+ i1^RIHt^RIHtHt^RIHtHt^RIH t H t ^RI H t ^RI HtRRlt!RR ]4t!R R ]4t!R R ]4tR#))S)DefinedFunctionArgumentIndexError)Dummyuniquely_named_symbol)gammadigamma)catalan) conjugatecB^RIHpHpW#8Xd V!^4#V!WW#V4#)r)betaincmpf)mpmathr r)abx1x2regr rs&&&&& d/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/functions/special/beta_functions.pybetainc_mpmath_fixr s"# x1v qRS))cja]tRt^toRtRtRt]R Rl4tRt Rt Rt R t RR lt R tR tVtR#)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 TcVPwr#V^8Xd2\W#4\V4\W#,4, ,#V^8Xd2\W#4\V4\W#,4, ,#\W4h)argsrr r)selfargindexxys&& rfdiff beta.fdiffmsbyy q=:wqzGAEN:; ; ]:wqzGAEN:; ;$T4 4rNcVf \W4#VP'd0VP'd\WRR7P4#R#R#)NF)evaluate)r is_Numberdoit)clsr r!s&&&reval beta.evalxs? 9:  ;;;1;;;u-224 4';rc ^VP^,;r#\VP4^8HpV'dVP^,MVP^,;rVVPRR4'd%VP!R/VBpVP!R/VBpVP'gVP'd\ P #V\ PJd ^V, #V\ PJd ^V, #WR^,8Xd ^W%,\V4,, #W%,pVP'dCVP'd1VPRJd!VPRJd\ P#W#8XdWV8Xd V'gV#\W%4#)rdeepTF) rlengetr'is_zerorComplexInfinityOner is_integer is_negativeZeror)rhintsr xoldsingle_argumentr!yoldss&, rr' beta.doits$99Q<dii.A-#2499Q< ! D 99VT " "AA 999 $$ $ :Q3J :Q3J A:ac'!*n% % E LLLQ]]]q||u/D LLE !66M 9?KAzrc VPwr#\V4\V4,\W#,4, #N)rr)rr6r r!s&, r_eval_expand_funcbeta._eval_expand_funcs+yyQxa 5<//rcVP^,P;'dVP^,P#r)ris_realrs&r _eval_is_realbeta._eval_is_reals,yy|##<< ! (<(<)rr r! piecewisekwargss&&&&,r_eval_rewrite_as_gammabeta._eval_rewrite_as_gammas%%///rc ^RIHp\\RW.4P4pV!WQ^, ,^V, V^, ,,V^^34#r)Integraltsympy.integrals.integralsrQrrname)rr r!rLrQrRs&&&, r_eval_rewrite_as_Integralbeta._eval_rewrite_as_IntegralsJ6 'aV499 :E AEQU#33aAY??rr-r=)T)__name__ __module__ __qualname____firstlineno____doc__ unbranchedr" classmethodr)r'r>rDrHrMrV__static_attributes____classdictcell__ __classdict__s@rrrsRUlJ 555 00=M0@@rrcPa]tRt^toRt^tRtRtRtRt Rt Rt Rt R t VtR #) 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/ TcVPwr#rEV^8Xd-^V, V^, ,)WB^, ,,#V^8Xd,^V, V^, ,WR^, ,,#\W4h)rrrrrrrrs&& rr" betainc.fdiffsgyy b q=Vq1u%%bq5k1 1 ]Fa!e$Ra%[0 0$T4 4rc&\VP3#r=)rrrCs&r _eval_mpmathbetainc._eval_mpmaths!499,,rc\;QJd&RVP4F 'dK RM RM!RVP44'dR#R#)c38"TFqPxK R#5ir=rB.0args& r (betainc._eval_is_real..0is{{iFTNallrrCs&rrDbetainc._eval_is_real3 30dii03330dii0 0 0 1rcRVP!\\VP4!#r=rGmapr rrCs&rrHbetainc._eval_conjugate yy#i344rc ^RIHp\\RWW4.4P4pV!Wq^, ,^V, V^, ,,WsV34#rPrS)rrrrrrLrQrRs&&&&&, rrV!betainc._eval_rewrite_as_Integral sJ6 'aB^<AA BE AEQU#33aR[AArc ^RIHpWA,V!V^V, 3V^,3V4,W1,V!V^V, 3V^,3V4,, V, #r)hyper)sympy.functions.special.hyperr)rrrrrrLrs&&&&&, r_eval_rewrite_as_hyperbetainc._eval_rewrite_as_hypersX7q!a%j1q5(B77"%%APQE UVYZUZT\^`Ba:aaefffrr-N)rXrYrZr[r\nargsr]r"rjrDrHrVrr_r`ras@rr r s>FN EJ 5-5B ggrr cbaa]tRtRtoRt^tRtV3RltRtRt Rt Rt R t R t R tVtV;t#) 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/ Tc&<\SV`WW#V4#r=)super__new__)r(rrrr __class__s&&&&&rrbetainc_regularized.__new__cswsqb11rcB\.VPO\^4N53#r)rrrrCs&rrj betainc_regularized._eval_mpmathfs !#5TYY#5!#555rc@VPwr#rEV^8Xd=^V, V^, ,)WB^, ,,\W#4, #V^8Xd<^V, V^, ,WR^, ,,\W#4, #\W4hre)rrrrgs&& rr"betainc_regularized.fdiffisyyy b q=Vq1u%%bq5k1DJ> > ]Fa!e$Ra%[04:= =$T4 4rc\;QJd&RVP4F 'dK RM RM!RVP44'dR#R#)c38"TFqPxK R#5ir=rnros& rrr4betainc_regularized._eval_is_real..urtruFTNrvrCs&rrD!betainc_regularized._eval_is_realtryrcRVP!\\VP4!#r=r{rCs&rrH#betainc_regularized._eval_conjugatexr~rc ^RIHp\\RWW4.4P4pWq^, ,^V, V^, ,,pV!WW434p W!W^^34, #rPrS) rrrrrrLrQrR integrandexprs &&&&&, rrV-betainc_regularized._eval_rewrite_as_Integral{sb6 'aB^<AA BAJAQ//  r;/hya)444rc ^RIHpWA,V!V^V, 3V^,3V4,W1,V!V^V, 3V^,3V4,, V, pV\W4, #r)rrr)rrrrrrLrrs&&&&&, rr*betainc_regularized._eval_rewrite_as_hypersf7q!a%j1q5(B77"%%APQE UVYZUZT\^`Ba:aaeffd1j  rr-)rXrYrZr[r\rr]rrjr"rDrHrVrr_r` __classcell__)rrbs@@rrrs@DJ EJ26 555!!rrNrA) 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