a RG5dÌ-ã@s‚ddlmZddlmZmZddlmZddlmZm Z ddl m Z ddd„Z Gdd „d eƒZ Gd d „d eƒZGd d „d eƒZdS)é)ÚS)ÚFunctionÚArgumentIndexError)ÚDummy)ÚgammaÚdigamma)Ú conjugatecCs4ddlm}m}||kr |dƒS||||||ƒSdS)Nr)ÚbetaincÚmpf)Úmpmathr r )ÚaÚbÚx1Úx2Úregr r ©rúb/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/functions/special/beta_functions.pyÚbetainc_mpmath_fixsrc@sTeZdZdZdZdd„Zeddd„ƒZdd „Zd d „Z d d „Z ddd„Z dd„Z dS)Úbetaa× 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] http://mathworld.wolfram.com/BetaFunction.html .. [3] http://dlmf.nist.gov/5.12 TcCsd|j\}}|dkr0t||ƒt|ƒt||ƒS|dkrVt||ƒt|ƒt||ƒSt||ƒ‚dS)Néé)Úargsrrr)ÚselfÚargindexÚxÚyrrrÚfdiffls  z beta.fdiffNcCs:|durt||ƒS|tjur$d|S|tjur6d|SdS©Nr)rrÚOne)ÚclsrrrrrÚevalws    z beta.evalcKs&|j\}}t|ƒt|ƒt||ƒS©N)rr)rÚhintsrrrrrÚ_eval_expand_func€s zbeta._eval_expand_funccCs|jdjo|jdjS©Nrr)rÚis_real©rrrrÚ _eval_is_real„szbeta._eval_is_realcCs | |jd ¡|jd ¡¡Sr$)Úfuncrrr&rrrÚ_eval_conjugate‡szbeta._eval_conjugatecKs|jfi|¤ŽSr!)r#)rrrÚ piecewiseÚkwargsrrrÚ_eval_rewrite_as_gammaŠszbeta._eval_rewrite_as_gammacKs<ddlm}tdƒ}|||dd||d|ddfƒS©Nr)ÚIntegralÚtr©Úsympy.integrals.integralsr.r)rrrr+r.r/rrrÚ_eval_rewrite_as_Integrals zbeta._eval_rewrite_as_Integral)N)T) Ú__name__Ú __module__Ú __qualname__Ú__doc__Ú unbranchedrÚ classmethodr r#r'r)r,r2rrrrrsV   rc@sHeZdZdZdZdZdd„Zdd„Zdd „Zd d „Z d d „Z dd„Z 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/ éTcCsf|j\}}}}|dkr4d||d ||dS|dkrXd||d||dSt||ƒ‚dS©Nérr9)rr©rrr r rrrrrrás z betainc.fdiffcCs t|jfSr!)rrr&rrrÚ _eval_mpmathìszbetainc._eval_mpmathcCstdd„|jDƒƒrdSdS)Ncss|] }|jVqdSr!©r%©Ú.0ÚargrrrÚ ðóz(betainc._eval_is_real..T©Úallrr&rrrr'ïszbetainc._eval_is_realcCs|jtt|jƒŽSr!©r(Úmaprrr&rrrr)ószbetainc._eval_conjugatecKs<ddlm}tdƒ}|||dd||d|||fƒSr-r0)rr r rrr+r.r/rrrr2ös z!betainc._eval_rewrite_as_IntegralcKsTddlm}||||d|f|df|ƒ||||d|f|df|ƒ|S©Nr)Úhyperr)Úsympy.functions.special.hyperrI)rr r rrr+rIrrrÚ_eval_rewrite_as_hyperûs zbetainc._eval_rewrite_as_hyperN) r3r4r5r6Únargsr7rr=r'r)r2rKrrrrr –sG r c@sPeZdZdZdZdZdd„Zdd„Zdd „Zd d „Z d d „Z dd„Z dd„Z dS)Úbetainc_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/ .. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/ r9TcCst |||||¡Sr!)rÚ__new__)rr r rrrrrrNLszbetainc_regularized.__new__cCstg|j¢tdƒ‘RfSr)rrrr&rrrr=Osz betainc_regularized._eval_mpmathcCsz|j\}}}}|dkr>d||d ||dt||ƒS|dkrld||d||dt||ƒSt||ƒ‚dSr:)rrrr<rrrrRs (&zbetainc_regularized.fdiffcCstdd„|jDƒƒrdSdS)Ncss|] }|jVqdSr!r>r?rrrrB^rCz4betainc_regularized._eval_is_real..TrDr&rrrr']sz!betainc_regularized._eval_is_realcCs|jtt|jƒŽSr!rFr&rrrr)asz#betainc_regularized._eval_conjugatec KsTddlm}tdƒ}||dd||d}|||||fƒ} | |||ddfƒSr-r0) rr r rrr+r.r/Ú integrandÚexprrrrr2ds  z-betainc_regularized._eval_rewrite_as_IntegralcKsbddlm}||||d|f|df|ƒ||||d|f|df|ƒ|}|t||ƒSrH)rJrIr)rr r rrr+rIrPrrrrKks Hz*betainc_regularized._eval_rewrite_as_hyperN) r3r4r5r6rLr7rNr=rr'r)r2rKrrrrrMsE rMN)r)Ú sympy.corerÚsympy.core.functionrrÚsympy.core.symbolrÚ'sympy.functions.special.gamma_functionsrrÚ$sympy.functions.elementary.complexesrrrr rMrrrrÚs    m