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67 lines
2.7 KiB
67 lines
2.7 KiB
from sympy.core.function import (diff, expand_func)
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from sympy.core.numbers import I
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from sympy.core.symbol import (Dummy, symbols)
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from sympy.functions.elementary.complexes import conjugate
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from sympy.functions.special.beta_functions import (beta, betainc, betainc_regularized)
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from sympy.functions.special.gamma_functions import gamma
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from sympy.functions.special.hyper import hyper
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from sympy.integrals.integrals import Integral
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from sympy.functions.special.gamma_functions import polygamma
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from sympy.core.function import ArgumentIndexError
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from sympy.core.expr import unchanged
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from sympy.testing.pytest import raises
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def test_beta():
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x, y = symbols('x y')
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t = Dummy('t')
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assert unchanged(beta, x, y)
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assert beta(5, -3).is_real == True
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assert beta(3, y).is_real is None
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assert expand_func(beta(x, y)) == gamma(x)*gamma(y)/gamma(x + y)
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assert expand_func(beta(x, y) - beta(y, x)) == 0 # Symmetric
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assert expand_func(beta(x, y)) == expand_func(beta(x, y + 1) + beta(x + 1, y)).simplify()
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assert diff(beta(x, y), x) == beta(x, y)*(polygamma(0, x) - polygamma(0, x + y))
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assert diff(beta(x, y), y) == beta(x, y)*(polygamma(0, y) - polygamma(0, x + y))
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assert conjugate(beta(x, y)) == beta(conjugate(x), conjugate(y))
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raises(ArgumentIndexError, lambda: beta(x, y).fdiff(3))
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assert beta(x, y).rewrite(gamma) == gamma(x)*gamma(y)/gamma(x + y)
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assert beta(x).rewrite(gamma) == gamma(x)**2/gamma(2*x)
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assert beta(x, y).rewrite(Integral).dummy_eq(Integral(t**(x - 1) * (1 - t)**(y - 1), (t, 0, 1)))
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def test_betainc():
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a, b, x1, x2 = symbols('a b x1 x2')
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assert unchanged(betainc, a, b, x1, x2)
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assert unchanged(betainc, a, b, 0, x1)
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assert betainc(1, 2, 0, -5).is_real == True
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assert betainc(1, 2, 0, x2).is_real is None
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assert conjugate(betainc(I, 2, 3 - I, 1 + 4*I)) == betainc(-I, 2, 3 + I, 1 - 4*I)
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assert betainc(a, b, 0, 1).rewrite(Integral).dummy_eq(beta(a, b).rewrite(Integral))
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assert betainc(1, 2, 0, x2).rewrite(hyper) == x2*hyper((1, -1), (2,), x2)
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assert betainc(1, 2, 3, 3).evalf() == 0
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def test_betainc_regularized():
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a, b, x1, x2 = symbols('a b x1 x2')
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assert unchanged(betainc_regularized, a, b, x1, x2)
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assert unchanged(betainc_regularized, a, b, 0, x1)
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assert betainc_regularized(3, 5, 0, -1).is_real == True
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assert betainc_regularized(3, 5, 0, x2).is_real is None
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assert conjugate(betainc_regularized(3*I, 1, 2 + I, 1 + 2*I)) == betainc_regularized(-3*I, 1, 2 - I, 1 - 2*I)
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assert betainc_regularized(a, b, 0, 1).rewrite(Integral) == 1
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assert betainc_regularized(1, 2, x1, x2).rewrite(hyper) == 2*x2*hyper((1, -1), (2,), x2) - 2*x1*hyper((1, -1), (2,), x1)
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assert betainc_regularized(4, 1, 5, 5).evalf() == 0
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