test_meijerint.py 文件源码

def test_expint():
    """ Test various exponential integrals. """
    from sympy import (expint, unpolarify, Symbol, Ci, Si, Shi, Chi,
                       sin, cos, sinh, cosh, Ei)
    assert simplify(unpolarify(integrate(exp(-z*x)/x**y, (x, 1, oo),
                meijerg=True, conds='none'
                ).rewrite(expint).expand(func=True))) == expint(y, z)

    assert integrate(exp(-z*x)/x, (x, 1, oo), meijerg=True,
                     conds='none').rewrite(expint).expand() == \
        expint(1, z)
    assert integrate(exp(-z*x)/x**2, (x, 1, oo), meijerg=True,
                     conds='none').rewrite(expint).expand() == \
        expint(2, z).rewrite(Ei).rewrite(expint)
    assert integrate(exp(-z*x)/x**3, (x, 1, oo), meijerg=True,
                     conds='none').rewrite(expint).expand() == \
        expint(3, z).rewrite(Ei).rewrite(expint).expand()

    t = Symbol('t', positive=True)
    assert integrate(-cos(x)/x, (x, t, oo), meijerg=True).expand() == Ci(t)
    assert integrate(-sin(x)/x, (x, t, oo), meijerg=True).expand() == \
        Si(t) - pi/2
    assert integrate(sin(x)/x, (x, 0, z), meijerg=True) == Si(z)
    assert integrate(sinh(x)/x, (x, 0, z), meijerg=True) == Shi(z)
    assert integrate(exp(-x)/x, x, meijerg=True).expand().rewrite(expint) == \
        I*pi - expint(1, x)
    assert integrate(exp(-x)/x**2, x, meijerg=True).rewrite(expint).expand() \
        == expint(1, x) - exp(-x)/x - I*pi

    u = Symbol('u', polar=True)
    assert integrate(cos(u)/u, u, meijerg=True).expand().as_independent(u)[1] \
        == Ci(u)
    assert integrate(cosh(u)/u, u, meijerg=True).expand().as_independent(u)[1] \
        == Chi(u)

    assert integrate(expint(1, x), x, meijerg=True
            ).rewrite(expint).expand() == x*expint(1, x) - exp(-x)
    assert integrate(expint(2, x), x, meijerg=True
            ).rewrite(expint).expand() == \
        -x**2*expint(1, x)/2 + x*exp(-x)/2 - exp(-x)/2
    assert simplify(unpolarify(integrate(expint(y, x), x,
                 meijerg=True).rewrite(expint).expand(func=True))) == \
        -expint(y + 1, x)

    assert integrate(Si(x), x, meijerg=True) == x*Si(x) + cos(x)
    assert integrate(Ci(u), u, meijerg=True).expand() == u*Ci(u) - sin(u)
    assert integrate(Shi(x), x, meijerg=True) == x*Shi(x) - cosh(x)
    assert integrate(Chi(u), u, meijerg=True).expand() == u*Chi(u) - sinh(u)

    assert integrate(Si(x)*exp(-x), (x, 0, oo), meijerg=True) == pi/4
    assert integrate(expint(1, x)*sin(x), (x, 0, oo), meijerg=True) == log(2)/2