Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$

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Numbers

$x$

$\operatorname{Chi}(x)$

1/6:

-1.207591317367339626249574479683833899002242802359144661817307175422061665637234023727338507700237520

1/5:

-1.022205566043146701994041723737200249411163668070784823364855556145125213734332356044234526856755716

1/4:

-0.7934129495528259620349850064832977068760617624364569731910469157066336165931008451130284146289795323

1/3:

-0.4934899271598711732168921436967939842838951588319580753807497516824912048695693476572174430324480676

2/5:

-0.2988074501231688426770496150640922679198149225046883072831717678050094580874435699952858729356900906

1/2:

-0.05277684495649361591313606332614143497272066798185573504273771620959350095180600636208104695094389929

3/5:

0.1577508933739786644685745456603097846473084989931964282738256563331130121185055902566014996136515139

2/3:

0.2849397246227166602117959380703929315295341631425884556762278146514652188822926904399016505366666771

3/4:

0.4334960015449959374764533066578370746761638594815490842917916551251090244801046419050718482382875735

4/5:

0.5183999848333914517320859141139820146228981824930815448139561241820140059430923255011999720679508812

5/6:

0.5736069370259858137286427134468897942862663530693676785020653257558818620944194108851692923340777707

0.8378669409802082408946785794357563099930066441628996920341249094044725679873313044133452295455008535

7/6:

1.091536763717242437966406357700062799568180233483625895421794114851595679123265668664140942551911402

6/5:

1.141841924170594539152070646797815557351544238168922321007993835488560359092962434849688677766070549

5/4:

1.217317300914782821509178535915928289101897008411005936163808948748848738386845497100675202863229522

4/3:

1.343596069152569651378138705320672740035513881103548546822214123021892627430221114175878664443728375

7/5:

1.445494075789643825413848411900525349925697669683707087098009430793318098056789030123132253496884708

3/2:

1.600632933361582593027750988111537904697438489656535757856173454890321383238712135027520056702799358

8/5:

1.759505807660964711569168486508673497463609246906509303275046923605514482242629473523141722934477838

5/3:

1.868011915285728161431194528114765440242260540161284620921051211969035779924884539780231479698229334

7/4:

2.007082487236159284593163752309333774714340815121914949742965283879490094065700230646083638491499276

9/5:

2.092577214062032322293195597831661856029758907896975266013888781937444126515625120997549366299740499

11/6:

2.150504285575902580306746381682569343125871792376542660574478022674290975867092413236739548576887896

2.452666922646914521906132647499492876601780688728509469099886190429507492292127499801914005422699261

13/6:

2.779095540210184097551161046079246232056225666820689831593851187293638485680120223206231197805203458

11/5:

2.847711781381224001298380610031392003448092884419848833698166637000901362150305136809208514329560447

9/4:

2.952903500958946600004200588506038764783377598498962063755718041902102810979618579447792879208974737

7/3:

3.134613096123788234783267359330398783188439130318531421767208974799856680820313485883576314486312217

12/5:

3.286115007707371821002092355137102162013663172740255557842008131420361753869241168273825155481501760

5/2:

3.524425488354165488213961975088201455525235860489474234612118550361586258020156385992437958479384808

13/5:

3.777132273991758228137893089999195456508876805721357315964356111419086764960977037463133409113065180

8/3:

3.954262936292527344633086601978395893157352715472555704753812130914491497223927689678239949011540654

11/4:

4.186159144905301277258986445893916094285031959911557423683633333493779238044508583929365400025214997

14/5:

4.331221215681974604615395406209173067443016129289709312443461139817126576283295966184554518907856353

17/6:

4.430518558492103907183793527828064699646871889992072608767249121365779892191071631132876042635310145

4.960392094765609760297917636694060120021087694453679259258017517050818531449201090062687159949886161

19/6:

5.552044886758841440633628561414454132133669476801303429577774326218742646714678298966479019385278749

16/5:

5.678584832234690239069783170591909170948276633337670139237058438810257183593730711504297495341843545

13/4:

5.873893642944718238987132196781902041888304300670991095624879122398975803597284397742647491479232543

10/3:

6.214771357307947840068979013591753482292750260879979613456693028534741381744079853571720724726318026

17/5:

6.502092165317440072555404432182484499891685237727731125063868710782672909238277022974164822935568101

7/2:

6.959191927647393449340445622960704611475079937427877992895968469344625281272301982868152254310261562

18/5:

7.450046842602646582011639867975907858127161657459471987239324521905858050575423849189705150664133480

11/3:

7.797472382235368636434293855330910459067520133260454678223024922427652710981468654144799340526132207

15/4:

8.256095497995344376743436136051192077871930102840256294397834991848625053088727613617968541198168390

19/5:

8.544991009172201507742302046860416586457636982697410799928141092120605793720632814099133314481180351

23/6:

8.743575029267397011094860675041033646455779407620288160344597134048561659006630892617625028830565920

9.813547558823185558083422709795686219140181602055331293896656094138488730719392606485884661630856802

25/6:

11.02585624812285647008775212864245406893569838412540757248048003421486264984418098859923319375401999

21/5:

11.28721594280007062022548554177546368004454148142985295617646035087271670228517616141504952002201544

17/4:

11.69191227546423934692607178913353713228946955132319333437036063101491119562590784627455726570959602

13/3:

12.40177110128044834832839669747739175675884088432405053082575609060884812128620102965229535327523367

22/5:

13.00331863078224507431660181784252919286507971450363415363819682082821225875443616873321869942475366

9/2:

13.96581164859243300731391139191147089258363875999012820289012910863888455983028114190694883882680629

23/5:

15.00612914532922753388600539777011223350371157051086224963828639587856429288907555107990126079497941

14/3:

15.74626348670481580034449051324214977409909754069192387203667699227971305926017656649473243303623365

19/4:

16.72765817136152522047829980986865639995777435998892823877028855068807507305822198192994300818694053

24/5:

17.34821843991922465145135943758862544640342092919606100030575037656454939521359195435651698834252034

29/6:

17.77577724301391893528101267530386944668624880327520336359938352493802270762238335320774224805906184

20.09206353010595106464704561591302368667141065186261628370901487879953068734568057387233989814625291

Definition

The hyperbolic cosine integral is $\operatorname{Chi}(x)=\gamma+\log x+\int_0^x(\cosh t-1)/t\,\mathrm{d}t$ [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].

Comments

(4)

The constant $\gamma$ in the definition is the Euler-Mascheroni 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)).Chi()

(P2)

Python

import mpmath

mpmath.mp.dps = 100
mpmath.chi(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 A099284: decimal expansion of $\operatorname{Chi}(1)$

Similar tables

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

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

Values of the cosine integral $\operatorname{Ci}(x)$ — the same integral with $\cos$ in place of $\cosh$

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