ó ¡¼™\c@ s\ddlmZmZddlmZmZddlmZmZdefd„ƒYZ dS(iÿÿÿÿ(tprint_functiontdivision(tFunctiontArgumentIndexError(tgammatdigammatbetacB sMeZdZdZeZd„Zed„ƒZd„Z d„Z d„Z RS(s1 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. Beta function or Euler's first integral is closely associated with gamma function. The Beta function 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)!} Examples ======== >>> from sympy import I, pi >>> from sympy.abc import x,y The Beta function obeys the mirror symmetry: >>> from sympy import beta >>> from sympy import conjugate >>> conjugate(beta(x,y)) beta(conjugate(x), conjugate(y)) Differentiation with respect to both x and y is supported: >>> from sympy import beta >>> from sympy import diff >>> diff(beta(x,y), x) (polygamma(0, x) - polygamma(0, x + y))*beta(x, y) >>> from sympy import beta >>> from sympy import diff >>> diff(beta(x,y), y) (polygamma(0, y) - polygamma(0, x + y))*beta(x, y) We can numerically evaluate the gamma function to arbitrary precision on the whole complex plane: >>> from sympy import beta >>> beta(pi,pi).evalf(40) 0.02671848900111377452242355235388489324562 >>> beta(1+I,1+I).evalf(20) -0.2112723729365330143 - 0.7655283165378005676*I See Also ======== sympy.functions.special.gamma_functions.gamma: Gamma function. sympy.functions.special.gamma_functions.uppergamma: Upper incomplete gamma function. sympy.functions.special.gamma_functions.lowergamma: Lower incomplete gamma function. sympy.functions.special.gamma_functions.polygamma: Polygamma function. sympy.functions.special.gamma_functions.loggamma: Log Gamma function. sympy.functions.special.gamma_functions.digamma: Digamma function. sympy.functions.special.gamma_functions.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 icC s„|j\}}|dkr@t||ƒt|ƒt||ƒS|dkrqt||ƒt|ƒt||ƒSt||ƒ‚dS(Nii(targsRRR(tselftargindextxty((sElib/python2.7/site-packages/sympy/functions/special/beta_functions.pytfdiffXs  % %cC sdS(N((tclsR R ((sElib/python2.7/site-packages/sympy/functions/special/beta_functions.pytevalcscK s1|j\}}t|ƒt|ƒt||ƒS(N(RR(RthintsR R ((sElib/python2.7/site-packages/sympy/functions/special/beta_functions.pyt_eval_expand_funcgscC s|jdjo|jdjS(Nii(Rtis_real(R((sElib/python2.7/site-packages/sympy/functions/special/beta_functions.pyt _eval_is_realkscC s*|j|jdjƒ|jdjƒƒS(Nii(tfuncRt conjugate(R((sElib/python2.7/site-packages/sympy/functions/special/beta_functions.pyt_eval_conjugatens( t__name__t __module__t__doc__tnargstTruet unbranchedR t classmethodRRRR(((sElib/python2.7/site-packages/sympy/functions/special/beta_functions.pyR sJ   N( t __future__RRtsympy.core.functionRRt'sympy.functions.special.gamma_functionsRRR(((sElib/python2.7/site-packages/sympy/functions/special/beta_functions.pyts