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from sympy.core.function import Function |
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from sympy.core.numbers import (I, Rational, pi) |
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from sympy.core.singleton import S |
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from sympy.core.symbol import Symbol |
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from sympy.functions.combinatorial.factorials import factorial |
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from sympy.functions.elementary.exponential import (exp, log) |
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from sympy.functions.elementary.hyperbolic import tanh |
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from sympy.functions.elementary.miscellaneous import sqrt |
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from sympy.functions.elementary.trigonometric import (cot, sin, tan) |
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from sympy.series.residues import residue |
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from sympy.testing.pytest import XFAIL, raises |
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from sympy.abc import x, z, a, s, k |
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def test_basic1(): |
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assert residue(1/x, x, 0) == 1 |
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assert residue(-2/x, x, 0) == -2 |
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assert residue(81/x, x, 0) == 81 |
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assert residue(1/x**2, x, 0) == 0 |
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assert residue(0, x, 0) == 0 |
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assert residue(5, x, 0) == 0 |
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assert residue(x, x, 0) == 0 |
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assert residue(x**2, x, 0) == 0 |
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def test_basic2(): |
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assert residue(1/x, x, 1) == 0 |
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assert residue(-2/x, x, 1) == 0 |
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assert residue(81/x, x, -1) == 0 |
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assert residue(1/x**2, x, 1) == 0 |
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assert residue(0, x, 1) == 0 |
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assert residue(5, x, 1) == 0 |
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assert residue(x, x, 1) == 0 |
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assert residue(x**2, x, 5) == 0 |
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def test_f(): |
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f = Function("f") |
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assert residue(f(x)/x**5, x, 0) == f(x).diff(x, 4).subs(x, 0)/24 |
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def test_functions(): |
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assert residue(1/sin(x), x, 0) == 1 |
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assert residue(2/sin(x), x, 0) == 2 |
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assert residue(1/sin(x)**2, x, 0) == 0 |
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assert residue(1/sin(x)**5, x, 0) == Rational(3, 8) |
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def test_expressions(): |
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assert residue(1/(x + 1), x, 0) == 0 |
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assert residue(1/(x + 1), x, -1) == 1 |
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assert residue(1/(x**2 + 1), x, -1) == 0 |
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assert residue(1/(x**2 + 1), x, I) == -I/2 |
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assert residue(1/(x**2 + 1), x, -I) == I/2 |
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assert residue(1/(x**4 + 1), x, 0) == 0 |
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assert residue(1/(x**4 + 1), x, exp(I*pi/4)).equals(-(Rational(1, 4) + I/4)/sqrt(2)) |
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assert residue(1/(x**2 + a**2)**2, x, a*I) == -I/4/a**3 |
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@XFAIL |
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def test_expressions_failing(): |
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n = Symbol('n', integer=True, positive=True) |
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assert residue(exp(z)/(z - pi*I/4*a)**n, z, I*pi*a) == \ |
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exp(I*pi*a/4)/factorial(n - 1) |
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def test_NotImplemented(): |
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raises(NotImplementedError, lambda: residue(exp(1/z), z, 0)) |
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def test_bug(): |
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assert residue(2**(z)*(s + z)*(1 - s - z)/z**2, z, 0) == \ |
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1 + s*log(2) - s**2*log(2) - 2*s |
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def test_issue_5654(): |
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assert residue(1/(x**2 + a**2)**2, x, a*I) == -I/(4*a**3) |
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assert residue(1/s*1/(z - exp(s)), s, 0) == 1/(z - 1) |
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assert residue((1 + k)/s*1/(z - exp(s)), s, 0) == k/(z - 1) + 1/(z - 1) |
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def test_issue_6499(): |
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assert residue(1/(exp(z) - 1), z, 0) == 1 |
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def test_issue_14037(): |
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assert residue(sin(x**50)/x**51, x, 0) == 1 |
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def test_issue_21176(): |
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f = x**2*cot(pi*x)/(x**4 + 1) |
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assert residue(f, x, -sqrt(2)/2 - sqrt(2)*I/2).cancel().together(deep=True)\ |
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== sqrt(2)*(1 - I)/(8*tan(sqrt(2)*pi*(1 + I)/2)) |
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def test_issue_21177(): |
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r = -sqrt(3)*tanh(sqrt(3)*pi/2)/3 |
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a = residue(cot(pi*x)/((x - 1)*(x - 2) + 1), x, S(3)/2 - sqrt(3)*I/2) |
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b = residue(cot(pi*x)/(x**2 - 3*x + 3), x, S(3)/2 - sqrt(3)*I/2) |
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assert a == r |
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assert (b - a).cancel() == 0 |
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