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)
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)
def test_elliptic_k():
assert elliptic_k(0) == pi / 2
assert elliptic_k(Rational(
1, 2)) == 8 * pi**Rational(3, 2) / gamma(-Rational(1, 4))**2
assert elliptic_k(1) == zoo
assert elliptic_k(-1) == gamma(Rational(1, 4))**2 / (4 * sqrt(2 * pi))
assert elliptic_k(oo) == 0
assert elliptic_k(-oo) == 0
assert elliptic_k(I * oo) == 0
assert elliptic_k(-I * oo) == 0
assert elliptic_k(zoo) == 0
assert elliptic_k(z).diff(z) == (elliptic_e(z) -
(1 - z) * elliptic_k(z)) / (2 * z *
(1 - z))
assert td(elliptic_k(z), z)
pytest.raises(ArgumentIndexError, lambda: elliptic_k(z).fdiff(2))
zi = Symbol('z', extended_real=False)
assert elliptic_k(zi).conjugate() == elliptic_k(zi.conjugate())
zr = Symbol('z', extended_real=True, negative=True)
assert elliptic_k(zr).conjugate() == elliptic_k(zr)
assert elliptic_k(z).conjugate() == conjugate(elliptic_k(z),
evaluate=False)
assert elliptic_k(z).rewrite(hyper) == \
(pi/2)*hyper((Rational(1, 2), Rational(1, 2)), (1,), z)
assert tn(elliptic_k(z), (pi / 2) * hyper((Rational(1, 2), Rational(1, 2)),
(1, ), z))
assert elliptic_k(z).rewrite(meijerg) == \
meijerg(((Rational(1, 2), Rational(1, 2)), []), ((0,), (0,)), -z)/2
assert tn(
elliptic_k(z),
meijerg(((Rational(1, 2), Rational(1, 2)), []),
((0, ), (0, )), -z) / 2)
assert elliptic_k(z).series(z) == pi/2 + pi*z/8 + 9*pi*z**2/128 + \
25*pi*z**3/512 + 1225*pi*z**4/32768 + 3969*pi*z**5/131072 + 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))
rx, ry = randcplx(), randcplx()
assert td(elliptic_pi(n, rx, ry), n)
assert td(elliptic_pi(rx, z, ry), z)
assert td(elliptic_pi(rx, ry, m), m)
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)