def test_elliptic_f():
    assert elliptic_f(z, 0) == z
    assert elliptic_f(0, m) == 0
    assert elliptic_f(pi * i / 2, m) == i * elliptic_k(m)
    assert elliptic_f(z, oo) == 0
    assert elliptic_f(z, -oo) == 0

    assert elliptic_f(-z, m) == -elliptic_f(z, m)

    assert elliptic_f(z, m).diff(z) == 1 / sqrt(1 - m * sin(z)**2)
    assert elliptic_f(z, m).diff(m) == elliptic_e(z, m)/(2*m*(1 - m)) - elliptic_f(z, m)/(2*m) - \
        sin(2*z)/(4*(1 - m)*sqrt(1 - m*sin(z)**2))
    r = randcplx()
    assert td(elliptic_f(z, r), z)
    assert td(elliptic_f(r, m), m)
    pytest.raises(ArgumentIndexError, lambda: elliptic_f(z, m).fdiff(3))

    mi = Symbol('m', extended_real=False)
    assert elliptic_f(z, mi).conjugate() == elliptic_f(z.conjugate(),
                                                       mi.conjugate())
    mr = Symbol('m', extended_real=True, negative=True)
    assert elliptic_f(z, mr).conjugate() == elliptic_f(z.conjugate(), mr)
    assert elliptic_f(z, m).conjugate() == conjugate(elliptic_f(z, m),
                                                     evaluate=False)

    assert elliptic_f(z, m).series(z) == \
        z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)
def test_elliptic_e():
    assert elliptic_e(z, 0) == z
    assert elliptic_e(0, m) == 0
    assert elliptic_e(i * pi / 2, m) == i * elliptic_e(m)
    assert elliptic_e(z, oo) == zoo
    assert elliptic_e(z, -oo) == zoo
    assert elliptic_e(0) == pi / 2
    assert elliptic_e(1) == 1
    assert elliptic_e(oo) == I * oo
    assert elliptic_e(-oo) == oo
    assert elliptic_e(zoo) == zoo

    assert elliptic_e(-z, m) == -elliptic_e(z, m)

    assert elliptic_e(z, m).diff(z) == sqrt(1 - m * sin(z)**2)
    assert elliptic_e(
        z, m).diff(m) == (elliptic_e(z, m) - elliptic_f(z, m)) / (2 * m)
    assert elliptic_e(z).diff(z) == (elliptic_e(z) - elliptic_k(z)) / (2 * z)
    r = randcplx()
    assert td(elliptic_e(r, m), m)
    assert td(elliptic_e(z, r), z)
    assert td(elliptic_e(z), z)
    pytest.raises(ArgumentIndexError, lambda: elliptic_e(z, m).fdiff(3))
    pytest.raises(ArgumentIndexError, lambda: elliptic_e(z).fdiff(2))

    mi = Symbol('m', extended_real=False)
    assert elliptic_e(z, mi).conjugate() == elliptic_e(z.conjugate(),
                                                       mi.conjugate())
    assert elliptic_e(mi).conjugate() == elliptic_e(mi.conjugate())
    mr = Symbol('m', extended_real=True, negative=True)
    assert elliptic_e(z, mr).conjugate() == elliptic_e(z.conjugate(), mr)
    assert elliptic_e(mr).conjugate() == elliptic_e(mr)
    assert elliptic_e(z, m).conjugate() == conjugate(elliptic_e(z, m))
    assert elliptic_e(z).conjugate() == conjugate(elliptic_e(z))

    assert elliptic_e(z).rewrite(hyper) == (pi / 2) * hyper(
        (Rational(-1, 2), Rational(1, 2)), (1, ), z)
    assert elliptic_e(z, m).rewrite(hyper) == elliptic_e(z, m)
    assert tn(elliptic_e(z), (pi / 2) * hyper(
        (Rational(-1, 2), Rational(1, 2)), (1, ), z))
    assert elliptic_e(z).rewrite(meijerg) == \
        -meijerg(((Rational(1, 2), Rational(3, 2)), []), ((0,), (0,)), -z)/4
    assert elliptic_e(z, m).rewrite(meijerg) == elliptic_e(z, m)
    assert tn(
        elliptic_e(z), -meijerg(((Rational(1, 2), Rational(3, 2)), []),
                                ((0, ), (0, )), -z) / 4)

