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

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Numbers

$x$

$\operatorname{Si}(x)$

$1\pi$:

1.851937051982466170361053370157991363345809728981154909804783781876981890166348358532710336502954758

comment: The argument is specified by (4).

$2\pi$:

1.418151576132628450245780162299749429142453349295016271842720229696107098746422014491365975822099711

comment: The argument is specified by (4).

$3\pi$:

1.674761798979961265948438707462519696047617874413992612007099292532840013577240409226193917203643059

comment: The argument is specified by (4).

$4\pi$:

1.492161225584460055488343913762587569146229260210940642292980723606750032698574567973502743885528682

comment: The argument is specified by (4).

$5\pi$:

1.633964846102835213309961782908481827771852573714004017025186451888936522091188789407562294464365351

comment: The argument is specified by (4).

$6\pi$:

1.518033961467180605477875888801471065781094877711235316530227918441945164648895287819897816943237672

comment: The argument is specified by (4).

$7\pi$:

1.616085517451643988592416710549903262825260099824897695373647623122853305224166899379998676962460509

comment: The argument is specified by (4).

$8\pi$:

1.531131284990665745904025604867600571529753282019257185691222067479716953735516784302856369694362854

comment: The argument is specified by (4).

$9\pi$:

1.606076891817979873675037375973869236651052572086139921349920353704638863277099743549751472102391213

comment: The argument is specified by (4).

$10\pi$:

1.539029079577564460350171633023995496979320588140006975591435424875253410536566637801019907102081789

comment: The argument is specified by (4).

$11\pi$:

1.599685602332216993004012783260176952234338954126906428362489552616917070986082961128183570163233702

comment: The argument is specified by (4).

$12\pi$:

1.544307522339320502658562724818195410649567255187603188438109505648659422126423931945315928934898200

comment: The argument is specified by (4).

$13\pi$:

1.595252550589564585049113364131838318647842907393123185042206811265783394800260499806319299890175073

comment: The argument is specified by (4).

$14\pi$:

1.548083269679785785928161519155941139410330541657108865059983890061979792194413888035857924008882036

comment: The argument is specified by (4).

$15\pi$:

1.591997975769490242043067109395700438633871333154532594092573389855594754033261516594886688129344344

comment: The argument is specified by (4).

$16\pi$:

1.550917632803829298070867058086181747890355043938936995850303686953711691431751701419924609950948012

comment: The argument is specified by (4).

$17\pi$:

1.589507363352956638819447501720695560896838545259595282747152163578340971049270379779838562819692684

comment: The argument is specified by (4).

$18\pi$:

1.553123463303973272495930062989848044359641951457843686084849001890324984684601082030958858412770984

comment: The argument is specified by (4).

$19\pi$:

1.587540105959460521882208005064919854052889256357608495524223557068437909390957398668207131143078367

comment: The argument is specified by (4).

$20\pi$:

1.554888871044744746881454206584365834196949516967622880974057729362381371338029299923284861440132611

comment: The argument is specified by (4).

1/4:

0.2491335703197571640954025664870223741307663926517214583611496950623570857846851751656936622594611292

comment: A near-origin or moderate-argument reference; see (5).

1/2:

0.4931074180430666891616267075727646536413371384287211316602426140302374769327562878716402493512687074

comment: A near-origin or moderate-argument reference; see (5).

0.9460830703671830149413533138231796578123379547381117904714547735666870365407979180887021330817407112

comment: A near-origin or moderate-argument reference; see (5).

1.605412976802694848576720148198588940848583422328499660289006306564529357172666149845344797162787032

comment: A near-origin or moderate-argument reference; see (5).

1.758203138949053058105559303358501617209579460962609792592209216499274979704482197242669622646187422

comment: A near-origin or moderate-argument reference; see (5).

1.574186821706942052082971451206658507176789221924863859521719101132446140774178978380897708626580123

comment: A near-origin or moderate-argument reference; see (5).

$1\pi/2$:

1.370762168154488480069678288381575329990861335564368679635232143130035361067956225720065240813900561

comment: A complementary oscillation-phase reference; see (6).

$3\pi/2$:

1.608372753954010735832722620413597198898821524844609350477303824099711044680408997330423219195174563

comment: A complementary oscillation-phase reference; see (6).

$5\pi/2$:

1.555830987211027456993804561674194182175499388012615642293986748418933084633306941486541743648757784

comment: A complementary oscillation-phase reference; see (6).

$7\pi/2$:

1.578709224085772851390395387196504729518699173455683829291905126207218646978199556724010435094911075

comment: A complementary oscillation-phase reference; see (6).

Definition

The sine integral is $\operatorname{Si}(x)=\int_0^x\sin(t)/t\,\mathrm{d}t$ [4]. This table gives its values at intrinsic and bounded reference arguments.

Parameters

$x$

— argument ($x\geq0$)

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.

(4)

Differentiation gives $\operatorname{Si}^{\prime}(x)=\sin(x)/x$ [8]; hence its positive extrema occur at $x=n\pi$, $n\geq1$. Their heights include the Gibbs-phenomenon value at $\pi$.

(5)

The exploratory reference set $x=1/4,1/2,1,2,4,8$ provides sparse positive inputs near the origin and into the oscillatory regime. It complements the extrema without claiming a special use for every input. The cosine integral uses its real branch on $x>0$ [9].

(6)

The additional arguments $(2n-1)\pi/2$, $1\leq n\leq4$, sample the complementary phase positions between successive stationary points. They are selected from the phase of the defining sine or cosine integrand, with a short noncanonical cutoff, rather than from a decimal mesh [9].

(7)

The current generate.py reproduces the complete attached selection, including the retained entries. Their original written values and entry comments are preserved rather than reassigned to a new source. The selection manifest records the predecessor revision and source hashes; earlier generators and checks remain historical provenance, not the enumeration of this extension.

Programs

(P1)

Python

from generate import generator
from numberdb._write import to_text

for params, result in generator.all_entries():
    print(params["x"], to_text(result["number"], generator.digits_for(params)))

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 $\operatorname{Si}(1)$

[8]

DLMF: definitions and natural argument families

[9]

DLMF 6.2(ii): sine and cosine integrals

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 the origin and the twenty extremal heights at $x=n\pi$, $1\leq n\leq20$; the selection is not exhaustive over the real domain. In addition, it includes $x=1/4,1/2,1,2,4,8$ and the complementary phase arguments $(2n-1)\pi/2$ for $1\leq n\leq4$. The exploratory families are identification references with practical finite bounds, not claims of documented applications for every input. All prior entries are retained.)

How they were obtained:

All retained values and metadata are preserved, including their precision. New nonexact values have 100 significant digits. balanced_values.py reproduces the complete selection in balanced-selection.json with Arb balls, including uncertainty in exact irrational arguments and certified roots. A 640-bit guard is supplemented by measured accuracy checks and precision escalation.

Every written interval contains a freshly recomputed certified enclosure; every value is also checked with independent mpmath computations at 50 and 100 decimal guard digits. Exact origin anchors stay exact. Structural identities, root endpoint signs where applicable, and selected transformed definite integrals supplement these checks. See balanced-checks-evidence.json. Earlier checks-evidence.json and source attachments describe earlier selections and remain for provenance.