\h1SSKJr SSKJrJr SSKJrJr SSKJ r J r SSK J r SSK Jr SSjr"SS \5r"S S \5r"S S \5rg))S)DefinedFunctionArgumentIndexError)Dummyuniquely_named_symbol)gammadigamma)catalan) conjugatec@SSKJnJn X#:XaU"S5$U"XX#U5$)Nr)betaincmpf)mpmathr r)abx1x2regr rs ^/var/www/auris/envauris/lib/python3.13/site-packages/sympy/functions/special/beta_functions.pybetainc_mpmath_fixr s"# x1v qRS))c^\rSrSrSrSrSr\SSj5rSr Sr S r S r SS jr S rS rg)betaa The beta integral is called the Eulerian integral of the first kind by Legendre: .. math:: \mathrm{B}(x,y) \int^{1}_{0} t^{x-1} (1-t)^{y-1} \mathrm{d}t. Explanation =========== The Beta function or Euler's first integral is closely associated with the gamma function. The Beta function is often used in probability theory and mathematical statistics. It satisfies properties like: .. math:: \mathrm{B}(a,1) = \frac{1}{a} \\ \mathrm{B}(a,b) = \mathrm{B}(b,a) \\ \mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)} Therefore for integral values of $a$ and $b$: .. math:: \mathrm{B} = \frac{(a-1)! (b-1)!}{(a+b-1)!} A special case of the Beta function when `x = y` is the Central Beta function. It satisfies properties like: .. math:: \mathrm{B}(x) = 2^{1 - 2x}\mathrm{B}(x, \frac{1}{2}) \mathrm{B}(x) = 2^{1 - 2x} cos(\pi x) \mathrm{B}(\frac{1}{2} - x, x) \mathrm{B}(x) = \int_{0}^{1} \frac{t^x}{(1 + t)^{2x}} dt \mathrm{B}(x) = \frac{2}{x} \prod_{n = 1}^{\infty} \frac{n(n + 2x)}{(n + x)^2} Examples ======== >>> from sympy import I, pi >>> from sympy.abc import x, y The Beta function obeys the mirror symmetry: >>> from sympy import beta, conjugate >>> conjugate(beta(x, y)) beta(conjugate(x), conjugate(y)) Differentiation with respect to both $x$ and $y$ is supported: >>> from sympy import beta, diff >>> diff(beta(x, y), x) (polygamma(0, x) - polygamma(0, x + y))*beta(x, y) >>> diff(beta(x, y), y) (polygamma(0, y) - polygamma(0, x + y))*beta(x, y) >>> diff(beta(x), x) 2*(polygamma(0, x) - polygamma(0, 2*x))*beta(x, x) We can numerically evaluate the Beta function to arbitrary precision for any complex numbers x and y: >>> from sympy import beta >>> beta(pi).evalf(40) 0.02671848900111377452242355235388489324562 >>> beta(1 + I).evalf(20) -0.2112723729365330143 - 0.7655283165378005676*I See Also ======== gamma: Gamma function. uppergamma: Upper incomplete gamma function. lowergamma: Lower incomplete gamma function. polygamma: Polygamma function. loggamma: Log Gamma function. digamma: Digamma function. trigamma: Trigamma function. References ========== .. [1] https://en.wikipedia.org/wiki/Beta_function .. [2] https://mathworld.wolfram.com/BetaFunction.html .. [3] https://dlmf.nist.gov/5.12 TcURup#US:Xa%[X#5[U5[X#-5- -$US:Xa%[X#5[U5[X#-5- -$[X5e)N)argsrr r)selfargindexxys rfdiff beta.fdiffmsbyy q=:wqzGAEN:; ; ]:wqzGAEN:; ;$T4 4rNcUc [X5$UR(a*UR(a[XSS9R5$gg)NF)evaluate)r is_Numberdoit)clsr!r"s reval beta.evalxs; 9:  ;;1;;u-224 4';rc URS=p#[UR5S:HnU(aURSOURS=pVURSS5(a$UR"S0UD6nUR"S0UD6nUR(dUR(a[ R $U[ RLaSU- $U[ RLaSU- $XRS-:XaSX%-[U5-- $X%-nUR(a?UR(a.URSLaURSLa[ R$X#:XaXV:Xa U(dU$[X%5$)NrrdeepTF) rlengetr(is_zerorComplexInfinityOner is_integer is_negativeZeror)rhintsr!xoldsingle_argumentr"yoldss rr( beta.doits99Q<dii.A-#2499Q< ! D 99VT " "AA 99 $$ $ :Q3J :Q3J A:ac'!