    assert elliptic_e(z, m).series(z) == \
        z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
    assert elliptic_e(z).series(z) == pi/2 - pi*z/8 - 3*pi*z**2/128 - \
        5*pi*z**3/512 - 175*pi*z**4/32768 - 441*pi*z**5/131072 + O(z**6)
示例#3
0
def test_P():
    assert P(0, z, m) == F(z, m)
    assert P(1, z, m) == F(z, m) + \
        (sqrt(1 - m*sin(z)**2)*tan(z) - E(z, m))/(1 - m)
    assert P(n, i*pi/2, m) == i*P(n, m)
    assert P(n, z, 0) == atanh(sqrt(n - 1)*tan(z))/sqrt(n - 1)
    assert P(n, z, n) == F(z, n) - P(1, z, n) + tan(z)/sqrt(1 - n*sin(z)**2)
    assert P(oo, z, m) == 0
    assert P(-oo, z, m) == 0
    assert P(n, z, oo) == 0
    assert P(n, z, -oo) == 0
    assert P(0, m) == K(m)
    assert P(1, m) == zoo
    assert P(n, 0) == pi/(2*sqrt(1 - n))
    assert P(2, 1) == -oo
    assert P(-1, 1) == oo
    assert P(n, n) == E(n)/(1 - n)
    assert P(oo, m) == 0
    assert P(n, oo) == 0

    assert P(n, -z, m) == -P(n, z, m)

    ni, mi = Symbol('n', extended_real=False), Symbol('m', extended_real=False)
    assert P(ni, z, mi).conjugate() == \
        P(ni.conjugate(), z.conjugate(), mi.conjugate())
    nr, mr = Symbol('n', extended_real=True, negative=True), \
        Symbol('m', extended_real=True, negative=True)
    assert P(nr, z, mr).conjugate() == P(nr, z.conjugate(), mr)
    assert P(n, m).conjugate() == P(n.conjugate(), m.conjugate())

    assert P(n, z, m).diff(n) == (E(z, m) + (m - n)*F(z, m)/n +
        (n**2 - m)*P(n, z, m)/n - n*sqrt(1 -
            m*sin(z)**2)*sin(2*z)/(2*(1 - n*sin(z)**2)))/(2*(m - n)*(n - 1))
    assert P(n, z, m).diff(z) == 1/(sqrt(1 - m*sin(z)**2)*(1 - n*sin(z)**2))
    assert P(n, z, m).diff(m) == (E(z, m)/(m - 1) + P(n, z, m) -
        m*sin(2*z)/(2*(m - 1)*sqrt(1 - m*sin(z)**2)))/(2*(n - m))
    assert P(n, m).diff(n) == (E(m) + (m - n)*K(m)/n +
        (n**2 - m)*P(n, m)/n)/(2*(m - n)*(n - 1))
    assert P(n, m).diff(m) == (E(m)/(m - 1) + P(n, m))/(2*(n - m))
    rx, ry = randcplx(), randcplx()
    assert td(P(n, rx, ry), n)
    assert td(P(rx, z, ry), z)
    assert td(P(rx, ry, m), m)
    pytest.raises(ArgumentIndexError, lambda: P(n, z, m).fdiff(4))
    pytest.raises(ArgumentIndexError, lambda: P(n, m).fdiff(3))

