Values of the Fresnel cosine integral $C(x)$

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Numbers

$x$

$C(x)$

0.7798934003768228294742064136526901366306257081363209601031335831780717609791088901087787073052785282

comment: The argument is specified by (9).

$\sqrt{3}$:

0.3210561864106780695711492519171907716293812522996196132842750762216990735320068739850774503808286338

comment: The argument is specified by (9).

$\sqrt{5}$:

0.6408068404452539449387670340885006660177122693723308229559796340905947187761399908107259016327543861

comment: The argument is specified by (9).

$\sqrt{7}$:

0.3803906937680257451035588977632938813772351339087800676994345687499072937408533364280265396350848459

comment: The argument is specified by (9).

0.6057207892976856295561610742871546971452241199383242385818462301649969705618914967816623916536859336

comment: The argument is specified by (9).

$\sqrt{11}$:

0.4042604972448340779400038126726447548156404215231559553035908060459098365012747549669279739971657592

comment: The argument is specified by (9).

$\sqrt{13}$:

0.5881276455282872548646524186250949450738871230291278892437900235246723574684006773325911921468084086

comment: The argument is specified by (9).

$\sqrt{15}$:

0.4179220936249201378284074258490084379661098672609348201685381174276825144262370111424256314158458830

comment: The argument is specified by (9).

$\sqrt{17}$:

0.5771212523718251084740861852596818168962387269255203493649043764189391410558947581961932005675837707

comment: The argument is specified by (9).

$\sqrt{19}$:

0.4270356033980088055932374181660509270695730411676597064179221173923750555440937922442829066109424313

comment: The argument is specified by (9).

$\sqrt{21}$:

0.5694134117957716337797564849753450163255333018391686653159065150622861348396013410183539703990334806

comment: The argument is specified by (9).

$\sqrt{23}$:

0.4336656870944328144943376219021306701444712568373861617383444500322118261770097690401696294329194234

comment: The argument is specified by (9).

0.5636311887040122311021074044130139641207537623099921078616593412498859935331646182126086015708039804

comment: The argument is specified by (9).

$\sqrt{27}$:

0.4387666536862618884704143093441028206788304624221276256833151276375718284109515681055274809890590196

comment: The argument is specified by (9).

$\sqrt{29}$:

0.5590873872857030846153188143420749372326999429851543935638780557459802347888707036795777760097121916

comment: The argument is specified by (9).

$\sqrt{31}$:

0.4428478735796019249207394400063621720815766851985477627300621355516295226680562182282684903053117154

comment: The argument is specified by (9).

$\sqrt{33}$:

0.5553952224507595341698130967698284950006714479468794057988204997807302572784322283739065705496595753

comment: The argument is specified by (9).

$\sqrt{35}$:

0.4462091213596228693474148195351134201532962204434839918355272509430023108055108602483919943679147763

comment: The argument is specified by (9).

$\sqrt{37}$:

0.5523182338562316027242533644802253660769846969368950882698527012535737982339124596715620634572970172

comment: The argument is specified by (9).

$\sqrt{39}$:

0.4490397881441164040677890845348449408279643167677654547996577837464688235114934933474097077103523832

comment: The argument is specified by (9).

$\sqrt{2}$:

0.5288915951112465925568670379223205183661948137201804434346521640933897729889879694687112508190553661

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

0.4882534060753407545002235033572610376883671545092153829475964426552750821354195848976660360266022599

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

$\sqrt{6}$:

0.5066415640626164768562060577083338267447818625568330346873869256708312064182908691057964275464882563

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

$\sqrt{8}$:

0.4956196980956746332815740528168459429258915480604570457032555596174794882007245938443022932720795872

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

$\sqrt{10}$:

0.5031581047232053626001088552226006006804106408905129160679762467461640264393283885269251211679602222

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

$\sqrt{12}$:

0.4975872742888735682098619879159253930134103201906127776078039175417850902102777393713515516891710053

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

$\sqrt{14}$:

0.5019196883695551822957099235995135550299109378810951916448616868391966477082837080643474907586855463

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

0.4984260330381776155307095868245097300350135211824269469752137903190608739367562440331957986548427861

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

$\sqrt{1/2}$:

0.6647169317774971062421108635871436685418032702125249400532885991000729886650559725134181751678952585

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

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

0.7217059242926050877715858156119067071717932981792008804439748635733424719649134799372303728068514873

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

$\sqrt{3/2}$:

0.6987932112819265361584551348467379235674862815426546730722502512028371207840897581798178307691106353

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

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

0.3681929762809747963106624017460055313715683095106596109382454766741134981461776062195709031405430141

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

$\sqrt{5/2}$:

0.3781520881155292484301156673523986720651224231088983094500858183493144087541099301334846082838567699

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

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

0.5608039810639548648690870450175753068019808627040126373638455097657307372343216104844510926866821862

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

$\sqrt{7/2}$:

0.3721723365491987891415795946529392440428777120470093329340724850477908026871490549059716233536203720

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

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

0.4743107173203279931730365277474657820253877052046737013509757099870111701511661273305742096418884617

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

1/2:

0.4923442258714463928788436651566816377660951457715012532946526193191076795983945814165027047148093055

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

Definition

The Fresnel cosine integral is $C(x)=\int_0^x\cos(\pi t^2/2)\,\mathrm{d}t$ [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)

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

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, $C(0)=0$, and is stored as the integer rather than as a hundred places of one.

(9)

From the defining integral, $C^{\prime}(x)=\cos(\pi x^2/2)$, so the positive extrema occur at $x=\sqrt{2n-1}$, $n\geq1$. 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 A099290: decimal expansion of the Fresnel cosine 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 sine integral $S(x)$ — 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 $x=\sqrt{2n-1}$, $1\leq n\leq20$; 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.