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Mini Shell
Mini Shell
from sympy import (
Rational, gammasimp, factorial, gamma, binomial, pi, S,
sin, exp, Piecewise, powsimp, sqrt, simplify, symbols, cos, rf)
from sympy.abc import x, y, n, k
def test_gammasimp():
R = Rational
# was part of test_combsimp_gamma() in test_combsimp.py
assert gammasimp(gamma(x)) == gamma(x)
assert gammasimp(gamma(x + 1)/x) == gamma(x)
assert gammasimp(gamma(x)/(x - 1)) == gamma(x - 1)
assert gammasimp(x*gamma(x)) == gamma(x + 1)
assert gammasimp((x + 1)*gamma(x + 1)) == gamma(x + 2)
assert gammasimp(gamma(x + y)*(x + y)) == gamma(x + y + 1)
assert gammasimp(x/gamma(x + 1)) == 1/gamma(x)
assert gammasimp((x + 1)**2/gamma(x + 2)) == (x + 1)/gamma(x + 1)
assert gammasimp(x*gamma(x) + gamma(x + 3)/(x + 2)) == \
(x + 2)*gamma(x + 1)
assert gammasimp(gamma(2*x)*x) == gamma(2*x + 1)/2
assert gammasimp(gamma(2*x)/(x - S.Half)) == 2*gamma(2*x - 1)
assert gammasimp(gamma(x)*gamma(1 - x)) == pi/sin(pi*x)
assert gammasimp(gamma(x)*gamma(-x)) == -pi/(x*sin(pi*x))
assert gammasimp(1/gamma(x + 3)/gamma(1 - x)) == \
sin(pi*x)/(pi*x*(x + 1)*(x + 2))
assert gammasimp(factorial(n + 2)) == gamma(n + 3)
assert gammasimp(binomial(n, k)) == \
gamma(n + 1)/(gamma(k + 1)*gamma(-k + n + 1))
assert powsimp(gammasimp(
gamma(x)*gamma(x + S.Half)*gamma(y)/gamma(x + y))) == \
2**(-2*x + 1)*sqrt(pi)*gamma(2*x)*gamma(y)/gamma(x + y)
assert gammasimp(1/gamma(x)/gamma(x - Rational(1, 3))/gamma(x + Rational(1, 3))) == \
3**(3*x - Rational(3, 2))/(2*pi*gamma(3*x - 1))
assert simplify(
gamma(S.Half + x/2)*gamma(1 + x/2)/gamma(1 + x)/sqrt(pi)*2**x) == 1
assert gammasimp(gamma(Rational(-1, 4))*gamma(Rational(-3, 4))) == 16*sqrt(2)*pi/3
assert powsimp(gammasimp(gamma(2*x)/gamma(x))) == \
2**(2*x - 1)*gamma(x + S.Half)/sqrt(pi)
# issue 6792
e = (-gamma(k)*gamma(k + 2) + gamma(k + 1)**2)/gamma(k)**2
assert gammasimp(e) == -k
assert gammasimp(1/e) == -1/k
e = (gamma(x) + gamma(x + 1))/gamma(x)
assert gammasimp(e) == x + 1
assert gammasimp(1/e) == 1/(x + 1)
e = (gamma(x) + gamma(x + 2))*(gamma(x - 1) + gamma(x))/gamma(x)
assert gammasimp(e) == (x**2 + x + 1)*gamma(x + 1)/(x - 1)
e = (-gamma(k)*gamma(k + 2) + gamma(k + 1)**2)/gamma(k)**2
assert gammasimp(e**2) == k**2
assert gammasimp(e**2/gamma(k + 1)) == k/gamma(k)
a = R(1, 2) + R(1, 3)
b = a + R(1, 3)
assert gammasimp(gamma(2*k)/gamma(k)*gamma(k + a)*gamma(k + b))
3*2**(2*k + 1)*3**(-3*k - 2)*sqrt(pi)*gamma(3*k + R(3, 2))/2
# issue 9699
assert gammasimp((x + 1)*factorial(x)/gamma(y)) == gamma(x + 2)/gamma(y)
assert gammasimp(rf(x + n, k)*binomial(n, k)).simplify() == Piecewise(
(gamma(n + 1)*gamma(k + n + x)/(gamma(k + 1)*gamma(n + x)*gamma(-k + n + 1)), n > -x),
((-1)**k*gamma(n + 1)*gamma(-n - x + 1)/(gamma(k + 1)*gamma(-k + n + 1)*gamma(-k - n - x + 1)), True))
A, B = symbols('A B', commutative=False)
assert gammasimp(e*B*A) == gammasimp(e)*B*A
# check iteration
assert gammasimp(gamma(2*k)/gamma(k)*gamma(-k - R(1, 2))) == (
-2**(2*k + 1)*sqrt(pi)/(2*((2*k + 1)*cos(pi*k))))
assert gammasimp(
gamma(k)*gamma(k + R(1, 3))*gamma(k + R(2, 3))/gamma(k*R(3, 2))) == (
3*2**(3*k + 1)*3**(-3*k - S.Half)*sqrt(pi)*gamma(k*R(3, 2) + S.Half)/2)
# issue 6153
assert gammasimp(gamma(Rational(1, 4))/gamma(Rational(5, 4))) == 4
# was part of test_combsimp() in test_combsimp.py
assert gammasimp(binomial(n + 2, k + S.Half)) == gamma(n + 3)/ \
(gamma(k + R(3, 2))*gamma(-k + n + R(5, 2)))
assert gammasimp(binomial(n + 2, k + 2.0)) == \
gamma(n + 3)/(gamma(k + 3.0)*gamma(-k + n + 1))
# issue 11548
assert gammasimp(binomial(0, x)) == sin(pi*x)/(pi*x)
e = gamma(n + Rational(1, 3))*gamma(n + R(2, 3))
assert gammasimp(e) == e
assert gammasimp(gamma(4*n + S.Half)/gamma(2*n - R(3, 4))) == \
2**(4*n - R(5, 2))*(8*n - 3)*gamma(2*n + R(3, 4))/sqrt(pi)
i, m = symbols('i m', integer = True)
e = gamma(exp(i))
assert gammasimp(e) == e
e = gamma(m + 3)
assert gammasimp(e) == e
e = gamma(m + 1)/(gamma(i + 1)*gamma(-i + m + 1))
assert gammasimp(e) == e
p = symbols("p", integer=True, positive=True)
assert gammasimp(gamma(-p+4)) == gamma(-p+4)
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