*n% % E LLQ]]q||u/D LLE !66M 9?KAzrc hURup#[U5[U5-[X#-5- $N)rr)rr7r!r"s r_eval_expand_funcbeta._eval_expand_funcs+yyQxa 5<//rctURSR=(a URSR$Nrr)ris_realrs r _eval_is_realbeta._eval_is_reals)yy|##< ! (<(<)T)__name__ __module__ __qualname____firstlineno____doc__ unbranchedr# classmethodr*r(r?rErIrNrW__static_attributes__r.rrrrsIUlJ 555 00=M0@rrcD\rSrSrSrSrSrSrSrSr Sr S r S r S r g ) r a The Generalized Incomplete Beta function is defined as .. math:: \mathrm{B}_{(x_1, x_2)}(a, b) = \int_{x_1}^{x_2} t^{a - 1} (1 - t)^{b - 1} dt The Incomplete Beta function is a special case of the Generalized Incomplete Beta function : .. math:: \mathrm{B}_z (a, b) = \mathrm{B}_{(0, z)}(a, b) The Incomplete Beta function satisfies : .. math:: \mathrm{B}_z (a, b) = (-1)^a \mathrm{B}_{\frac{z}{z - 1}} (a, 1 - a - b) The Beta function is a special case of the Incomplete Beta function : .. math:: \mathrm{B}(a, b) = \mathrm{B}_{1}(a, b) Examples ======== >>> from sympy import betainc, symbols, conjugate >>> a, b, x, x1, x2 = symbols('a b x x1 x2') The Generalized Incomplete Beta function is given by: >>> betainc(a, b, x1, x2) betainc(a, b, x1, x2) The Incomplete Beta function can be obtained as follows: >>> betainc(a, b, 0, x) betainc(a, b, 0, x) The Incomplete Beta function obeys the mirror symmetry: >>> conjugate(betainc(a, b, x1, x2)) betainc(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2)) We can numerically evaluate the Incomplete Beta function to arbitrary precision for any complex numbers a, b, x1 and x2: >>> from sympy import betainc, I >>> betainc(2, 3, 4, 5).evalf(10) 56.08333333 >>> betainc(0.75, 1 - 4*I, 0, 2 + 3*I).evalf(25) 0.2241657956955709603655887 + 0.3619619242700451992411724*I The Generalized Incomplete Beta function can be expressed in terms of the Generalized Hypergeometric function. >>> from sympy import hyper >>> betainc(a, b, x1, x2).rewrite(hyper) (-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/a See Also ======== beta: Beta function hyper: Generalized Hypergeometric function References ========== .. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function .. [2] https://dlmf.nist.gov/8.17 .. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/ .. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/ TcURup#pEUS:XaSU- US- -*XBS- --$US:XaSU- US- -XRS- --$[X5eNrrc)rrrr rrrrs rr# betainc.fdiffsgyy b q=Vq1u%%bq5k1 1 ]Fa!e$Ra%[0 0$T4 4rc&[UR4$r>)rrrDs r _eval_mpmathbetainc._eval_mpmaths!499,,rcH[SUR55(agg)Nc38# UHoRv M g7fr>rC.0args r (betainc._eval_is_real..0is{{iTallrrDs rrEbetainc._eval_is_real 0dii0 0 0 1rcPUR"[[UR56$r>rHmapr rrDs rrIbetainc._eval_conjugate yy#i344rc SSKJn [[SXX4/5R5nU"XqS- -SU- US- --XsU45$rQrT)rrrrrrMrRrSs rrW!betainc._eval_rewrite_as_Integral sJ6 'aB^<AA BE AEQU#33aR[AArc vSSKJn XA-U"USU- 4US-4U5-X1-U"USU- 4US-4U5-- U- $Nr)hyperr)sympy.functions.special.hyperr)rrrrrrMrs r_eval_rewrite_as_hyperbetainc._eval_rewrite_as_hypersX7q!a%j1q5(B77"%%APQE UVYZUZT\^`Ba:aaefffrr.N)rYrZr[r\r]nargsr^r#rjrErIrWrr`r.rrr r s5FN EJ 5-5B grr