def test_expressions_failing():
assert residue(1 / (x**4 + 1), x,
exp(I * pi / 4)) == -(Rational(1, 4) + I / 4) / sqrt(2)
n = Symbol('n', integer=True, positive=True)
assert residue(exp(z)/(z - pi*I/4*a)**n, z, I*pi*a) == \
exp(I*pi*a/4)/factorial(n - 1)
assert residue(1 / (x**2 + a**2)**2, x, a * I) == -I / 4 / a**3
def test_sympyissue_21177():
e1 = cot(pi * x) / ((x - 1) * (x - 2) + 1)
e2 = cot(pi * x) / (x**2 - 3 * x + 3)
pt = Rational(3, 2) - sqrt(3) * I / 2
ans = -sqrt(3) * tanh(sqrt(3) * pi / 2) / 3
assert residue(e1, x, pt) == ans
assert residue(e2, x, pt) == ans
def test_basic1():
assert residue(1/x, x, 0) == 1
assert residue(-2/x, x, 0) == -2
assert residue(81/x, x, 0) == 81
assert residue(1/x**2, x, 0) == 0
assert residue(0, x, 0) == 0
assert residue(5, x, 0) == 0
assert residue(x, x, 0) == 0
assert residue(x**2, x, 0) == 0
def test_basic2():
assert residue(1/x, x, 1) == 0
assert residue(-2/x, x, 1) == 0
assert residue(81/x, x, -1) == 0
assert residue(1/x**2, x, 1) == 0
assert residue(0, x, 1) == 0
assert residue(5, x, 1) == 0
assert residue(x, x, 1) == 0
assert residue(x**2, x, 5) == 0
def test_basic1():
assert residue(1 / x, x, 0) == 1
assert residue(-2 / x, x, 0) == -2
assert residue(81 / x, x, 0) == 81
assert residue(1 / x**2, x, 0) == 0
assert residue(0, x, 0) == 0
assert residue(5, x, 0) == 0
assert residue(x, x, 0) == 0
assert residue(x**2, x, 0) == 0
def test_expressions():
assert residue(1 / (x + 1), x, 0) == 0
assert residue(1 / (x + 1), x, -1) == 1
assert residue(1 / (x**2 + 1), x, -1) == 0
assert residue(1 / (x**2 + 1), x, I) == -I / 2
assert residue(1 / (x**2 + 1), x, -I) == I / 2
assert residue(1 / (x**4 + 1), x, 0) == 0
assert residue(1 / (x**4 + 1), x, exp(I * pi / 4)) == -root(-1, 4) / 4
assert residue(1 / (x**2 + a**2)**2, x, a * I) == -I / 4 / a**3
def test_expressions():
assert residue(1/(x + 1), x, 0) == 0
assert residue(1/(x + 1), x, -1) == 1
assert residue(1/(x**2 + 1), x, -1) == 0
assert residue(1/(x**2 + 1), x, I) == -I/2
assert residue(1/(x**2 + 1), x, -I) == I/2
assert residue(1/(x**4 + 1), x, 0) == 0
assert residue(1/(x**4 + 1), x, exp(I*pi/4)) == -root(-1, 4)/4
assert residue(1/(x**2 + a**2)**2, x, a*I) == -I/4/a**3
def test_basic2():
assert residue(1 / x, x, 1) == 0
assert residue(-2 / x, x, 1) == 0
assert residue(81 / x, x, -1) == 0
assert residue(1 / x**2, x, 1) == 0
assert residue(0, x, 1) == 0
assert residue(5, x, 1) == 0
assert residue(x, x, 1) == 0
assert residue(x**2, x, 5) == 0
def test_sympyissue_21176():
e = x**2 * cot(pi * x) / (x**4 + 1)
pt = -sqrt(2) / 2 - sqrt(2) * I / 2
assert residue(
e, x,
pt) == sqrt(2) * I / Mul(2,
-2 + 2 * I,
tan(sqrt(2) * pi / 2 + sqrt(2) * I * pi / 2),
evaluate=False)
def test_expressions():
assert residue(1 / (x + 1), x, 0) == 0
assert residue(1 / (x + 1), x, -1) == 1
assert residue(1 / (x**2 + 1), x, -1) == 0
assert residue(1 / (x**2 + 1), x, I) == -I / 2
assert residue(1 / (x**2 + 1), x, -I) == I / 2
assert residue(1 / (x**4 + 1), x, 0) == 0
def test_sympyissue_5654():
assert residue(1 / (x**2 + a**2)**2, x, a * I) == -I / (4 * a**3)
def test_bug():
assert residue(2**(z)*(s + z)*(1 - s - z)/z**2, z, 0) == \
1 + s*log(2) - s**2*log(2) - 2*s
def test_NotImplemented():
pytest.raises(NotImplementedError, lambda: residue(exp(1 / z), z, 0))
def test_expressions_failing():
n = Symbol('n', integer=True, positive=True)
assert residue(exp(z)/(z - pi*I/4*a)**n, z, I*pi*a) == \
exp(I*pi*a/4)/factorial(n - 1)
def test_functions():
assert residue(1 / sin(x), x, 0) == 1
assert residue(2 / sin(x), x, 0) == 2
assert residue(1 / sin(x)**2, x, 0) == 0
assert residue(1 / sin(x)**5, x, 0) == Rational(3, 8)
def test_functions():
assert residue(1/sin(x), x, 0) == 1
assert residue(2/sin(x), x, 0) == 2
assert residue(1/sin(x)**2, x, 0) == 0
assert residue(1/sin(x)**5, x, 0) == Rational(3, 8)
def test_bug():
assert residue(2**(z)*(s + z)*(1 - s - z)/z**2, z, 0) == \
1 + s*log(2) - s**2*log(2) - 2*s
def test_NotImplemented():
pytest.raises(NotImplementedError, lambda: residue(exp(1/z), z, 0))
def test_expressions_failing():
n = Symbol('n', integer=True, positive=True)
assert residue(exp(z)/(z - pi*I/4*a)**n, z, I*pi*a) == \
exp(I*pi*a/4)/factorial(n - 1)
def test_sympyissue_6499():
assert residue(1 / (exp(z) - 1), z, 0) == 1
def test_f():
f = Function('f')
assert residue(f(x) / x**5, x, 0) == f(x).diff((x, 4)).subs({x: 0}) / 24
def test_sympyissue_5654():
assert residue(1/(x**2 + a**2)**2, x, a*I) == -I/(4*a**3)
def test_f():
f = Function("f")
assert residue(f(x)/x**5, x, 0) == f(x).diff(x, 4).subs({x: 0})/24
def test_sympyissue_6499():
assert residue(1/(exp(z) - 1), z, 0) == 1
def test_f():
f = Function("f")
assert residue(f(x) / x**5, x, 0) == f(x).diff(x, 4).subs(x, 0) / 24