Values of the Fresnel sine integral $S(x)$

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Numbers

$x$

$S(x)$

$\sqrt{2}$:

0.7139722140219396136290864445540744192342533790480201933811346920912513446963638290807381025179679407

comment: The argument is specified by (9).

0.3434156783636982421953008159580684568865418122025247675792689204494767226635844744099255427628399726

comment: The argument is specified by (9).

$\sqrt{6}$:

0.6289396585401117728184466575898857835874301338936196998637469273323838305080700340663560956522276463

comment: The argument is specified by (9).

$\sqrt{8}$:

0.3879689926370840423209953257839401150018733339393444152152127756611889373032020868579735141144298043

comment: The argument is specified by (9).

$\sqrt{10}$:

0.6003623872514163464779588105343786203600684863438193020577173987264135270737114835527168827742940624

comment: The argument is specified by (9).

$\sqrt{12}$:

0.4083013319319968570379499122099649942978256588378485617160563073987127479917838335555500988093646671

comment: The argument is specified by (9).

$\sqrt{14}$:

0.5849422386136541946868267916839699121961824205281904499357795383487733235914215476562012442766841036

comment: The argument is specified by (9).

0.4205157542469284244453431407430670454544259302864021845554070415663643292186552030928116047704261302

comment: The argument is specified by (9).

$\sqrt{18}$:

0.5749567208876167799931603659147529666048873790377506055172862103995629335255432934215582284021834083

comment: The argument is specified by (9).

$\sqrt{20}$:

0.4288773650821645290696683451403967023839926508586979752457802425779177214065409837210050940399018833

comment: The argument is specified by (9).

$\sqrt{22}$:

0.5678215816628569341150982648203626622941759718028589348540872907775714240624968279649289143067358422

comment: The argument is specified by (9).

$\sqrt{24}$:

0.4350593471289110965049178989093155295073294649502747174279547759943820670655513323157249296453731065

comment: The argument is specified by (9).

$\sqrt{26}$:

0.5623977800501405327456753471952112038685322742519742929638418685727195406822260146952488455276330837

comment: The argument is specified by (9).

$\sqrt{28}$:

0.4398683042084832856073014250571730763360476471780328030302272835044351972717823267421940526678955363

comment: The argument is specified by (9).

$\sqrt{30}$:

0.5580956171490010672785171184348441602130753347160347476803279836081552584665452675365664513631747049

comment: The argument is specified by (9).

$\sqrt{32}$:

0.4437468760363048524201111322404436987723092003962895510148114661587833289900239189764184624205075748

comment: The argument is specified by (9).

$\sqrt{34}$:

0.5545753847289788636977468556487570590149290201425765148013079619708216148319234519267217496410269646

comment: The argument is specified by (9).

0.4469607612369302776239202878407789191062455470831531284957966923293313013942181764645977320961782948

comment: The argument is specified by (9).

$\sqrt{38}$:

0.5516258390307205989322703581893023061982391186240658964944412567888346734474818098199650297421255342

comment: The argument is specified by (9).

$\sqrt{40}$:

0.4496803282386250622499088906389720100133757472398374737225140262248550140297305944553645525759220908

comment: The argument is specified by (9).

0.4382591473903547660767566966251526374937865724524165673344073262658059382112998771734606741411548457

comment: The other coordinate at an axis-tangent point of Cornu's spiral; see (10).

$\sqrt{3}$:

0.5173051218636263692865268341903558004908133781956532570786082207998195831074112372552057974477605276

comment: The other coordinate at an axis-tangent point of Cornu's spiral; see (10).

$\sqrt{5}$:

0.4913925389676197244898030041321475744945508032451102284731633810662074267559242146613869308212418187

comment: The other coordinate at an axis-tangent point of Cornu's spiral; see (10).

$\sqrt{7}$:

0.5053187400454302931243238882155469598340715189174239330898240136406119544839547009484947263853019955

comment: The other coordinate at an axis-tangent point of Cornu's spiral; see (10).

0.4963129989673750360976122652991121038564670345832411060867864797255800015977318607582479312692337430

comment: The other coordinate at an axis-tangent point of Cornu's spiral; see (10).

$\sqrt{11}$:

0.5027439987156609692253284278140212642193522490029064823362984686378981537916294828492829177479882176

comment: The other coordinate at an axis-tangent point of Cornu's spiral; see (10).

$\sqrt{13}$:

0.4978571112962106184277526444403698652816675246600007998938489752127164164510747353592429112831382469

comment: The other coordinate at an axis-tangent point of Cornu's spiral; see (10).

$\sqrt{15}$:

0.5017326018844271236175481903437571521237023306669420539000452089034640536155130360410471239129964652

comment: The other coordinate at an axis-tangent point of Cornu's spiral; see (10).

$\sqrt{1/2}$:

0.1771219699791394139583863788921418849485832604950947617630038904394809417932690012063221991163564907

comment: A diagonal-tangent point of Cornu's spiral; see (10).

$1\sqrt{2/\pi}$:

0.2475582876516108426099050144052166355678178520505607433029455487383778682978538406490113083526747232

comment: A unit-quadratic-phase reference; see (11).