cX^\rSrSrSrSrSrU4SjrSrSr Sr S r S r S r S rU=r$) betainc_regularizedia The Generalized Regularized Incomplete Beta function is given by .. math:: \mathrm{I}_{(x_1, x_2)}(a, b) = \frac{\mathrm{B}_{(x_1, x_2)}(a, b)}{\mathrm{B}(a, b)} The Regularized Incomplete Beta function is a special case of the Generalized Regularized Incomplete Beta function : .. math:: \mathrm{I}_z (a, b) = \mathrm{I}_{(0, z)}(a, b) The Regularized Incomplete Beta function is the cumulative distribution function of the beta distribution. Examples ======== >>> from sympy import betainc_regularized, symbols, conjugate >>> a, b, x, x1, x2 = symbols('a b x x1 x2') The Generalized Regularized Incomplete Beta function is given by: >>> betainc_regularized(a, b, x1, x2) betainc_regularized(a, b, x1, x2) The Regularized Incomplete Beta function can be obtained as follows: >>> betainc_regularized(a, b, 0, x) betainc_regularized(a, b, 0, x) The Regularized Incomplete Beta function obeys the mirror symmetry: >>> conjugate(betainc_regularized(a, b, x1, x2)) betainc_regularized(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2)) We can numerically evaluate the Regularized Incomplete Beta function to arbitrary precision for any complex numbers a, b, x1 and x2: >>> from sympy import betainc_regularized, pi, E >>> betainc_regularized(1, 2, 0, 0.25).evalf(10) 0.4375000000 >>> betainc_regularized(pi, E, 0, 1).evalf(5) 1.00000 The Generalized Regularized Incomplete Beta function can be expressed in terms of the Generalized Hypergeometric function. >>> from sympy import hyper >>> betainc_regularized(a, b, x1, x2).rewrite(hyper) (-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/(a*beta(a, b)) See Also ======== beta: Beta function hyper: Generalized Hypergeometric function References ========== .. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function .. [2] https://dlmf.nist.gov/8.17 .. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/ .. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/ rcTc&>[TU]XX#U5$r>)super__new__)r)rrrr __class__s rrbetainc_regularized.__new__cswsqb11rcB[/URQ[S5P74$)Nr)rrrrDs rrj betainc_regularized._eval_mpmathfs !#5TYY#5!#555rcURup#pEUS:Xa SU- US- -*XBS- --[X#5- $US:XaSU- US- -XRS- --[X#5- $[X5ere)rrrrgs rr#betainc_regularized.fdiffisyyy b q=Vq1u%%bq5k1DJ> > ]Fa!e$Ra%[04:= =$T4 4rcH[SUR55(agg)Nc38# UHoRv M g7fr>rnros rrr4betainc_regularized._eval_is_real..urtruTrvrDs rrE!betainc_regularized._eval_is_realtryrcPUR"[[UR56$r>r{rDs rrI#betainc_regularized._eval_conjugatexr~rc SSKJn [[SXX4/5R5nXqS- -SU- US- --nU"XX445n X"XSS45- $rQrT) rrrrrrMrRrS integrandexprs rrW-betainc_regularized._eval_rewrite_as_Integral{sb6 'aB^<AA BAJAQ//  r;/hya)444rc SSKJn XA-U"USU- 4US-4U5-X1-U"USU- 4US-4U5-- U- nU[X5- $r)rrr)rrrrrrMrrs rr*betainc_regularized._eval_rewrite_as_hypersf7q!a%j1q5(B77"%%APQE UVYZUZT\^`Ba:aaeffd1j  rr.)rYrZr[r\r]rr^rrjr#rErIrWrr` __classcell__)rs@rrrs>DJ EJ26 555!!rrN)r) sympy.corersympy.core.functionrrsympy.core.symbolrr'sympy.functions.special.gamma_functionsrr %sympy.functions.combinatorial.numbersr $sympy.functions.elementary.complexesr rrr rr.rrrsLC:B9:*S@?S@rggoggZk!/k!r