116254604824584124830965809091797213580670478764533665

23/6:

8.748203112697635771633116966309580694662014716466737316615646589484083141218682048000479381818215885

9.817326911233034464562297569928152633991898149097820874257390114452073336891755011184962633527426188

25/6:

11.02894651406205278632777199323743822004927283480571473469750003821696305364479093909655945638372556

21/5:

11.29018470498201256801663234049955125656304294610928210934712880307044295309033553222279010913281126

17/4:

11.69470792817651330537874275182661557265534656557968649484520773642910113447206928085536040956658452

13/3:

12.40430096103017491890151106624614303688810139939906065886957143766543362232496411974097811287553712

22/5:

13.00565464082290090331070562191065787056091714897888332840250862661261870933415757370133051425910342

9/2:

13.96788504934714762174679698578105068107257073256027250327600315272689063985471377289273768375231002

23/5:

15.00797015114646903695731822749109683591171245782321713117023193288484042326947210705401509107911438

14/3:

15.74796456384632243728005873017823253212112743678842979261363936735388830621450360978607955440512887

19/4:

16.72919953606678911572880525211126151934504003884037413525274113278677761474750402597030523149284772

24/5:

17.34967143385610297567155312242627091935881201356855158166104521167616358985488532847576131811036906

29/6:

17.77717422114520171258660079042074437578346191157383984831671298957729229736071903604804693237863291

20.09321182569722639044437617788284340874767674733331415047386037284311412658254000478545980377124624

Definition

The hyperbolic sine integral is $\operatorname{Shi}(x)=\int_0^x\sinh(t)/t\,\mathrm{d}t$ [4]. This table holds it at rational $x>0$.

Parameters

$x$

— argument ($x>0$)

Formulas

(1)

For $x>0$, $\operatorname{Shi}(x)=\int_0^x\sinh(t)/t\,\mathrm{d}t$ and $\operatorname{Chi}(x)=\gamma+\log x+\int_0^x(\cosh(t)-1)/t\,\mathrm{d}t$ [4].

(2)

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

(3)

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].

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)).Shi()

(P2)

Python

import mpmath

mpmath.mp.dps = 100
mpmath.shi(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 A099283: decimal expansion of $\operatorname{Shi}(1)$

Similar tables

Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$ — the companion hyperbolic integral; their sum is $\operatorname{Ei}$ (2)

Values of the exponential integral — $\operatorname{Ei}=\operatorname{Shi}+\operatorname{Chi}$

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

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