U Zʹj1@sddlmZddlmZmZddlmZmZddlm Z m Z ddl m Z ddl mZdddZGd d d eZGd d d eZGd ddeZdS))S)FunctionArgumentIndexError)Dummyuniquely_named_symbol)gammadigamma)catalan) conjugatecCs4ddlm}m}||kr |dS||||||SdS)Nr)betaincmpf)Zmpmathr r )abx1x2regr r re/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/functions/special/beta_functions.pybetainc_mpmath_fix src@s\eZdZdZdZddZedddZdd Zd d Z d d Z ddZ dddZ ddZ dS)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 TcCsd|j\}}|dkr0t||t|t||S|dkrVt||t|t||St||dS)N)argsrrr)selfargindexxyrrrfdiffms  z beta.fdiffNcCs4|dkrt||S|jr0|jr0t||ddSdS)NF)evaluate)rZ is_Numberdoit)clsrrrrrevalxs  z beta.evalcKs|jd}}t|jdk}|r*|jdn|jd}}|ddr\|jf|}|jf|}|jsh|jrntjS|tjkrd|S|tjkrd|S||dkrd||t|S||}|j r|j r|j dkr|j dkrtj S||kr||kr|s|St ||S)NrrdeepTF) rlengetris_zerorZComplexInfinityZOner is_integerZ is_negativeZZeror)rhintsrZxoldZsingle_argumentrZyoldsrrrrs,       z beta.doitcKs&|j\}}t|t|t||SN)rr)rr'rrrrr_eval_expand_funcs zbeta._eval_expand_funccCs|jdjo|jdjSNrr)ris_realrrrr _eval_is_realszbeta._eval_is_realcCs ||jd|jdSr+)funcrr r-rrr_eval_conjugateszbeta._eval_conjugatecKs |jf|Sr))r*)rrrZ piecewisekwargsrrr_eval_rewrite_as_gammaszbeta._eval_rewrite_as_gammacKsHddlm}ttd||gj}|||dd||d|ddfSNr)IntegraltrZsympy.integrals.integralsr4rrname)rrrr1r4r5rrr_eval_rewrite_as_Integrals zbeta._eval_rewrite_as_Integral)N)T)__name__ __module__ __qualname____doc__ unbranchedr classmethodr!rr*r.r0r2r8rrrrrsV   rc@sHeZdZdZdZdZddZddZdd Zd d Z d d Z ddZ dS)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/ TcCsf|j\}}}}|dkr4d||d ||dS|dkrXd||d||dSt||dSNrr?)rrrrr rrrrrrrs z betainc.fdiffcCs t|jfSr))rrr-rrr _eval_mpmathszbetainc._eval_mpmathcCstdd|jDrdSdS)Ncss|] }|jVqdSr)r,.0argrrr sz(betainc._eval_is_real..Tallrr-rrrr.szbetainc._eval_is_realcCs|jtt|jSr)r/mapr rr-rrrr0 szbetainc._eval_conjugatecKsLddlm}ttd||||gj}|||dd||d|||fSr3r6)rr rrrr1r4r5rrrr8 s z!betainc._eval_rewrite_as_IntegralcKsTddlm}||||d|f|df|||||d|f|df||SNr)hyperr)sympy.functions.special.hyperrN)rr rrrr1rNrrr_eval_rewrite_as_hypers zbetainc._eval_rewrite_as_hyperN) r9r:r;r<nargsr=rrCr.r0r8rPrrrrr sG r c@sPeZdZdZdZdZddZddZdd Zd d Z d d Z ddZ ddZ dS)betainc_regularizeda 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/ .. 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