    assert P(n, z, m).series(z) == z + z**3*(m/6 + n/3) + \
        z**5*(3*m**2/40 + m*n/10 - m/30 + n**2/5 - n/15) + O(z**6)
示例#4
0
def test_E():
    assert E(z, 0) == z
    assert E(0, m) == 0
    assert E(i*pi/2, m) == i*E(m)
    assert E(z, oo) == zoo
    assert E(z, -oo) == zoo
    assert E(0) == pi/2
    assert E(1) == 1
    assert E(oo) == I*oo
    assert E(-oo) == oo
    assert E(zoo) == zoo

    assert E(-z, m) == -E(z, m)

    assert E(z, m).diff(z) == sqrt(1 - m*sin(z)**2)
    assert E(z, m).diff(m) == (E(z, m) - F(z, m))/(2*m)
    assert E(z).diff(z) == (E(z) - K(z))/(2*z)
    r = randcplx()
    assert td(E(r, m), m)
    assert td(E(z, r), z)
    assert td(E(z), z)
    pytest.raises(ArgumentIndexError, lambda: E(z, m).fdiff(3))
    pytest.raises(ArgumentIndexError, lambda: E(z).fdiff(2))

    mi = Symbol('m', extended_real=False)
    assert E(z, mi).conjugate() == E(z.conjugate(), mi.conjugate())
    assert E(mi).conjugate() == E(mi.conjugate())
    mr = Symbol('m', extended_real=True, negative=True)
    assert E(z, mr).conjugate() == E(z.conjugate(), mr)
    assert E(mr).conjugate() == E(mr)

    assert E(z).rewrite(hyper) == (pi/2)*hyper((-S.Half, S.Half), (S.One,), z)
    assert E(z, m).rewrite(hyper) == E(z, m)
    assert tn(E(z), (pi/2)*hyper((-S.Half, S.Half), (S.One,), z))
    assert E(z).rewrite(meijerg) == \
        -meijerg(((S.Half, Rational(3, 2)), []), ((S.Zero,), (S.Zero,)), -z)/4
    assert E(z, m).rewrite(meijerg) == E(z, m)
    assert tn(E(z), -meijerg(((S.Half, Rational(3, 2)), []), ((S.Zero,), (S.Zero,)), -z)/4)

    assert E(z, m).series(z) == \
        z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
    assert E(z).series(z) == pi/2 - pi*z/8 - 3*pi*z**2/128 - \
        5*pi*z**3/512 - 175*pi*z**4/32768 - 441*pi*z**5/131072 + O(z**6)
示例#5
0
def test_F():
    assert F(z, 0) == z
    assert F(0, m) == 0
    assert F(pi*i/2, m) == i*K(m)
    assert F(z, oo) == 0
    assert F(z, -oo) == 0

    assert F(-z, m) == -F(z, m)

    assert F(z, m).diff(z) == 1/sqrt(1 - m*sin(z)**2)
    assert F(z, m).diff(m) == E(z, m)/(2*m*(1 - m)) - F(z, m)/(2*m) - \
        sin(2*z)/(4*(1 - m)*sqrt(1 - m*sin(z)**2))
    r = randcplx()
    assert td(F(z, r), z)
    assert td(F(r, m), m)

    mi = Symbol('m', extended_real=False)
    assert F(z, mi).conjugate() == F(z.conjugate(), mi.conjugate())
    mr = Symbol('m', extended_real=True, negative=True)
    assert F(z, mr).conjugate() == F(z.conjugate(), mr)

