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Values of the incomplete beta function $B(x;a,b)$
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Numbers
$x$
$a$
$b$ 
$B(x;a,b)$
1/2
1/2
1/2:
1.570796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534
1/2
1/2
3/2:
1.285398163397448309615660845819875721049292349843776455243736148076954101571552249657008706335529267
1/2
1
1:
1/2
1/2
1
2:
3/8
1/2
3/2
3/2:
0.1963495408493620774039152114549689302623230874609441138109340370192385253928880624142521765838823167
1/2
3/2
1/2:
0.2853981633974483096156608458198757210492923498437764552437361480769541015715522496570087063355292670
1/2
3/2
5/2:
0.1398414370913477053686242723941511317978282103971387235721336851762859293631106978737927549586078250
1/2
2
2:
1/12
1/2
2
1:
1/8
1/2
2
3:
11/192
1/2
5/2
5/2:
0.03681553890925538951323410214780667442418557889892702133955013194110722351116651170267228310947793439
1/2
5/2
3/2:
0.05650810375801437203529093906081779846449487706380539023880035184295259602977736454045942162527449171
1/2
5/2
7/2:
0.02465776945462769475661705107390333721209278944946351066977506597055361175558325585133614155473896720
1/2
3
3:
1/60
1/2
3
2:
5/192
1/2
3
4:
7/640
1/2
7/2
7/2:
0.007669903939428206148590437947459723838371995603943129445739610821064004898159689938056725647807902998
1/2
7/2
5/2:
0.01215776945462769475661705107390333721209278944946351066977506597055361175558325585133614155473896720
1/2
7/2
9/2:
0.004951023398285531645723790402301290490614569230542993294298376839103431020508416397599791395332522928
1/2
4
4:
1/280
1/2
4
3:
11/1920
1/2
4
5:
163/71680
1/2
9/2
9/2:
0.001677791486749920095004158301006814589643874038362559566255539867107751071472432173949908735457978781
1/2
9/2
7/2:
0.002718880541142674502866647545158433347757426373400136151441233981960573877651273540456934252475380070
1/2
9/2
11/2:
0.001055909632263848936390968039392296183710825908070168672016658822442764424625104975863843256617878279
1/2
5
5:
1/1260
1/2
5
4:
93/71680
1/2
5
6:
319/645120
1/2
11/2
11/2:
0.0003775030845187320213759356177265332826698716586315759024074964700992439910812972391387294654780452257
1/2
11/2
9/2:
0.0006218818544860711586131902616145184059330481302923908942388810446649866468473271980860654788401005015
1/2
11/2
13/2:
0.0002331407468048205561425132634087211867894812838612424966582936895950765409951940741148192781935680674
1/2
6
6:
1/5544
1/2
6
5:
193/645120
1/2
6
7:
1255/11354112
1/2
13/2
13/2:
0.00008651112353554275489865191239566387727851225510306947763505127439774341462279728396929216917205203089
1/2
13/2
11/2:
0.0001443623377139114652334223543178120958803903747703334057492027805041674500861031650239101872844771583
1/2
7
7:
1/24024
1/2
7
6:
793/11354112
1/2
15/2
15/2:
0.00002008293939217956810147276537756482865394034493464112873670833155661900696600651235001425355779779288
1/2
8
8:
1/102960
1/2
17/2
17/2:
0.000004706938920042086273782679385366756715767268344056514547666015208582579757657776332034590677608857707
1/2
9
9:
1/437580
1/2
19/2
19/2:
0.000001111360578343270370198688188211595335667271692346677045976698035359775776113641633952611687768758070
1/2
10
10:
1/1847560
2/3
1
1/2:
0.8452994616207484709817024389960850887047964974597462479627953470320446553941333086125692088285009495
2/3
2
1:
2/9
1/3
1/2
1:
1.154700538379251529018297561003914911295203502540253752037204652967955344605866691387430791171499050
1/3
1
2:
5/18
3/4
3/2
1/2:
0.6141848493043784227723528757166995363300218196724401166443631192396222145348696569390583950913935454
3/4
1/2
1/2:
2.094395102393195492308428922186335256131446266250070547316629728205210937524139332418689883561411379
1/4
1/2
3/2:
0.9566114774905181964589688159230519057685628800151127938431091769142859886082348423749590175796649886
1/4
1/2
1/2:
1.047197551196597746154214461093167628065723133125035273658314864102605468762069666209344941780705689
4/5
2
1/2:
0.4985346217334118466739084970069902054355024790783637296055316950467388544285528375519519542052546173
1/5
1/2
2:
0.8347987115999214866594248363263431278978308542549696037278016382865944789047804957813813791280787160
5/6
5/2
1/2:
0.4279054980082395108721772795415983391138720768929283068404281504149630391241286894049266727214085149
1/6
1/2
5/2:
0.7501917470879329535513139891882152424600664478727363760251760717004681132331996850805863867818853856
6/7
3
1/2:
0.3796455293328876523012325273348781479593962548885999142911380296418604571747990424318979793333352257
1/7
1/2
3:
0.6870211373337790143654341393317885187072704117780667523755286370248062094918676242347686873333314410
7/8
7/2
1/2:
0.3442171037944167382594801624912357610011617795961027815749345604938260956681770866690140384414268515
1/8
1/2
7/2:
0.6375306004523936487600958947836088903104536577086177874797356246023665312962632254022468444779847322
8/9
4
1/2:
0.3168854921941341694428114181200600953687373440459860212946632699719119472205891958978378731465151218
1/9
1/2
4:
0.5974002220915801162714742961656541903455483702397282644196224443138023384936965183878764125677705925
9/10
9/2
1/2:
0.2950219067802953891257359016380961610581879553835372682330222388734424812625826316174424165814781341
1/10
1/2
9/2:
0.5640073344356636995163931484773929088394755522580932296898141730857260673313026414449108559730070016
10/11
5
1/2:
0.2770433075216740552530897759340963139723600709090827165238049528899245762902569217387429872766093608
1/11
1/2
5:
0.5356551051767386431596086367643163844403383417893299818888934598084881221224414909596697111360890519
11/12
11/2
1/2:
0.2619357344791756396384871376548889262755177858972712878311154758052904411658207669050659143448177433
1/12
1/2
11/2:
0.5111905826151875401394290074490512366323793709801961602994372949579612525686759788510520309542188789
12/13
6
1/2:
0.2490174858185212835842310418132394808898856937163699723342022073411768021905078240519050749429009993
1/13
1/2
6:
0.4897992529982175331545856970034993358489310451003688444046145314755620145483089147648337417959157395
3/8
3/2
5/2:
0.1050755162724852061707012741172275270851235860288625848028833473741545114081984413309188663764715359
5/8
5/2
3/2:
0.09127402457687687123321393733774140317719950143208152900805068964508401398468962108333331020741078083
2/5
2
3:
82/1875
3/5
3
2:
99/2500
5/12
5/2
7/2:
0.01909644412995479537411362849676643861669635049434285899926184004070185954093003784265745451587284774
7/12
7/2
5/2:
0.01771909477930059413912047365104023580748922840458416234028829190040536397023647386001482859360508665
3/7
3
4:
20187/2352980
4/7
4
3:
14272/1764735
7/16
7/2
9/2:
0.003927234957069855264850413618279062436131271784518118620000341784779185531897136015600014632842971923
9/16
9/2
7/2:
0.003742668982358350883740024329180661402240723819425010825739269036284819366262553922456711014964931075
4/9
4
5:
2744512/1506635235
5/9
5
4:
4218125/2410616376
9/20
9/2
11/2:
0.0008532626038482854330892487239328882272245052372340075574893432023762760837930231366083385449874003634
11/20
11/2
9/2:
0.0008245288829016346619149095770739263624193688011285520087661966647314749876794090373415701904705784174
5/11
5
6:
2632173125/6536230999452
6/11
6
5:
354904416/907809861035
11/24
11/2
13/2:
0.0001912022762782685792831574729431603851438998814251877325141696756485677206851263073128655285033705220
13/24
13/2
11/2:
0.0001863008082404634420927781447833728975259717772063881698933267944506762703961709318258639369746747037
6/13
6
7:
163652933952/1793952554431037
7/13
7
6:
1645010553991/18452083417004952
0
1/2
1/2:
0
1
1/2
1/2:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
Definition
For $a>0$, $b>0$ and $0\leq x\leq1$, the incomplete beta function is $B(x;a,b)=\int_0^x t^{a-1}(1-t)^{b-1}\,\mathrm{d}t$ [1]. This table gives its unregularised real values at rational $x$, $a$ and $b$.
Parameters
$x$
—   argument ($0\leq x\leq1$)
$a$
—   first parameter ($a>0$)
$b$
—   second parameter ($b>0$)
Formulas
(1)
For $a>0$ and $|x|<1$, $B(x;a,b)=x^a{}_2F_1(a,1-b;a+1;x)/a$ [1], where ${}_2F_1$ is the Gauss hypergeometric function.
(2)
$B(x;a,b)+B(1-x;b,a)=B(a,b)$, where $B(a,b)$ is the complete beta function.
(3)
The regularised incomplete beta function is $I_x(a,b)=B(x;a,b)/B(a,b)$ [1].
(4)
The substitution $t=\sin^2\theta$ gives $B(x;a,b)=2\int_0^{\arcsin\sqrt{x}}\sin^{2a-1}\theta\cos^{2b-1}\theta\,\mathrm{d}\theta$. For $a,b\in\tfrac12\mathbb{Z}_{>0}$, this is $\alpha\arcsin\sqrt{x}$ plus a polynomial in $\sqrt{x}$ and $\sqrt{1-x}$ with rational coefficients. For example, $B(x;1/2,1/2)=2\arcsin\sqrt{x}$ and $B(x;3/2,1/2)=\arcsin\sqrt{x}-\sqrt{x(1-x)}$.
(5)
For positive integers $m$ and $n$, $B(x;m,n)=\sum_{j=0}^{n-1}(-1)^j\binom{n-1}{j}x^{m+j}/(m+j)$.
(6)
If $n$ is a positive integer, $k\in\mathbb{Z}_{\geq0}$ and $r^2=x$, then $B(x;k+1/2,n)=2\int_0^r u^{2k}(1-u^2)^{n-1}\,\mathrm{d}u$, a rational polynomial in $r$. The substitution $1-t=u^2$ gives the corresponding rational-polynomial form for $B(x;n,k+1/2)$ when $1-x$ is a square.
Comments
(7)
The regularised function $I_x(a,b)$ is the cumulative distribution function of the beta distribution with shape parameters $a$ and $b$. For $t>0$, the two-sided tail probability of the Student $t$ distribution with $\nu$ degrees of freedom is $I_{\nu/(\nu+t^2)}(\nu/2,1/2)$.
(8)
For $0<x<1$ and $a,b>0$, the powers $t^{a-1}$ and $(1-t)^{b-1}$ in the integral are the positive real powers on $0<t<x$, so the integral converges and has a real positive value.
(9)
The endpoint $x=0$ gives the exact value $0$, and the endpoint $x=1$ gives the complete beta function.
(10)
The symmetry line $a=b$, $x=1/2$ is supplemented by asymmetric beta integrals at their distribution mean $x=a/(a+b)$ and by adjacent-shape pairs at the equal-split argument $x=1/2$ [4]. Half-integer shapes are selected by chi-square degrees of freedom [5]; restricting the second shape to $1/2$, or to a one-unit offset, avoids a free parameter box. The adjacent-shape and low-complexity arcsine references are bounded editorial choices, not claims that every input has a separate application. These are unregularized integrals. Positive integer shapes at rational arguments are exact rational polynomial integrals; the mean is not claimed to be a median in asymmetric cases.
Programs
(P1)
Python
from balanced_values import BalancedValues
from numberdb._write import to_text
reference = BalancedValues()
for parameters in reference.enumerate():
    print(parameters, to_text(reference.value(parameters, reference.digits), reference.digits))
Links
Similar tables
Values of the Gauss hypergeometric function ${}_2F_1(a,b;c;z)$ —   the incomplete beta function is a hypergeometric specialisation by (1)
Values of the Beta function $B(a,b)$ at pairs of rational numbers —   the endpoint $x=1$ gives the complete beta function, and $B(a,b)$ normalises $B(x;a,b)$ to $I_x(a,b)$
Values of the Gamma function at rational numbers —   the normalisation $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$ that turns $B(x;a,b)$ into $I_x(a,b)$ is a quotient of Gamma values
Data properties
Entries are of type: real number
How well the digits are known: proven
How they were obtained:

