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| 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.elementary.trigonometric import (cos, sin) | |
| from sympy.functions.special.delta_functions import (DiracDelta, Heaviside) | |
| from sympy.integrals.deltafunctions import change_mul, deltaintegrate | |
| f = Function("f") | |
| x_1, x_2, x, y, z = symbols("x_1 x_2 x y z") | |
| def test_change_mul(): | |
| assert change_mul(x, x) == (None, None) | |
| assert change_mul(x*y, x) == (None, None) | |
| assert change_mul(x*y*DiracDelta(x), x) == (DiracDelta(x), x*y) | |
| assert change_mul(x*y*DiracDelta(x)*DiracDelta(y), x) == \ | |
| (DiracDelta(x), x*y*DiracDelta(y)) | |
| assert change_mul(DiracDelta(x)**2, x) == \ | |
| (DiracDelta(x), DiracDelta(x)) | |
| assert change_mul(y*DiracDelta(x)**2, x) == \ | |
| (DiracDelta(x), y*DiracDelta(x)) | |
| def test_deltaintegrate(): | |
| assert deltaintegrate(x, x) is None | |
| assert deltaintegrate(x + DiracDelta(x), x) is None | |
| assert deltaintegrate(DiracDelta(x, 0), x) == Heaviside(x) | |
| for n in range(10): | |
| assert deltaintegrate(DiracDelta(x, n + 1), x) == DiracDelta(x, n) | |
| assert deltaintegrate(DiracDelta(x), x) == Heaviside(x) | |
| assert deltaintegrate(DiracDelta(-x), x) == Heaviside(x) | |
| assert deltaintegrate(DiracDelta(x - y), x) == Heaviside(x - y) | |
| assert deltaintegrate(DiracDelta(y - x), x) == Heaviside(x - y) | |
| assert deltaintegrate(x*DiracDelta(x), x) == 0 | |
| assert deltaintegrate((x - y)*DiracDelta(x - y), x) == 0 | |
| assert deltaintegrate(DiracDelta(x)**2, x) == DiracDelta(0)*Heaviside(x) | |
| assert deltaintegrate(y*DiracDelta(x)**2, x) == \ | |
| y*DiracDelta(0)*Heaviside(x) | |
| assert deltaintegrate(DiracDelta(x, 1), x) == DiracDelta(x, 0) | |
| assert deltaintegrate(y*DiracDelta(x, 1), x) == y*DiracDelta(x, 0) | |
| assert deltaintegrate(DiracDelta(x, 1)**2, x) == -DiracDelta(0, 2)*Heaviside(x) | |
| assert deltaintegrate(y*DiracDelta(x, 1)**2, x) == -y*DiracDelta(0, 2)*Heaviside(x) | |
| assert deltaintegrate(DiracDelta(x) * f(x), x) == f(0) * Heaviside(x) | |
| assert deltaintegrate(DiracDelta(-x) * f(x), x) == f(0) * Heaviside(x) | |
| assert deltaintegrate(DiracDelta(x - 1) * f(x), x) == f(1) * Heaviside(x - 1) | |
| assert deltaintegrate(DiracDelta(1 - x) * f(x), x) == f(1) * Heaviside(x - 1) | |
| assert deltaintegrate(DiracDelta(x**2 + x - 2), x) == \ | |
| Heaviside(x - 1)/3 + Heaviside(x + 2)/3 | |
| p = cos(x)*(DiracDelta(x) + DiracDelta(x**2 - 1))*sin(x)*(x - pi) | |
| assert deltaintegrate(p, x) - (-pi*(cos(1)*Heaviside(-1 + x)*sin(1)/2 - \ | |
| cos(1)*Heaviside(1 + x)*sin(1)/2) + \ | |
| cos(1)*Heaviside(1 + x)*sin(1)/2 + \ | |
| cos(1)*Heaviside(-1 + x)*sin(1)/2) == 0 | |
| p = x_2*DiracDelta(x - x_2)*DiracDelta(x_2 - x_1) | |
| assert deltaintegrate(p, x_2) == x*DiracDelta(x - x_1)*Heaviside(x_2 - x) | |
| p = x*y**2*z*DiracDelta(y - x)*DiracDelta(y - z)*DiracDelta(x - z) | |
| assert deltaintegrate(p, y) == x**3*z*DiracDelta(x - z)**2*Heaviside(y - x) | |
| assert deltaintegrate((x + 1)*DiracDelta(2*x), x) == S.Half * Heaviside(x) | |
| assert deltaintegrate((x + 1)*DiracDelta(x*Rational(2, 3) + Rational(4, 9)), x) == \ | |
| S.Half * Heaviside(x + Rational(2, 3)) | |
| a, b, c = symbols('a b c', commutative=False) | |
| assert deltaintegrate(DiracDelta(x - y)*f(x - b)*f(x - a), x) == \ | |
| f(y - b)*f(y - a)*Heaviside(x - y) | |
| p = f(x - a)*DiracDelta(x - y)*f(x - c)*f(x - b) | |
| assert deltaintegrate(p, x) == f(y - a)*f(y - c)*f(y - b)*Heaviside(x - y) | |
| p = DiracDelta(x - z)*f(x - b)*f(x - a)*DiracDelta(x - y) | |
| assert deltaintegrate(p, x) == DiracDelta(y - z)*f(y - b)*f(y - a) * \ | |
| Heaviside(x - y) | |