$\sqrt{3/2}$:

0.6416988905685876555386160768886808197986603362322564210426885087684141849303626152728441350292573586

comment: A diagonal-tangent point of Cornu's spiral; see (10).

$2\sqrt{2/\pi}$:

0.6421187357445146953318002859307074484957773008887281670851493375808421115681758776309379344297222532

comment: A unit-quadratic-phase reference; see (11).

$\sqrt{5/2}$:

0.6528316466610344384981099211184946284077567439713134421345108170156171348770379077471801522012700249

comment: A diagonal-tangent point of Cornu's spiral; see (10).

$3\sqrt{2/\pi}$:

0.6172135970241896124400929011078847161701404510433965710768269050712555055410471690686542627762873057

comment: A unit-quadratic-phase reference; see (11).

$\sqrt{7/2}$:

0.3920993952502802379065070005989045767225246071522609779108043953645512623032170588527719323598715926

comment: A diagonal-tangent point of Cornu's spiral; see (10).

$4\sqrt{2/\pi}$:

0.5961265594980171973648629640371573960022357548787927199570128930780753240127125371453396930460013208

comment: A unit-quadratic-phase reference; see (11).

1/2:

0.06473243285999927761148051223061476765072591849351249278758894564828689767392197589943934122214205058

comment: A bounded arc-length reference; see (12).

Definition

The Fresnel sine integral is $S(x)=\int_0^x\sin(\pi t^2/2)\,\mathrm{d}t$ [4], in the DLMF normalisation. This table gives its values at intrinsic and bounded reference arguments.

Parameters

$x$

— argument ($x\geq0$)

Formulas

(1)

The Fresnel integrals in this table are $S(x)=\int_0^x\sin(\pi t^2/2)\,\mathrm{d}t$ and $C(x)=\int_0^x\cos(\pi t^2/2)\,\mathrm{d}t$ [4].

(2)

With $S$ and $C$ as in (1), $C(x)+iS(x)=\frac{1+i}{2}\operatorname{erf}((\sqrt{\pi}/2)(1-i)x)$ [3].

(3)

If $S_2(x)=\int_0^x\sin(t^2)\,\mathrm{d}t$ and $C_2(x)=\int_0^x\cos(t^2)\,\mathrm{d}t$, then $S_2(x)=\sqrt{\pi/2}\,S(\sqrt{2/\pi}\,x)$ and $C_2(x)=\sqrt{\pi/2}\,C(\sqrt{2/\pi}\,x)$.

(4)

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

(5)

$\int_0^n\sin(t^2)\,dt=\sqrt{\pi/2}\,S(n\sqrt{2/\pi})$.

Comments

(6)

The phase is $\pi t^2/2$, as in (1). (3) converts the alternative integrals with $\sin(t^2)$ and $\cos(t^2)$; their arguments must also be rescaled.

(7)

Negative arguments are not listed separately. Values at negative arguments follow from (4).

(8)

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

(9)

From the defining integral, $S^{\prime}(x)=\sin(\pi x^2/2)$, so the positive extrema occur at $x=\sqrt{2n}$, $n\geq1$. These are the horizontal or vertical tangent points of Cornu's spiral [8].

(10)

Cornu's spiral has coordinates $(C(x),S(x))$, arc length $x$, and tangent angle $\pi x^2/2$ [9]. The reference set includes both coordinates at the axis-tangent arguments $\sqrt{2n}$ and $\sqrt{2n-1}$ through $n=8$, retaining the longer existing extrema prefix. The four diagonal-tangent arguments $\sqrt{(2n-1)/2}$, $1\leq n\leq4$, have tangent slope $\pm1$.

(11)

The inputs $x=n\sqrt{2/\pi}$, $1\leq n\leq4$, correspond to the unit-phase integrals with $\sin(t^2)$ or $\cos(t^2)$ and simple integer upper limits. This bounded exploratory family makes the normalization change explicit [10].

(12)

The additional arc lengths $x=1/2,1,2,3,4$, where not already included as tangent points, are a short exploratory reference for coordinates of Cornu's spiral. They are not an interpolation mesh.

(13)

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: Error function

[2]

Wikipedia: Dawson function

[3]

Wikipedia: Fresnel integral

[4]

DLMF 7.2: Definitions

[5]

OEIS A099289: decimal expansion of the Fresnel sine integral at 1

[6]

MathWorld: Probability Integral

[7]

DLMF 7.3: Graphics

[8]

DLMF: definitions and natural argument families

[9]

DLMF 7.20(ii): Cornu's spiral

[10]

DLMF 7.2(iii): Fresnel normalization

Similar tables

Values of the error function $\operatorname{erf}(x)$ — $C(x)+iS(x)$ is an error function of a complex argument (2)

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

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

Values of Dawson's integral $F(x)$ — another member of the error-function family

Values of the Fresnel cosine integral $C(x)$ — the companion Fresnel integral; together they trace the Cornu spiral

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 the origin and the twenty extremal heights at $x=\sqrt{2n}$, $1\leq n\leq20$; the selection is not exhaustive over the real domain. In addition, it includes the paired axis-tangent coordinates through $n=8$, four diagonal-tangent arguments, the four unit-quadratic references and simple arc lengths $1/2,1,2,3,4$, as described in (10), (11) and (12). 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.