from sympy import residue, Symbol, Function, sin, S, I, pi, exp, log, pi, factorial
from sympy.utilities.pytest import XFAIL, raises
from sympy.abc import x, y, z, a, s


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_f():
    # FIXME: we get infinite recursion here:
    f = Function("f")
    assert residue(f(x)/x**5, x, 0) == f.diff(x, 4)/24


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) == S(3)/8


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


@XFAIL
def test_expressions_failing():
    assert residue(1/(x**4 + 1), x, exp(I*pi/4)) == -(S(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_NotImplemented():
    raises(NotImplementedError, lambda: residue(exp(1/z), z, 0))


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_issue_2555():
    assert residue(1/(x**2 + a**2)**2, x, a*I) == -I/(4*a**3)


def test_issue_3400():
    assert residue(1/(exp(z) - 1), z, 0) == 1

    # github issue 2519:
    assert residue((z**3 + 5)/((z**4 - 1)*(z + 1)), z, -1) == -S(9)/4