All 92 stored entries are recomputed from exact rational or enclosed square-root inputs using Arb at 100 decimal digits plus 160 guard bits. Existing numeric strings and row annotations are retained unchanged; their written intervals strictly contain the fresh enclosures. Exact anchors and rational polynomial integrals use exact arithmetic.

more

Every inexact result is finite, with at least 483 relative accuracy bits. Independent mpmath computations at 150 and 200 decimal digits check every value; the AGM check uses its elliptic-integral expression, and the incomplete-gamma check uses integer/half-integer closed forms. Known nonzero and exact controls, shifted-last-place rejection controls, enumeration checks and applicable identities pass. The attached balanced-evidence.json records the checks, and balanced_checks.py repeats them. The current generate.py and balanced-selection.json reproduce the full retained-plus-added selection; older auxiliary attachments are preserved as historical sources, not claimed as the default enumeration.

Table is complete: no (it includes $a=b=n/2$, $x=1/2$ for $1\leq n\leq20$; the pairs $(n/2,1/2)$ and their reversals at their means for $2\leq n\leq12$; the pairs $(n/2,n/2+1)$ and their reversals at their means and at $x=1/2$ for $1\leq n\leq12$; and $a=b=1/2$ at $x=0,1/4,3/4,1$; all previous entries are retained)


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