image-handler/image-handler10.0/venv/lib/site-packages/sympy/simplify/tests/test_gammasimp.py

from sympy.core.function import Function
from sympy.core.numbers import (Rational, pi)
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.combinatorial.factorials import (rf, binomial, factorial)
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.functions.special.gamma_functions import gamma
from sympy.simplify.gammasimp import gammasimp
from sympy.simplify.powsimp import powsimp
from sympy.simplify.simplify import simplify

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)


def test_issue_22606():
    fx = Function('f')(x)
    eq = x + gamma(y)
    # seems like ans should be `eq`, not `(x*y + gamma(y + 1))/y`
    ans = gammasimp(eq)
    assert gammasimp(eq.subs(x, fx)).subs(fx, x) == ans
    assert gammasimp(eq.subs(x, cos(x))).subs(cos(x), x) == ans
    assert 1/gammasimp(1/eq) == ans
    assert gammasimp(fx.subs(x, eq)).args[0] == ans