    assert F(z, m).series(z) == \
        z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)
def test_elliptic_pi():
    assert elliptic_pi(0, z, m) == elliptic_f(z, m)
    assert elliptic_pi(1, z, m) == elliptic_f(z, m) + \
        (sqrt(1 - m*sin(z)**2)*tan(z) - elliptic_e(z, m))/(1 - m)
    assert elliptic_pi(n, i * pi / 2, m) == i * elliptic_pi(n, m)
    assert elliptic_pi(n, z, 0) == atanh(sqrt(n - 1) * tan(z)) / sqrt(n - 1)
    assert elliptic_pi(n, z, n) == elliptic_f(z, n) - elliptic_pi(
        1, z, n) + tan(z) / sqrt(1 - n * sin(z)**2)
    assert elliptic_pi(oo, z, m) == 0
    assert elliptic_pi(-oo, z, m) == 0
    assert elliptic_pi(n, z, oo) == 0
    assert elliptic_pi(n, z, -oo) == 0
    assert elliptic_pi(0, m) == elliptic_k(m)
    assert elliptic_pi(1, m) == zoo
    assert elliptic_pi(n, 0) == pi / (2 * sqrt(1 - n))
    assert elliptic_pi(2, 1) == -oo
    assert elliptic_pi(-1, 1) == oo
    assert elliptic_pi(n, n) == elliptic_e(n) / (1 - n)
    assert elliptic_pi(oo, m) == 0
    assert elliptic_pi(n, oo) == 0

    assert elliptic_pi(n, -z, m) == -elliptic_pi(n, z, m)

    ni, mi = Symbol('n', extended_real=False), Symbol('m', extended_real=False)
    assert elliptic_pi(ni, z, mi).conjugate() == \
        elliptic_pi(ni.conjugate(), z.conjugate(), mi.conjugate())
    nr, mr = Symbol('n', extended_real=True, negative=True), \
        Symbol('m', extended_real=True, negative=True)
    assert elliptic_pi(nr, z,
                       mr).conjugate() == elliptic_pi(nr, z.conjugate(), mr)
    assert elliptic_pi(n,
                       m).conjugate() == elliptic_pi(n.conjugate(),
                                                     m.conjugate())
    assert elliptic_pi(n, z, m).conjugate() == conjugate(elliptic_pi(n, z, m))

    assert elliptic_pi(
        n, z,
        m).diff(n) == (elliptic_e(z, m) + (m - n) * elliptic_f(z, m) / n +
                       (n**2 - m) * elliptic_pi(n, z, m) / n -
                       n * sqrt(1 - m * sin(z)**2) * sin(2 * z) /
                       (2 * (1 - n * sin(z)**2))) / (2 * (m - n) * (n - 1))
    assert elliptic_pi(n, z, m).diff(z) == 1 / (sqrt(1 - m * sin(z)**2) *
                                                (1 - n * sin(z)**2))
    assert elliptic_pi(
        n, z,
        m).diff(m) == (elliptic_e(z, m) /
                       (m - 1) + elliptic_pi(n, z, m) - m * sin(2 * z) /
                       (2 * (m - 1) * sqrt(1 - m * sin(z)**2))) / (2 * (n - m))
    assert elliptic_pi(
        n, m).diff(n) == (elliptic_e(m) + (m - n) * elliptic_k(m) / n +
                          (n**2 - m) * elliptic_pi(n, m) / n) / (2 * (m - n) *
                                                                 (n - 1))
    assert elliptic_pi(
        n, m).diff(m) == (elliptic_e(m) /
                          (m - 1) + elliptic_pi(n, m)) / (2 * (n - m))

    # workaround fredrik-johansson/mpmath#571, suggested by Kalevi Suominen
    # in https://github.com/sympy/sympy/issues/20933#issuecomment-779077562
    bounds = {'a': -0.9, 'b': -0.9, 'c': 0.9, 'd': 0.9}
    rx, ry = randcplx(**bounds), randcplx(**bounds)
    assert td(elliptic_pi(n, rx, ry), n, **bounds)
    assert td(elliptic_pi(rx, z, ry), z, **bounds)
    assert td(elliptic_pi(rx, ry, m), m, **bounds)

    pytest.raises(ArgumentIndexError, lambda: elliptic_pi(n, z, m).fdiff(4))
    pytest.raises(ArgumentIndexError, lambda: elliptic_pi(n, m).fdiff(3))

    assert elliptic_pi(n, z, m).series(z) == z + z**3*(m/6 + n/3) + \
        z**5*(3*m**2/40 + m*n/10 - m/30 + n**2/5 - n/15) + O(z**6)